Spherical and Plane Integral Operators for PDEs: Construction, Analysis, and Applications 9783110315332, 9783110315295

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Karl K. Sabelfeld, Irina A. Shalimova: Spherical and Plane Integral Operators for PDEs — 2013/9/11 — 9:13 — page i

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Karl K. Sabelfeld, Irina A. Shalimova Spherical and Plane Integral Operators for PDEs

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Karl K. Sabelfeld, Irina A. Shalimova: Spherical and Plane Integral Operators for PDEs — 2013/9/11 — 9:13 — page iii

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Karl K. Sabelfeld, Irina A. Shalimova

Spherical and Plane Integral Operators for PDEs | Construction, Analysis, and Applications

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Karl K. Sabelfeld, Irina A. Shalimova: Spherical and Plane Integral Operators for PDEs — 2013/9/11 — 9:13 — page iv

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Mathematics Subject Classification 2010: 35-02, 65-02, 65C30, 65C05, 65N35, 65N75, 65N80 Authors: Prof. Dr. Karl K. Sabelfeld Novosibirsk University (NSU) Russian Academy of Sciences Siberian Branch Institute of Computational Mathematics and Geophysics Lavrentjeva 6 630090 Novosibirsk RUSSIA [email protected] Dr. Irina A. Shalimova Novosibirsk University (NSU) Russian Academy of Sciences Siberian Branch Institute of Computational Mathematics and Geophysics Lavrentjeva 6 630090 Novosibirsk RUSSIA [email protected]

ISBN 978-3-11-031529-5 e-ISBN 978-3-11-031533-2 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.dnb.de. © 2013 Walter de Gruyter GmbH, Berlin/Boston Typesetting: le-tex publishing services GmbH, Leipzig Printing and binding: CPI buch bücher.de GmbH, Birkach ♾Printed on acid-free paper Printed in Germany www.degruyter.com

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Karl K. Sabelfeld, Irina A. Shalimova: Spherical and Plane Integral Operators for PDEs — 2013/9/11 — 9:13 — page v

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Preface The monograph is devoted to spherical and plane integral operators for high-dimen­ sional boundary value problems of mathematical physics. The derived integral opera­ tors are used to provide equivalent integral formulations of the boundary value prob­ lems. Direct and converse mean value theorems are proved for scalar elliptic equa­ tions such as the Laplace, Helmholtz, and diffusion equations, parabolic equations, high-order elliptic equations, e.g. the biharmonic and metaharmonic equations, and systems of elliptic equations like the Lamé equation and other systems of elasticity equations. These results are presented in the first part of the book, which includes Chapters 1–8 and follow mainly our book Spherical Means for PDEs published in 1997 by VSP [167]. The second part, Chapters 9–13, deals with the applications of the devel­ oped integral operator relations to numerics for PDEs. The integral operators defined on disks, spheres, 2D half-planes, 3D half-spaces, and some other domains are used to construct new numerical methods for solving relevant boundary value problems for a wide class of domains. We consider mainly two basic approaches: the first is devel­ oped for domains that can be composed as a union of overlapped disks, spheres, halfplanes and half-spaces. The second is similar to the method of fundamental solutions, but is based on a numerical inversion of the integral operators by a randomized spec­ tral projection method. We also show how the integral operators can be used to solve PDEs with random loads and stochastic boundary conditions. The book is written for mathematicians who work in the field of partial differential and integral equations, physicists, and engineers dealing with computational meth­ ods and applied probability, for students and postgraduates studying mathematical physics and numerical mathematics.

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Contents Preface | v 1

Introduction | 1

2 2.1 2.1.1 2.1.2 2.1.3 2.1.4 2.2 2.2.1 2.2.2 2.3 2.4 2.4.1 2.4.2 2.4.3 2.5

Scalar second-order PDEs | 5 Spherical mean value relations for the Laplace equation | 5 Direct spherical mean value relation | 5 Converse mean value theorem | 11 Integral equation equivalent to the Dirichlet problem | 12 Poisson–Jensen formula | 14 The diffusion and Helmholtz equations | 15 Diffusion equation | 15 Helmholtz equation | 17 Generalized second-order elliptic equations | 18 Parabolic equations | 20 Heat equation | 20 Parabolic equations with variable coefficients | 25 Expansion of the parabolic means | 27 Wave equation | 29

3 3.1 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.3 3.4 3.4.1 3.4.2

High-order elliptic equations | 32 Balayage operator | 32 Biharmonic equation | 34 Direct spherical mean value relation | 34 Generalized Poisson formula | 35 Rigid fixing of the boundary | 39 Nonhomogeneous biharmonic equation | 42 Fourth-order equation governing the bending of a plate | 44 Metaharmonic equations | 48 Polyharmonic equation | 48 General case | 50

4 4.1 4.2 4.3

Triangular systems of elliptic equations | 55 One-component diffusion system | 55 Two-component diffusion system | 56 Coupled biharmonic–harmonic equation | 58

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Contents | vii

5 5.1 5.1.1 5.1.2 5.2 5.3

Systems of elasticity theory | 60 Lamé equation | 60 Direct spherical mean value theorem | 60 Converse spherical mean value theorem | 64 Pseudovibration elastic equation | 66 Thermoelastic equation | 73

6 6.1 6.1.1 6.1.2 6.2 6.3

The generalized Poisson formula for the Lamé equation | 74 Plane elasticity | 74 Poisson formula for the displacements in rectangular coordinates | 74 Poisson formula for displacements in polar coordinates | 83 Generalized spatial Poisson formula for the Lamé equation | 86 An alternative derivation of the Poisson formula | 98

7 7.1 7.2 7.2.1 7.2.2 7.3

Spherical means for the stress and strain tensors | 102 Spherical means for the displacements | 102 Mean value relations for the stress and strain tensors | 105 Mean value relation for the strain components | 105 Mean value relation for the stress components | 110 Mean value relations for the stress components | 111

8 8.1 8.2 8.3 8.3.1 8.3.2 8.4 8.5 8.5.1 8.5.2 8.5.3 8.5.4 8.5.5 8.6

Random Walk on Spheres method | 120 Spherical mean as a mathematical expectation | 120 Iterations of the spherical mean operator | 121 The Random Walk on Spheres algorithm | 122 The Random Walk on Spheres process for the Dirichlet problem | 122 Inhomogeneous case | 130 Biharmonic equation | 132 Isotropic elastostatics governed by the Lamé equation | 134 Naive generalization | 134 Modification of the algorithm | 135 Nonisotropic Random Walk on Spheres | 137 Branching process | 139 Analytical continuation with respect to the spectral parameter | 141 Alternative Schwarz procedure | 144

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viii | Contents 9 9.1 9.2 9.2.1 9.2.2 9.2.3 9.2.4 9.3 9.4 9.4.1 9.4.2 9.5 9.5.1 9.5.2 9.5.3 9.5.4 9.6 9.6.1 9.6.2 9.7

Random Walk on Fixed Spheres for Laplace and Lamé equations | 148 Introduction | 148 Laplace equation | 150 Integral formulation of the Dirichlet problem | 150 Approximation by linear algebraic equations | 157 Set of overlapping disks | 158 Estimation of the spectral radius | 163 Isotropic elastostatics | 165 Iteration methods | 168 Stochastic iterative procedure with optimal random parameters | 168 SOR method | 173 Discrete Random Walk algorithms | 176 Discrete Random Walk based on the iteration method | 176 Discrete Random Walk method based on SOR | 177 Sampling from discrete distribution | 178 Variance of stochastic methods | 179 Numerical simulations | 181 Laplace equation | 181 Lamé equation | 182 Conclusion and discussion | 184

10 Stochastic spectral projection method for solving PDEs | 186 10.1 Introduction | 186 10.2 Laplace equation | 187 10.2.1 Two overlapping disks | 187 10.2.2 Neumann boundary conditions | 192 10.2.3 Overlapping of a half-plane with a set of disks | 194 10.3 Extension to the isotropic elasticity: Lamè equation | 197 10.3.1 Elastic disk | 197 10.3.2 Elastic half-plane | 199 10.4 Extension to 3D problems | 200 10.4.1 A sphere | 200 10.4.2 Elastic half-space | 201 10.5 Stochastic projection method for large linear systems | 203 11 11.1 11.2 11.3

Stochastic boundary collocation and spectral methods | 205 Introduction | 205 Surface and volume potentials | 206 Random Walk on Boundary Algorithm | 208

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Contents | ix

11.4 11.4.1 11.4.2 11.4.3 11.5 11.5.1 11.5.2 11.5.3 11.6 11.7 11.8 11.8.1 11.8.2 11.8.3 11.9

General scheme of the method of fundamental solutions (MFS) | 210 Kupradze–Aleksidze’s method based on first-kind integral equation | 212 MFS for Laplace and Helmholz equations | 213 Biharmonic equation | 214 MFS with separable Poisson kernel | 214 Dirichlet problem for the Laplace equation | 215 Evaluation of the Green function and solving inhomogeneous problems | 217 Evaluation of derivatives on the boundary and construction of the Poisson integral formulae | 219 Hydrodynamics friction and the capacitance of a chain of spheres | 220 Lamé equation: plane elasticity problem | 225 SVD and randomized versions | 229 SVD background | 229 Randomized SVD algorithm | 230 Using SVD for the linear least squares solution | 232 Numerical experiments | 233

12 Solution of 2D elasticity problems with random loads | 241 12.1 Introduction | 241 12.2 Lamé equation with nonzero body forces | 244 12.3 Random loads | 249 12.4 Random Walk methods and Double Randomization | 251 12.4.1 General description | 251 12.4.2 Green-tensor integral representation for the correlations | 252 12.5 Simulation results | 254 12.5.1 Testing the simulation procedure for random loads | 254 12.5.2 Testing the Random Walk algorithm for nonzero body forces | 254 12.5.3 Calculation of correlations for the displacement vector | 255 13 Boundary value problems with random boundary conditions | 260 13.1 Introduction | 260 13.1.1 Spectral representations | 261 13.1.2 Karhunen–Loève expansion | 263 13.2 Stochastic boundary value problems for the 2D Laplace equation | 265 13.2.1 Dirichlet problem for a 2D disk: white noise excitations | 267 13.2.2 General homogeneous boundary excitations | 273 13.2.3 Neumann boundary conditions | 274 13.2.4 Upper half-plane | 276

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x | Contents 13.3 13.4 13.5 13.5.1 13.5.2 13.6 13.6.1 13.6.2 13.6.3 13.6.4 13.6.5 13.6.6 13.6.7 13.6.8 13.6.9 13.6.10 13.6.11

3D Laplace equation | 279 Biharmonic equation | 282 Lamé equation: plane elasticity problem | 285 White noise excitations | 285 General case of homogeneous excitations | 293 Response of an elastic 3D half-space to random excitations | 297 Introduction | 297 System of Lamé equations governing an elastic half-space with no tangential surface forces | 298 Stochastic boundary value problem: correlation tensor | 299 Spectral representations for partially homogeneous random fields | 301 Displacement correlations for the white noise excitations | 303 Homogeneous excitations | 305 Conclusions and discussion | 308 Appendix A: the Poisson formula | 309 Appendix B: some 2D Fourier transform formulae | 311 Appendix C: some 2D integrals | 312 Appendix D: some further Fourier transform formulae | 314

Bibliography | 317 Index | 327

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1 Introduction It is well known that many boundary value problems for partial differential equations can be reformulated in the integral form. We mention the classical example from po­ tential theory where the original boundary value problems are reformulated in terms of equivalent integral equations (on the boundary, or in the volume, as in the Green formula, see, e.g. [33; 75; 93; 154]). The advantages of the integral formulation are well known: the solution is obtained on the whole, the boundary conditions are automati­ cally taken into account, and the questions of existence and uniqueness are resolved on the basis of the well-developed Fredholm theory. In this book, we deal with integral formulation of PDEs in the form of local integral equations whose kernels are defined on disks, spheres, planes, and other standard domains. Let us recall the famous mean value relation for harmonic functions. Assume for ¯ be a sphere of radius r simplicity that G is a bounded domain in Rn and let S(x, r) ⊂ G centered at a point x. Let u (x) be a regular harmonic function in G, i.e. Δu (x) = 0 ,

x ∈ G,

u ∈ C 2 (G) .

(1.1)

This means that the function u (x) satisfies the Laplace equation in G and the second ¯ the follow­ derivatives of u are continuous in G. Then for all x ∈ G and all S(x, r) ⊂ G ing spherical mean value relation is true:  1 u(x) = u (y)dS(y) . (1.2) ω n r n −1 S ( x,r )

On the right-hand side is the spherical mean of u, i.e. the integral of u over the sphere S(x, r) with respect to the surface element measure dS(y); ω n is the surface area of a unit sphere in Rn . The following converse mean value relation is well known [33]. Theorem 1.1 (Weak converse mean value theorem). If a function u(x) ∈ C2 (G) satis­ ¯ then u (x) is fies the mean value relation (1.2) for all x ∈ G and for all S(x, r) ⊂ G, harmonic in G. Less known is the following remarkable converse mean value theorem. Theorem 1.2 (The strong converse mean value theorem). Assume that the Dirichlet problem ¯) , Δu (x) = 0 , u |Γ = φ , u ∈ C2 (G) ∩ C(G (1.3) ¯) G bounded, has a solution for any continuous function φ. Then if a function v ∈ C(G ¯ for all x ∈ G, then satisfies the mean value relation for at least one sphere S(x, r) ⊂ G, v(x) is harmonic in G.

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2 | 1 Introduction This exciting statement can be found in the book of Courant and Hilbert [33] where examples were given showing that all the conditions of the theorem are necessary. However, in [33], there was no attempt to derive an integral formulation of the Dirichlet problem from this approach. In this book, we deal with the integral formulations of the boundary value prob­ lems for various PDEs which we derive from the strong converse spherical mean value relations. In Chapter 2, we give the results for simple equations such as the Laplace equation, the diffusion and Helmholtz equations, and the heat equation. Chapter 3 includes some high-order equations (biharmonic, polyharmonic, and metaharmonic equations), and Chapter 4 deals with some elliptic systems. Chapters 5, 6, and 7 are devoted to the Lamé equation, pseudovibration elastic, and thermoelastic equations, respectively. It should be noted that different mean value relations can be derived: we can re­ late the spherical means of the solution and its derivatives to the values of these func­ tions at the center of the sphere. There are many relations of this kind (e.g. see [189]) and we will use some of them. However, the most interesting are the mean value rela­ tions which provide equivalent integral formulations of the original differential equa­ tion such as the strong converse mean value theorem described above for the Laplace equation. Therefore, our general objective is to give equivalent integral formulations of the corresponding PDEs. In Chapter 8, we give some applications: these are probabilistic numerical algo­ rithms for solving PDEs (the so-called Random Walk on Spheres algorithms) which we construct on the basis of the integral formulations. The solution in such methods is represented in the form of expectations over Markov processes generated in some sense by the spherical means. The algorithms are simple enough and provide effective numerical solutions to the boundary value problems for complicated domains of high dimension. In addition, the implementation of the algorithms can easily be carried out for parallel computers. The spherical mean value relations can also be applied to solving different prob­ lems of mathematical physics, for example, getting information about eigenvalues, constructing high-order finite difference schemes, finding asymptotics of the solu­ tions, solving inverse problems of the potential theory, etc. The probabilistic representations described in Chapter 8 assume minimum a pri­ ori information about the smoothness of the solution (e.g. the integral equation equiv­ ¯ Ac­ alent to the Laplace equation assumes only that the solution is continuous in G). tually an approach based on spherical mean value relations enables us to construct generalized solutions (see, e.g. [141]). Another interesting advantage of these represen­ tations is the possibility of solving numerically the exterior boundary value problems in complicated high-dimensional domains. It should be noted that there is not much literature concerning spherical mean value relations for PDEs. First, there is the classical book of Courant and Hilbert [33]. In [189] (see also the references in this book) a short review of the works in this field has

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1 Introduction

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| 3

been carried out. The book [124] contains a set of mean value relations for metahar­ monic equations. A series of articles by Diaz and Payne [39] deal with the mean value relations for elasticity problems. In [97], the mean value relations for some parabolic equations, and in [67], the mean value relations for telegraph equations were derived. We also mention the mean value relations where the integration is taken over other domains, for instance, ellipsoids [51]. There is also an interesting class of approximate mean value relations [58; 141; 132] which characterize the relevant equations. Special mean value relations are used in the Monte Carlo Random Walk on Spheres methods and can be found in the corre­ sponding literature [51], [160]. In Chapters 9 and 10, we present further applications of the spherical and plane integral operators described in Chapters 1–7. The approach presented in Chapters 9 and 10 are based on the Poisson-type integral representations for disks, spheres, halfplanes, half-spaces, and other standard domains. The original differential boundary value problem is equivalently reformulated in the form of a system of integral equa­ tions defined on the intersection surfaces of these standard domains. Then, we invert the system of integral equations by a spectral expansion of the kernels. In Chapter 11, we present a stochastic boundary method which can be considered as a randomized version of the method of fundamental solutions (MFS). We suggest to solve the large system of linear equations for the weights in the expansion over the fun­ damental solutions by a randomized singular value decomposition method we intro­ duced in [178]. In addition, we also deal with solving inhomogeneous problems where we use the integral representation through the Green integral formula. The relevant volume integrals are calculated by a Monte Carlo integration technique which uses the symmetry of the Green function. We also construct a stochastic boundary method based on the spectral inversion of the Poisson formula representing the solution in a disk. This is done for the Laplace equation, and the system of elasticity equations. We stress that the stochastic boundary method proposed is of high generality, and it can be applied to any bounded and unbounded domain with any boundary condition provided the existence and uniqueness of the solution are proven. We present a series of numerical results that illustrate the performance of the suggested methods. Chapter 12 deals with an elasticity problem with random loads. In Chapter 13, we study boundary value problems with stochastic boundary condi­ tions. We construct exact proper orthogonal decomposition for some classical bound­ ary value problems, for a disk, ball, half-plane, and a half-space, with a Dirichlet and Neumann boundary conditions, where the boundary functions are white noise or ho­ mogeneous (2π-periodic) random processes. In case the boundary function is a white noise, the solutions are treated as generalized random fields with the convergence in the proper spaces and relevant generalized treatment of boundary conditions. In the last section of this chapter we study a response of an elastic half-space to ran­ dom excitations of displacements on the boundary under the condition of no shear­ ing forces. We analyze the white noise excitations and general random fluctuations of

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4 | 1 Introduction displacements prescribed on the boundary. We consider the case of partially ordered defects on the boundary whose positions are governed by an exponential–cosine-type correlation function. The analysis is based on a Poisson-type integral formula which we derive here for the case of zero shearing forces on the boundary. We obtain exact representations for the displacement correlation tensor and the Karhunen–Loève ex­ pansion for the solution of the Lamé equation itself and analyze some features of the correlation structure of the displacements. The Monte Carlo technique developed can be applied to a wide class of differential equations with random boundary conditions.

Acknowledgment Support from the Russian Foundation for Basic Research, under Grants nos. 12-01-00635-a and N 12-01-00727-à, is kindly acknowledged.

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2 Scalar second-order PDEs 2.1 Spherical mean value relations for the Laplace equation and integral formulation of the Dirichlet problem We consider equations in a domain G whose boundary ∂G is denoted by Γ and the ¯ Note that the domain G may be unbounded. Throughout the whole book, closure by G. we use the following notations (partially already introduced in the previous chapter): – S(x, r) – a sphere of radius r centered at the point x, – B(x, r) – a ball of radius r centered at the point x, – Ω – the unit sphere S(0, 1), – dΩ – the surface element of Ω, – dS = do – the surface element of S(0, r). – ω m – the area of the surface of the unit sphere in Rm .  – N r u (x) = ω1m Ω u (x + rs)dΩ(s) – the spherical mean of the function u (x). Let us start with a simple case, the Dirichlet problem for the Laplace equation in a domain G ⊂ Rm : Δu (x) = 0 ,

x ∈ G,

u(y) = φ(y) ,

(2.1)

y ∈ Γ.

We seek a regular solution to (2.1) and (2.2), i.e. u ∈ C2 (G)

(2.2) 

¯ ). C(G

2.1.1 Direct spherical mean value relation It is well known that every regular solution to (2.1) satisfies the spherical mean value relation  1 u (x + rs) dΩ(s) (2.3) u ( x ) = N r u ( x ) := ωm Ω

 ¯ := G Γ. The same is true for for each x ∈ G and for all spheres S(x, r) contained in G the volume mean value relation (it can be obtained directly from (2.3) by integrating)  m u (y) dy. (2.4) u(x) = ωm rm B ( x,r )

The mean value relation (2.3) can be derived by different methods. For small r, it is pos­ sible to use the method based on the power expansion of the integrand. We present

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6 | 2 Scalar second-order PDEs this method here and we will use it later to derive the mean value relations for dif­ ferent equations. The following statement is very useful, in particular, to get power expansions of the spherical means.   α α We denote by D the differential operator D = −i ∂x∂ k and D α = D1 1 · · · D mm , 1≤ k ≤ m where α is the multiindex: ∂ , |α | = α 1 + · · · + α m . α = (α 1 , . . . , α m ) , α! = α 1 ! . . . α m ! , D k = −i ∂x k In [189], the following result is presented. ˆ be the Lemma 2.1. Let η be a measure in Rm with a compact support and let h(y) = η Fourier transform of η:  ˆ = exp{−i(y, s)}dη(s) . h(y) = η Rm

Then



u (x + ry) dη(y) = {h(−rD) u } (x)

(2.5)

for x, r, and u for which the left-hand side exists and the right-hand side converges. Proof. The integral exists for sufficiently small r > 0 and as a function of r is ana­ lytic; the same for the right-hand side. Thus it is sufficient to prove the statement for sufficiently small r. The expansion of the integrand and the integration yields   ∞  1 − i ( y,s ) [−i(y, s)]k dη (s) e dη(s) = k! k = 0 m R ⎫ ⎧  ∞ ⎬  (−i)k ⎨  k α α = y s dη ( s) ⎭ k! ⎩ | α|=k α k =0 ⎫ ⎧

 ∞ ⎬  (−1)k ⎨  k α α = y s dη(s)⎭ ⎩ α k! k =0 | α |= k    1 (−iy)α s α dη(s) . = α α! k k! = . α!(k − |α |)! α

Here

Let u ( x + z) =



a α (x)z α ,

a α (x) =

α

Then  u (x + rs)dη(s) =

  α

1 ∂|α| u(x) α . α1 α! ∂x1 · · · ∂x mm



a α (x)r

|α| α

s

dη(s) =



a α ( x ) r| α |





s α dη(s)

α

   1 (iD α )u (x)r| α| s α dη(s) . = α α!

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2.1 Spherical mean value relations for the Laplace equation | 7

We consider the case when η is the uniform measure dΩ. To formulate the result we define a function W m (·) of the dimension m. Let J ν (z) be the Bessel function  2k+ν ∞  (−1)k z . (2.6) J ν ( z) = Γ ( k + ν + 1 ) Γ ( k + 1 ) 2 k =0 Here Γ(·) is the Euler Gamma-function. Let us introduce the function     m /2 − 1 m 2 J m /2 − 1 ( z ) . W m ( z) = Γ 2 z Theorem 2.1. Let u (x) be a real-valued analytic function. Then   √  N r u (x) = W m ir Δ u (x)

(2.7)

(2.8)

for x, r, and u for which the left-hand side is defined and the right-hand side exists. Proof. We get (2.8) from (2.5) by choosing the measure η as dΩ/ω m and taking into account that the Fourier measure of dΩ/ω m is W m (|y|i), and |D| = (−Δ)1/2 . For example, in R3 1 h(iy) = 4π

2π

π

sin(θ)dθ exp(|y| cos θ) =

dφ 0

0

hence (2.5) has the form



N r u(x) =

sh(|y|) ; |y|

 √  sh r Δ √ u(x) . r Δ

(2.9)

Note that the last relation is the expansion in powers of the Laplace operator given in [33] ∞  r2k (2.10) Δ k u(x) u(x) = N r u(x) − (2k + 1)! k =1 since

∞     W m ir λ =

λ k r2k . 2k k! m(m + 2) · · · (m + 2k − 2) k =0

In R2 the “spherical” (a circular) mean has the representation 1 2π

2π

 2k ∞  √   −2 r u (x + re )dφ = J 0 r −Δ u (x) = (k!) Δ k u(x) . 2 k =0 iφ

0



Here J 0 (|x|) is the Bessel function that is obtained as the Fourier transform of the uniform measure dφ/2π. Note that if we choose the uniform measure in the disk whose Fourier transform is 2J 1 (|x|)/|x|, we get 1 π

2π 1

u (x + rte iα )tdtdα = 0 0

∞ 

k =0

1 k!(k + 1)!

 2k r Δ k u(x) . 2

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8 | 2 Scalar second-order PDEs This relation can be generalized by choosing a general distribution ρ on the inter­ val [0, 1] which leads to 1 2π

2π 1

u (x + rte iα )dρ (t)dα =

∞ 

(k!)−2

k =0

0 0

where

 2k r a k Δ k u(x) , 2

1 a k = (t/2)2k dρ (t) . 0

Remark 2.1. Note that the power expansions were given by Pizetti [137] (see also [33]): if a function u (x) = u (x1 , . . . , x m ) belongs to C2p+2 (G), G ⊂ Rm , then for all x ∈ G and all sufficiently small r N r u(x) =

p 

r2k C k u (x) + Q p (r)u (x) ,

k =0

and m N B u(x) ≡ ωm rm =

p 



u (x + y)dy B ( x,r )

r2k C k u (x) + Q p (r)u (x) ,

k =0

where C k and C k are the differential operators of the form Ck =

Γ(m/2) Δk , 22k k!Γ(m/2 + k )

C k =

m Ck , m + 2k

and

k ≥ 0.

The remainders can be estimated: for instance, in 3D  1 (r − |y|)2p+1 p+1 Δ u (x + y)dy , Q p ( r) u ( x ) = 4π(2p + 1)! r|y| B ( x,r )

hence |Q p (r)u (x)| ≤ c(p)r2p+2 u (2p+2) B ,

where the norm u (2p+2) B is defined by u

(2p+2)

B =

m  i 1 ,i 2 ,...,i ν

   ∂ ∂  ∂    sup  ··· u ( x ) .  ∂x ∂x ∂x i i i 1 2 ν =1 x ∈ B ( x,r )

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2.1 Spherical mean value relations for the Laplace equation |

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The spherical mean value relation for small radii for the harmonic functions fol­ ¯ can be lows from the obtained expansion (2.10). Generalization for all S(x, r) ⊂ G obtained by the analytical continuation or by using the Green formula. Let us illus­ trate this in the case Rm , m ≥ 3. The Green formula reads    ∂ ∂ u(x) = E(x, y) u (y) − u (y) E(x, y) do(y) ∂ν ∂ν ∂G  − Δu (z)E(x, z)dz . G

Here E(x, y) =

|x − y|2−m , (m − 2)ω m

ν is the outward normal vector to the boundary ∂G and do(y) = r m−1 dω m = ω m r m−1 dΩ . From this formula we immediately get the desired mean value relation, since ∂/∂ν = ∂/∂r for G = B(x, r), Δu = 0 and  ∂ udo = 0 . ∂ν S ( x,r )

The last equality follows from the well-known Green formula     ∂v ∂u (uΔv − vΔu )dx = u −v do , ∂ν ∂ν G

∂G

which is true for arbitrary functions u, v ∈ C2 (G). We would now like to answer the questions: Does the spherical mean value relation uniquely characterize the harmonic func­ tions? Is it possible to use the spherical mean value relation (in one form or another) to give an equivalent integral formulation of the Dirichlet problem for the Laplace equa­ tion? How do we solve numerically the integral equation generated by the relevant spherical mean value relation? In the next section, we shall give answers to these questions. In [224], the following general problem was studied: does the weak spherical mean value relation uniquely characterize the corresponding differential equation? We now present the relevant statements. Assume that u is a real-valued analytic function which satisfies the mean value relation  u (x0 ) = u (x0 + ry)dσ (y) , (2.11)

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10 | 2 Scalar second-order PDEs where σ is a measure with compact support. Then, taking in Lemma 2.1 the measure η = σ − δ x0 , δ x0 being the Dirac measure concentrated at the point x0 and  bα zα , h( z) = α

we get



(−1)| α| r| α| b α D α u (x) = 0 .

(2.12)

α

From this, for sufficiently small r > 0, we get that Q j (D)u = 0 , where Q j (D) =

j = 0, 1, 2, . . . , 

(2.13)

bα Dα .

| α |= j

Thus the spherical mean value relation (2.11) for a real-valued analytic function holds if and only if the function u satisfies the infinite system of differential equations (2.13). More generally, for continuous functions, the following statement holds (see [224]). Theorem 2.2. A function u ∈ C(G), G bounded, satisfies the mean value relation (2.11) for all x ∈ G and all 0 < r < d(x) if and only if the function u is a weak solution to the system (2.13). It is interesting to find out when the system (2.13) is equivalent to a single equation L(D)u = 0. In this case σ is a distribution characterizing the operator L(D). Theorem 2.3. The system (2.13) is equivalent to a single equation of the type L(D)u = 0 , where L(ξ ) is a homogeneous polynomial if and only if the polynomial L(ξ ) is divisible through all Q j and there is an integer k such that L = cQ k , c being a constant. Finally, the form of the characterizing distribution is given in the following statement. Theorem 2.4. For each homogeneous polynomial L there exists a mass with a compact support such that a function u ∈ C(G) is a weak solution to L(D)u = 0 if and only if the spherical mean value relation (2.11) holds for all x ∈ G and all 0 < r < d(x). Each mass ˆ (0) ≠ 0) has this of the form L(D)m (m is a distribution with a compact support and m property. In the next section, we deal with the strong converse mean value theorem which uniquely characterizes not only the equation but also the solution of the Dirichlet problem and provides an equivalent integral equation.

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2.1 Spherical mean value relations for the Laplace equation | 11

2.1.2 Converse mean value theorem The mean value relation (2.3) characterizes the solutions to (2.1). We start with the weak converse mean value theorem, whose proof is well known (e.g. see [33]). Proposition 2.1. Suppose that a function u (·) is continuous in G and for every sphere S(x, r) contained in G, u (·) satisfies the mean value relation (2.3). Then u (·) is harmonic in G. The proof follows from expansion (2.10). We formulate a stronger result which presents an equivalent formulation of the problem (2.1), (2.2). Theorem 2.5 (Strong converse mean value theorem). Let φ(·) be a given continuous and bounded function on Γ; if G is unbounded, we suppose (H) the solution u (·) to the Dirichlet problem associated with φ(·) tends to 0 at infinity for the dimensions m ≥ 3 and is bounded in the case m = 2. Assume that G is a domain for which the problem (2.1), (2.2) has a unique solution for any continuous and bounded function φ. Suppose that there exists a function  v ∈ C(G Γ), v|Γ = φ, such that the mean value relation (2.3) holds at every  point x ∈ G for at least one sphere S(x, r) ⊂ G Γ; if G is unbounded, we suppose that v(·) tends to 0 at infinity in dimensions m ≥ 3 and is bounded when m = 2. Then v(·) is the unique regular solution to the problem (2.1), (2.2). The same conclu­ sion holds if the volume mean value relation  m v(x) = v(y) dy σm rm B ( x,r )

holds for every point x ∈ G and at least one ball B(x, r) ⊂ G



Γ.

Proof. We use the same arguments as those given in [33], where the converse mean value theorem for the harmonic functions for bounded domains was given. We shall now give the proof for the volume mean value relation (the same argu­ ments work in the case of the spherical mean value relation). Let u be the solution to the Dirichlet problem and x be fixed in G. Since u satisfies the volume mean value relation for every ball B(y, r) contained in G, we conclude that the function u − v satisfies the volume mean value relation for B(x, r). Let F be the set of points of G where u − v attains its maximum M: even if G is unbounded, as u − v tends to 0 at infinity (or is bounded, in dimensions m = 2), F is a compact set; let x0 be a point of F whose distance to Γ is minimum. If x0 were an interior point of G, we could find a ball B(x0 , r0 ) ⊂ G for which the volume mean value relation holds and then u − v would be equal to M inside B(x0 , r0 ), which is a contradiction with the definition of x0 . Therefore, x0 belongs to Γ. We repeat the same argument for the minimum of u − v. Since, by hypothesis, (u − v)|Γ = 0, we conclude u ≡ v in G.

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12 | 2 Scalar second-order PDEs Using this converse mean value relation we can formulate an integral equation equiv­ alent to the Dirichlet problem.

2.1.3 Integral equation equivalent to the Dirichlet problem In this section, we use the strong converse mean value relation to derive an equivalent integral equation of the second kind. Let us denote by d(x) the distance from a point x ∈ G to the boundary Γ and let d∗ = sup d(x) . x∈G

We also introduce an “ε-boundary”: Γ ε = {x ∈ G : d ( x) ≤ ε} . Let δ x (y) = δ(|x − y| − d(x)) . For simplicity, we use this notation to indicate that δ x (y) is equal to 1/ω m if |x − y| = d(x) and to 0 otherwise. We define a kernel function k ε by ⎧ ⎨ δ x (y) if x ∈ G \ Γ ε , k ε (x, y) := ⎩ (2.14) 0 if x ∈ Γ ε , and define the integral operator K ε by



K ε ψ ( x ) :=

k ε (x, y)ψ(y)dy

(2.15)

G

for each ψ(·) ∈ C(G). We now fix the boundary function φ and suppose that the conditions of the con­ verse mean value relation are satisfied. Denote by u the solution to the Dirichlet prob­ lem corresponding to φ and by f ε (x) the function ⎧ ⎨0 if x ∈ G \ Γ ε , f ε (x) = ⎩ u (x) if x ∈ Γ ε . Consider the integral equation v(x) = K ε v(x) + f ε (x) .

(2.16)

Note that the mean value relation implies (2.16). Thus if we assume that u (x) is known in Γ ε , (2.16) presents the desired integral equation of the second kind, and the converse mean value theorem states that this equation has a unique solution which coincides with the solution to the Dirichlet problem for the Laplace equation. It remains to propose a method of calculation of the solution to (2.16). We show that the conventional successive iteration method applied to (2.16) is convergent.

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2.1 Spherical mean value relations for the Laplace equation |

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Theorem 2.6. For any ε > 0, the integral equation (2.16) has a unique solution given by the Neumann series ∞  v(x) = f ε (x) + K εi f ε (x) (2.17) i =1

and it coincides with the solution to (2.1) and (2.2) if the assumptions of the converse mean value relation are satisfied. Proof. Let ε be fixed. It is simple to show the convergence of the series f ε (x) +

N 

K εi f ε (x) ,

(2.18)

i =1

if d∗ < ∞. It is then sufficient to prove the existence of 0 < λ < 1 such that for any continuous and bounded function g K 2ε gL ∞ < λgL ∞

(this fact also implies the uniqueness of the solution to (2.17)). Let ν(ε) = ε2 /(4d∗2 ) . For x ∈ G \ Γ ε we have 



k ε (x, y) G

k ε (y, y )dy dy =



#∞

i =1

G



=

Let v(x) := f ε (x) +

⎞ ⎛  ⎟ ⎜ δ x (y) ⎝ δ y (y )dy ⎠ dy

G\Γ ε

G

(2.19)

δ x (y)dy ≤ 1 − ν(ε) < 1 . G\Γ ε

K εi f ε (x). It is clear that v satisfies v(x) = K ε v(x) + f ε (x) .

If x belongs to Γ ε , then, for any y, k ε (x, y) = 0 and thus v(x) = f ε (x) = u (x). On the other hand, if x belongs to G \ Γ ε , then the definition of k ε implies that v satisfies the mean value relation at x with the sphere of radius d(x). We conclude by using Theorem 2.5. Remark 2.2. Note that the condition d∗ < ∞ is not necessary for the convergence of the iteration method. However, it is difficult to study the convergence of the itera­ tion method in this general case. In the last chapter, we prove the convergence for the half-space R3+ and the exterior of a sphere. In the integral equation (2.16) and in the iteration method we used the spheres S(x, d(x)). However, the strong converse mean value relation implies that in each step of the iteration method we can use a sphere of an arbitrary radius, say, r x , such ¯ In the last chapter, we will see that this scheme with a special choice that S(x, r x ) ⊂ G. of r x is reasonable when solving the Poisson equation by the Random Walk method.

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14 | 2 Scalar second-order PDEs

2.1.4 Poisson–Jensen formula More generally, a local integral equation can be derived if we replace the mean value relation by the Poisson formula representing the solution at an arbitrary point x ∈ S(x0 , r) ⊂ R3 :  u(x) =

H (x, y)u (y) dS(y) ,

(2.20)

S ( x 0 ,r )

where H (x, y) =

r2 − x 2 , 4πr|x − y|3

if we suppose that this equality holds for every x0 ∈ G at least for one sphere ¯ then u (·) is the unique regular solution to the Dirichlet problem. S(x0 , r) ⊂ G, In 2D, there is a further generalization of formula (2.20) – the so-called Poisson– Jensen formula [108]. Let u (z) = u (r, θ) be a univalent function harmonic in a disk K (0, R) except for a set of singularities where u has logarithmic poles. We denote the nonzero poles by {ζ i }, i = 1, . . . , where the indexing is chosen so that |ζ1 | < |ζ2 | < · · · < |ζ i | < · · · . Let μ j ln |z − ζ j | be the principal part corresponding to ζ j . It is also convenient to introduce the principal part μ0 ln |z| corresponding to the point z = 0 taking μ 0 = 0 if this point is not a logarithmic pole of u (z). Let us consider the conform mapping of the disk |z| ≤ ρ on itself by l ζ ( z) = ρ 2

z−ζ , ρ 2 − ζ¯ z

where |z| ≤ ρ < R. This transformation maps the point ζ into the center of the disk. Since l ζ (z) is an analytic function in the disk |z| < ρ 2 /|ζ |, having a single simple zero point at z = ζ and |l ζ (z)| = ρ at all points of the circle |z| = ρ, then the function $ % 1 u ζ (z) = ln |l ζ (z)| ρ is harmonic if |z| < ρ 2 /|ζ |, z ≠ ζ and has in this disk a logarithmic pole at the point z = ζ with the principal part ln |z − ζ | and u = 0 on the boundary z = ρ. If one part of the logarithmic poles of u (z), namely, the points 0, ζ1 , . . . , ζ ν(ρ) lies inside this disk and the rest is outside of it then   ν(ρ) ν(ρ)  ρ(z − ζ )     k   μ k u ζ k ( z) = u ( z) − μ k ln  v( z) = u ( z) −  ρ 2 − ζ¯k z  k =0 k =0 is harmonic in all the points of a disk whose radius is larger than ρ (the radius is equal to the minimum of the two values: |ζ ν(ρ)+1 | and ρ 2 /|ζ ν (ρ )|), while on the boundary |z| = ρ this function coincides with u (z). Therefore, for all |z| = r < ρ we get 1 v(r, θ) = 2π

2π

v(ρ, α ) 0

r2 − ρ 2 dα ; r2 + ρ 2 − 2rρ cos(θ − α )

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hence 1 u (r, θ) = 2π

2π

v(ρ, α ) 0

r2 − ρ 2 dα r2 + ρ 2 − 2rρ cos(θ − α ) (2.21)

   ρ(z − ζ )   k   + μ k ln   ρ 2 − ζ¯k z  k =0 ν(ρ)



which is known as the Poisson–Jensen formula. If the function has no logarithmic poles, then all the values μ j are zeros, and we come to the Poisson formula.

2.2 The diffusion and Helmholtz equations The aim of this section is to treat the diffusion equation Δu (x) − λu (x) = 0 ,

u |Γ = φ ,

(2.22)

where λ is a nonnegative constant and the Helmholtz equation Δu (x) + λu (x) = 0 , u |Γ = φ .

(2.23)

Using the function W m (z) introduced by (2.7) in the previous section we define a new function  w m (r, λ) = W m (ir λ ) . Since I ν (z) = J ν (iz)/i ν , we get the expansion 

w m (r, λ) = Γ  =Γ =

m 2



2 √

 m /2 − 1

r λ  ∞ k =0

 I m /2 − 1 ( r λ )

λ k r2k + m/2)

22k k!Γ(k

∞ 

k =0

since

m 2

2k k!m(m

(2.24)

λ k r2k , + 2) · · · (m + 2k − 2)

    m m Γ k+ =Γ m(m + 2) · · · (m + 2k − 2)/2k . 2 2

2.2.1 Diffusion equation From (2.8), we deduce the mean value relation for the solution to (2.22): 1 u ( x ) = w− m ( r, λ) N r u ( x ) .

(2.25)

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16 | 2 Scalar second-order PDEs ¯ using a different ap­ We obtain this mean value relation for arbitrary S(x, r) ⊂ G proach.  Let 1 v(x, r) = N r u = u (x + sr)dΩ(s) . ωm Ω

Then the function v(x, r) satisfies the Darboux equation [33] m − 1 ∂v ∂2 v + − Δv = 0 ∂r2 r ∂r

(2.26)

with the initial conditions ∂v (x, 0) = 0 . ∂r

v(x, 0) = u (x),

Indeed, using the Gauss theorem we get ⎛ ⎞   m 1 ∂u ∂v(x, r) ⎝ si ⎠ dΩ = ∂r ωm ∂x i i = 1 Ω   1 ∂u 1 = = Δudy . do ω m r m −1 ∂ν ω m r m −1 S ( x,r )

(2.27)

B ( x,r )

 ∂ ∂ = si ∂ν i=1 ∂x i

Here

m

denotes the differentiation with respect to the outward normal vector to the sphere Ω and dy is the volume element. Differentiating once more yields the desired result   m−1 1 m − 1 ∂v ∂2 v =− Δudy + Δudo = − + Δv . ∂r2 ωm rm ω m r m −1 r ∂r B ( x,r )

S ( x,r )

The operator Δ and the spherical mean operator N r are permutable [33]; therefore, from (2.22), we get Δv = λv. Thus substituting Δv = λv in the Darboux equation we find that v = N r u (x) solves the problem m − 1 ∂v ∂2 v + − λv = 0 , ∂r2 r ∂r ∂v (x, 0) = 0 . v(x, 0) = u (x) , ∂r The solution to this problem is v(x, r) = v(x, 0) · w m (λ, r) , which is the desired spherical mean value relation. In G ⊂ R3 , we have from (2.9) √ λr √ N r u(x) . (2.28) u(x) = sinh ( λr)

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2.2 The diffusion and Helmholtz equations

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√ Note that (2.25) and (2.28) are true for complex λ such that Re λr < π. The diffusion equation is treated exactly according to the scheme described for the Laplace equation, since the integral formulation can also be given for (2.22).

Proposition 2.2. We suppose that the Dirichlet problem for the diffusion equation has a unique solution for any continuous function φ; if G is unbounded, we suppose (H).  Let φ(·) be a given continuous and bounded function on Γ. Assume that u ∈ C(G Γ) satisfies the mean value relation (2.25) for each x ∈ G at least for one sphere S(x, r x ) ⊂ G and v|Γ = φ. Then u (·) solves the Dirichlet problem (2.22). Proof. The proof repeats the arguments used in the case of the Laplace equation, since the strong maximum principle holds (this follows from the inequality |w m (r, λ)| ≥ 1 for nonnegative λ).

2.2.2 Helmholtz equation If λ < 0, we come to the Helmholtz equation. We suppose in this case that G ⊂ R3 is a bounded domain and 0 > λ > λ0 (G) where λ0 (G) is the principal eigenvalue of the Laplace operator in G. √ Note that λ < dπ∗ ; indeed, it is well known that if G2 ⊂ G1 , then λ0 (G2 ) ≥ λ0 (G1 ); in our case, for all x ∈ G, S(x, d∗ ) ⊂ G, and the eigenvalue λ0 (S(x, d∗ )) is equal to π2 d ∗2 . Then the solution to (2.23) can be written in R3 in the form √ λr √ N r u(x) , (2.29) u(x) = sin ( λr) for any r such that S(x, r) is included in G. Under these restrictions the maximum principle is true (see [56]) and we conclude that the converse spherical mean value theorem is true. Note that for the Helmholtz equation we can derive the volume mean value rela­ tion in a ball B(x, r) ⊂ Rn  1 √ u(x) = u (y)dV (y) , n V r 2n/2 Γ( 2 + 1)τ n/2 ( λr) B ( x,r )

where V r = π n/2 r n /Γ(n/2 + 1) is the volume of the ball B(x, r) and τ α (z) = z−α J α (z). We notice that the function τ α (z) is related to the function W α (z) by W α (z) = Γ(α /2)2α/2−1 τ α/2−1(z) . The above volume mean value relation is obtained by integrating the spherical mean value relation for the Helmholtz equation  1 u (x + rs)dΩ . u(x) = Wn ωn Ω

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18 | 2 Scalar second-order PDEs Remark 2.3. The weak spherical mean value relations characterize the high-order el­ liptic equations with constant coefficients. Let us mention the following result (see, e.g. [55] and [56]) which we briefly mentioned above. Let μ be a nonnegative Borel measure with total mass equal to 1, such that the support of μ is contained in the unit sphere of Rm and not contained in any hyperplane. If u is a continuous function on some open set G ⊂ Rm having the mean value property  u (x) = u (x + ry)μ (dy) for every x ∈ G and every positive r < d(x), then u ∈ C∞ (G) and  i 1 +···+ i m = n

A i1 ...i m

∂n u i

i

∂x11 · · · ∂x mm

= 0, n = 1, 2, . . . ,

where the coefficients are the moments  i A i1 ...i m = x11 · · · x imm μ (dx) . Conversely, every infinitely differentiable solution of the last system of differential equations has the above mean-value property.

2.3 Generalized second-order elliptic equations The expansions of the spherical mean of the type (2.10), (see also (3.10) below) sug­ gest a device for defining a generalized Laplacian of nondifferentiable functions. For example, if in Δv(x) = f the functions v and f are merely continuous, one can say that Δv = f in a generalized sense if 1 f lim 2 [N r v − v] = . r →0 r 2m An analogous point of view is reported in [141] and [187]. From an expansion that we will obtain in Chapter 5 it is possible to define gener­ alized solutions to the Lamé equation μΔu(x) + (λ + μ ) grad div u(x) = f(x) , where x ∈ Rn , u = (u 1 , . . . , u n ), as continuous functions satisfying the relation lim r →0

' 1 & 1 f Nr u − u = . r2 2(λ + μ (n + 1))

(2.30)

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Here, the averaging matrix operator N r1 is defined by   1  1 N r u i (x) = (a + bs2i )u i (x + rs) dΩ ωn Ω

n  b  + s i s j u j (x + rs) dΩ , ω n j=i

i = 1, . . . , n ,

Ω

where a = 1 − β, b = nβ, and β=

(n + 2)(λ + μ ) . 2(λ + μ (n + 1))

In [58], elliptic equations with nonconstant coefficients were considered: Lu = 0 , where L=

m  i,k =1

a ik

∂2 , ∂x i ∂x k

and the coefficients are supposed merely measurable and bounded and the matrix a(x) = (a ik (x)) is symmetric and positive definite. Fulks [58] (see also [141]) proved that  1 r2 u (x + ra(x)1/2 s)H m−1 (ds) − u (x) = (2.31) Lu (x) + O(r2 ) ωm 2m Ω

as r → 0, u being any twice continuously differentiable function. Here, a(x)1/2 denotes the positive square root of the matrix a(x) and H m−1 the (m − 1)-dimensional measure. From (2.31) it is convenient to pass to the mean over an ellipsoid. Multiplying (2.31) by mr m−1 and integrating with respect to r, we obtain, using an obvious change of variables in the m-fold repeated integral  1 r2 u (y)dy − u (x) = Lu (x) + O(r2 ) , (2.32) meas E(x, r) 2(m + 2) E( x,r )

where E(x, r) is the ellipsoid ( ) E(x, r) = y ∈ Rm : (a−1 (x)(y − x), y − x) < r2 , with the center at x and axes on the eigenvectors of the matrix a(x), while meas E(x, r) = (ω m /m)r m [det a(x)]1/2 . Note that the boundaries ∂E(x, r) of the ellipsoids E(x, r) are the level surfaces of the Levi function  −1  1 − m /2 1 L(x, y) = a (x)(y − x), y − x , 1 / 2 (m − 2)ω m [det a(x)] the fundamental solution of the constant elliptic operator L, x fixed. Note that the same approach was used in [51] when constructing the Random Walk on ellipsoids.

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20 | 2 Scalar second-order PDEs Remark 2.4. Let C = (c jk ) be a symmetric n × n-matrix and r ≠ 0, and let  ( u ) = u (x + y)do(y) , σ r,C x Er

where the integration is taken over the surface of the ellipsoid    Er = y : c jk y j y k = r2 . It can be shown (e.g. see [73]) that if u is the solution to the following equation with constant matrix of coefficients A = (a jk ): n 

a jk

j,k =1

∂2 u = 0, ∂x j ∂x k

then the ellipsoidal mean value relation r,C + tA σ r,C (u) x (u) = σ x

is true for all t. For the Laplace equation it means that the integrals of harmonic functions over the confocal ellipsoids do not depend on the parameter t (if the function is considered in r,C + tA (u ) lies in G). a domain G, then t is such that the convex hull of supp σ x

2.4 Parabolic equations 2.4.1 Heat equation Let Q be a domain in Rn+1 whose points will be denoted by (x, t) = (x1 , x2 , . . . , x n , t). We deal in this section with the heat equation ∂u = Δu (x, t) + f (x, t), ∂t

(x, t) ∈ Q . 

Let Z (x, t) = θ(t)[4πt]

− n /2

exp

|x|2 4t

(2.33)



be the fundamental solution to (2.33), where θ(t) is the Heaviside step function: θ(t) = 0 if t ≤ 0 and θ(t) = 1 if t > 0. Introduce the function Z (α) = Z (x, t) − (4πα )−n/2 , depending on a positive parameter α and define a family of domains B α (x, t):   B α (x, t) = (x , t ) ∈ Rn+1 : t < t, Z (α) (x − x , t − t ) > 0

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2.4 Parabolic equations | 21

with the boundaries

  ∂B α (x, t) = (x , t ) ∈ Rn+1 : t ≤ t, Z (α) (x − x , t − t ) = 0 . (β)

We also define a family of domains B α (x, t) depending on the parameter β, 0 < β ≤ α: (β) B α (x, t) = B α (x, t) ∩ Rn × (t − β, t) , and let

(β)

∂B α (x, t) = ∂B α (x, t) ∩ Rn × [t − β, t] . We call the domain B α (x, t) a balloid and the set ∂B α (x, t), a spheroid of radius α (β) (β) with the center at (x, t). The domains B α (x, t) and ∂B α (x, t) are called truncated balloid and truncated spheroid, respectively. We now obtain a mean value theorem where the solution of (2.33) at the point (x, t) is expressed in terms of its values integrated over a spheroid or a truncated spheroid with the center at (x, t). Then by averaging we derive the relations where the solution of (2.33) at the point (x, t) is expressed in terms of its values integrated over the spheroid and balloid with the center at (x, t). Note that the balloid B α (x, t) is situated between the planes t = t and t = t − α. The intersection of B α (x, t) and the plane t = t − τ (0 < τ < α ) is an n-dimensional ball B(x, R(τ)) where R(τ) = (2τn ln(α /τ))1/2 . The maximum of the radius R(τ) is attained at τ = α /e; its value is Rmax = (2nα /e)1/2 . Hence, (2.34) B α (x, t) ⊂ B(x, Rmax ) × (t − α, t) , and

(β)

∂B α (x, t) ⊂ B(x, Rmax ) × (t − β, t) . Relation (2.34) shows that the balloid B α (x, t) tends to the point (x, t) as α → 0. There­ fore, for each point (x, t) ∈ Q, there exists α such that B α (x, t) ⊂ Q .

(2.35)

Thus let (x, t) ∈ Q, and we suppose that α is chosen so that condition (2.35) is satis­ fied. Obviously, ∂ (α) Z (x − x , t − t ) + Δ x Z (α) (x − x , t − t ) = 0, ∂t

t < t ,

(2.36)

where Δ y is the Laplace operator (differentiation with respect to the variable y). From (2.33) and (2.36) we get Z (α) (x − x , t − t )Δ x u (x , t ) − u (x , t )Δ x Z (α) (x − x , t − t )  ∂  = Z (α) (x − x , t − t )u (x , t ) − Z (α) (x − x , t − t )f (x , t ) . ∂t

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22 | 2 Scalar second-order PDEs Integrating this equality in the ball B(x, R(t − t )) we get by the Green formula   ∂  (α) Z (x − x , t − t )u (x , t ) dx ∂t B ( x,R ( t − t ))    ∂u ∂ (α) = Z (α) − u (x , t ) Z dS x (2.37) ∂n x ∂n x ∂B ( x,R ( t −t ))    + Z (α) (x − x , t − t )f (x , t ) dx , B ( x,R ( t − t ))

where Z (α) = Z (α) (x − x , t − t ), n x is the exterior normal vector at x , and dS x is the surface element of the sphere ∂B(x, R(t − t )) at the point x . Notice that Z (α) (x − x , t − t ) = 0 for x ∈ ∂B(x, R(t − t )) (by definition); hence we can change the order of the operator ∂t∂ and the integral on the left-hand side of (2.37). Thus we get from (2.37) ⎧ ⎪ ∂ ⎨ ∂t ⎪ ⎩

⎫ ⎪ ⎬



B ( x,R ( t − t ))

Z (α) u (x , t )dx ⎪ ⎭ 

=

− ∂B ( x,R ( t − t ))

∂Z (α) u (x , t )dS x + ∂n x



Z (α) f (x , t )dx .

(2.38)

B ( x,R ( t − t ))

We now integrate (2.38) with respect to t over [t − β, t], (0 < β ≤ α ):  Z (α) (x − x , t − t )u (x , t )dx lim t →t

B ( x,R ( t − t ))



Z (α) (x − x , β )u (x , t − β )dx

− B ( x,R ( β))



=

− ( β)

∂Z (α) (x − x , t − t )u (x , t )dS x dt ∂n x

(2.39)

∂B α ( x,t )



Z (α) (x − x , t − t )f (x , t )dx dt .

+ ( β)

B α ( x,t )

We prove that 

lim

t →t

Z (α) (x − x , t − t )u (x , t )dx = u (x, t) .

(2.40)

B ( x,R ( t − t ))

Indeed, let



𝛾(ρ, τ) = (4πτ)

− n /2

ρ2 exp − 4τ

 − (4πα )−n/2

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and let τ = t − t . Then  Z (α) (x − x , τ)u (x , t − τ)dx B ( x,R ( τ )) R (τ)

=

𝛾(ρ, τ)u (x + ρω, t − τ)dΩ(ω) S (0,1)

0 R (τ)

=



ρ n−1 dρ

ρ

n −1

(2.41)



𝛾(ρ, τ)u (x, t − τ)dΩ(ω)

dρ S (0,1)

0 R (τ)

+



ρ n−1 dρ

𝛾(ρ, τ)[u (x + ρω, t − τ) − u (x, t − τ)]dΩ(ω) , S (0,1)

0

where dΩ(ω) is the surface element of the unit sphere S(0, 1). Now, R (τ)

ρ n−1𝛾(ρ, τ)ω n dρ

0 R (τ)

= 0 n 2

 2ρ n−1  (4τ )−n/2 exp{−ρ 2 /4τ } − (4α )−n/2 dρ Γ(n/2) α

ln τ

= 0

r n /2 − 1 Γ(n/2)



e

(2.42)

  n /2 τ − dr → 1 α

−τ

as τ → 0. We used here the fact that ω m = 2π n/2 /Γ(n/2). Analogously, we can prove that R (τ) ρ n 𝛾(ρ, τ)dρ → 0 (2.43) 0

as τ → 0. We get from (2.41) by (2.42) and (2.43) the desired relation (2.40). Thus, from (2.39) and in view of (2.40), we get the following mean value relation: (β)

Theorem 2.7. If the parameters α and β, 0 < β ≤ α, are chosen so that B α (x, t) ⊂ Q, then the regular solutions to the heat equation (2.33) satisfy the relation  Z (α) (x − x , β )u (x , t − β )dx u (x, t) = B ( x,R ( β))





∂Z (α) − ∂n x

+ ( β)

u (x , t )dS x dt

(2.44)

∂B α ( x,t )



+

Z (α) (x − x , t − t )f (x , t )dx dt .

( β)

B α ( x,t )

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24 | 2 Scalar second-order PDEs An important particular case of (2.44) is β = α: 

u (x, t) =

− ∂B α ( x,t )



+

∂Z (α) (x − x , t − t )u (x , t )dS x dt ∂n x

Z (α) (x − x , t − t )f (x , t )dx dt ,

B α ( x,t )

which was derived in [97]. Further mean value relations can be obtained by the inte­ gration of (2.44) with respect to β over the interval [0, α ]. This yields Theorem 2.8. Under the assumptions of the previous theorem, the following mean value relation holds:

   t − t ∂Z (α) 1− (x − x , t − t ) u (x , t )dS x dt u (x, t) = − α ∂n x ∂B α ( x,t ) (2.45)  1 (α) + Z (x − x , t − t )u (x , t )dx dt + F α (x, t) , α B α ( x,t )

where 

F α (x, t) = B α ( x,t )

  t − t 1− Z (α) (x − x , t − t )f (x , t )dx dt . α

(2.46)

Here we used the relations α



Z (α) (x − x , β )u (x , t − β )dx

dβ B ( x,R ( β))

0



=

Z (α) (x − x , t − t )u (x , t )dx dt ,

B α ( x,t )

α

 −

dβ ( β)

0

∂B α ( x,t )

∂Z (α) (x − x , t − t )u (x , t )dS x dt ∂n x



(α − (t − t )) −

= ∂B α ( x,t )

α



u (x , t )dS x dt ,

Z (α) (x − x , t − t )f (x , t )dx dt

dβ 0

∂Z (α) ∂n x

( β)

B α ( x,t )



=

(α − (t − t ))Z (α) (x − x , t − t )f (x , t )dx dt .

B α ( x,t )

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Remark 2.5. Let us consider the equation ∂u − Δu (x, t) + c(x, t)u = f (x, t) , ∂t

(x, t) ∈ Q

(2.47)

with 0 ≤ c(x, t) ≤ c0 . Suppose that the parameter α is chosen so that B α (x, t) ⊂ Q. Then the regular solutions to equation (2.47) satisfy the following mean value relation:

   t − t ∂Z (α) − 1− (x − x , t − t ) u (x , t )dS x dt u (x, t) = α ∂n x ∂B α ( x,t )  1 + [1 − (α − (t − t ))c(x , t )]Z (α) (x − x , t − t )u (x , t )dx dt α B α ( x,t )

+ F α (x, t) ,

where F α (x, t) is defined in (2.46). The proof follows from (2.45).

2.4.2 Parabolic equations with variable coefficients In this section, we generalize the results obtained for the heat equation to the follow­ ing equation: n  ∂u ∂u − μΔu (x, t) + b i (x, t) + c(x, t)u = f (x, t), ∂t ∂x i i =1

(x, t) ∈ Q .

(2.48)

We assume that the coefficients b i (x, t) and their derivatives ∂b i /∂x i are continuous and bounded on Q, and the coefficient c(x, t) and the function f (x, t) are also contin­ uous and bounded on Q. We employ in this section the summation convention (sum­ mation over the repeated index). Let   |x − x − (t − t )b (x, t)|2 θ(t − t ) Z (x, t; x , t ) = exp − , [4πμ (t − t )]n/2 4μ (t − t ) where θ(t) is the Heaviside step function, b (x, t) = (b 1 (x, t), . . . , b n (x, t)). The function Z (x, t; x t ) solves the equation −

∂Z ∂Z − μΔ x Z (x, t; x , t ) − b i (x, t) (x, t; x , t ) = 0, ∂t ∂x i

t < t .

Introduce the function Z (α) (x, t; x , t ) = Z (x, t; x , t ) − (4πμα )−n/2 .

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26 | 2 Scalar second-order PDEs Obviously, −

∂Z (α) ∂Z (α) (α) Z − μΔ ( x, t; x , t ) − b ( x, t ) (x, t; x , t ) = 0 , x i ∂t ∂x i

(2.49)

for t < t. As in the previous section, we define the families of domains:   B α (x, t) = (x , t ) ∈ Rn+1 ; t < t, Z (α) (x, t; x , t ) > 0 is a balloid,   ∂B α (x, t) = (x , t ) ∈ Rn+1 ; t ≤ t, Z (α) (x, t; x , t ) = 0

is a spheroid and the following two-parametric sets:   (β) B α (x, t) = (x , t ) ∈ Rn+1; t − β < t < t, Z (α) (x, t; x , t ) > 0 ,   (β) ∂B α (x, t) = (x , t ) ∈ Rn+1; t − β ≤ t ≤ t, Z (α) (x, t; x , t ) = 0 . Note that the structure of these balloids is somewhat more complicated: the inter­ section of B α (x, t) with the plane t = t − τ, (0 < τ < α) is the ball in Rn centered at the point x(τ) = x − τb (x, t) with the radius R(τ) = (2τnμ ln(α /τ))1/2. It is clear that (β) for the arbitrary point (x, t) ∈ Q there is a value of α such that B α (x, t) ⊂ Q. Theorem 2.9. Assume that the parameters α and β, 0 < β ≤ α, are chosen so that (β) B α (x, t) ⊂ Q. Then the regular solutions to equation (2.48) satisfy the following mean value relation:  Z (α) (x, t; x , t − β )u (x , t − β )dx u (x, t) = B ( x ( β) ,R ( β))



∂Z (α) −μ ∂n x

+ ( β)



u (x , t )dS x dt

∂B α ( x,t )



+

k (x, t; x , t )Z (α) (x, t; x , t )u (x , t )dx dt

( β)

B α ( x,t )



+

Z (α) (x, t; x , t )f (x , t )dx dt ,

( β)

B α ( x,t )

where k (x, t; x , t ) =

' b i (x, t) − b i (x , t ) & x i − x i − (t − t )b i (x, t) 2μ (t − t ) ∂b i + (x , t ) − c(x , t ) . ∂x i

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Proof. From (2.48) and (2.49) we find   μ Z (α) (x, t; x , t )Δ x u (x , t ) − u (x , t )Δ x Z (α) (x, t; x , t )   ∂  ∂  (α) = Z (α) (x, t; x , t )u (x , t ) + b ( x , t ) u ( x , t ) Z i ∂t ∂x i ( α )   ∂Z + b i (x, t) − b i (x , t ) (x, t; x , t )u (x , t ) ∂x i $ % ∂b i + c(x , t ) − (x , t ) Z (α) (x, t; x , t )u (x , t ) ∂x i

(2.50)

+ f (x , t )Z (α) (x, t; x , t ) .

Integrating this equality, first, with respect to x over the ball B(x(t − t ), R(t − t )), then with respect to t over (t − β, t), and using the Green formulae (keeping in mind (β) Z (α) (x, t; x , t ) ≡ 0 for (x , t ) ∈ ∂B α (x, t)), we get the desired result.

2.4.3 Expansion of the parabolic means It is possible to derive expansions of the means which generalize expansions (2.8). Let x ∈ Rn , t ∈ R1 and let α = (α ij ) be a constant positive definite symmetric n × n matrix whose entries α ij are real values. In Rn+1 = {(x, t) : t ∈ R1 } we consider the parabolic equation ∂u − Lu = 0 , (2.51) ∂t where the operator L is defined by Lu ≡

n  i,j =1

α ij

∂2 u . ∂x i ∂x j

Let us consider a set B r in Rn+1 which we call a balloid of radius r: ( ) B r = (y, τ) : yT α −1 y < 2(n + m)τ ln(r2 /τ)

(2.52)

where m, r are positive real numbers and the matrix α is defined above. It is clear that in (2.52) 0 < τ < r2 , and the vector y is so that yT α −1 y < [2(n + m)/e]r2 , i.e. the horizontal diameter of the balloid is a quantity of order r. As r → 0, the spheroids shrink monotonically to the point (0, 0). Let us consider the following parabolic mean:  (M r u )(x, t) = u (x + y, t − τ )K (y, τ )dydτ , Br

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28 | 2 Scalar second-order PDEs where K (y, τ) =

n+m |α |1/2 π n/2 2n+m−1 Γ(m/2)r n+m ' m /2 − 1 & × ln(r2 /τ ) 2(n + m)τ ln(r2 /τ ) − yT α −1 y .

Let p ≥ 0 be an integer. We say that u (x, t) ∈ C2p,p (G) if all the derivatives ∂ ∂ ∂ ··· ∂x i1 ∂x i2 ∂x i k



∂ ∂t

l

u (x, t)

exist and are continuous in G for all pairs (k, l) of integers satisfying the conditions k ≥ 0, l ≥ 0, k + 2l ≤ 2p . ¯ ), p ≥ 0. Then for all inner points Theorem 2.10. Assume that u (x, t) ∈ C2p+2,p+1(G (x, t) ∈ G and for all sufficiently small values r (such that 0 < r < d(x, t)) the following expansion is true: (M r u )(x, t) =

p 

r2i L i u (x, t) + Q p (r)u (x, t) ,

(2.53)

i =0

where the differential operators are defined by Li =

(−1)i i!



n+m n + m + 2i

(n+m)/2+1 

∂ n+m − L ∂t n + m + 2i

 i −1 

∂ −L ∂t



for i ≥ 1, L0 = 1, and Q p (r)u (x, t) = O(r2p+1 ) as r → 0. If u (x, t) ∈ C∞ (G) satisfies the condition lim Q p (r)u (x, t) = 0 ,

p→∞

then we can pass to the limit as p → ∞. Thus for the solutions to the equation Lu (x, t) satisfying (2.54), we have the following mean value relation: u (x, t) = (M r u )(x, t)

(2.54) ∂u ∂t

=

(2.55)

for all (x, t) ∈ G and all r, 0 < r < r(x, t). It is not difficult to show that this property is true for each solution of ∂u = Lu from the class C2,1 . ∂t Conversely, if a function u ∈ C4,2 satisfies (2.55), then we get from the expansion that 0 = r2 Lu (x, t) + O(r3 ) , or 0 = Lu (x, t)+ O(r). Then we get by r → 0: Lu (x, t) = 0. This implies the weak mean value theorem. A strong variant of this statement can be proved using the maximum principle according to the scheme given for the Laplace equation.

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Note that different expansions of the type (2.53) can be derived. For instance, choosing K (y, τ) =

1 2 |α |1/2 π n/2 2n+m Γ(m/2)r n+m +2m × τ (m

2

+ m )/2−1

[2(n + m2 + 2m)τ ln(r2 /τ ) − yT α −1 y]m/2−1

× [(m + 2)(2(n + m2 + 2m)τ ln(r2 /τ ) − yT α −1 y) + yT α −1 y]

and B r = {(y, τ) : yT α −1 y < 2(n + m2 + 2m)τ ln(r2 /τ)} , we come to an expansion of the type (2.53) with Li =

(−1)i [g(n, m)](n+m)/2+1 i!

where g(n, m) =



∂ − g(n, m)L ∂t

 i −1 

∂ −L ∂t



,

(2.56)

n + m2 + 2m . n + m2 + 2m + 2i

Note that both expansions with (2.53) and (2.56) result, when m → +0, in the expan­ sions with   n /2 + 1   i −1   (−1)i n ∂ n ∂ Li = × − −L . L i! n + 2i ∂t n + 2i ∂t This is the expansion of the average of the function u (x, t) over the surfaces (spheroids) {(y, τ ) : Z (x, t, y, τ ) = π −n/2 |α |−1/2 r−n } .

2.5 Wave equation Let us consider the simplest case, the one-dimensional wave equation for a function u (x, t), x ∈ R, t ≥ 0: u tt − c2 u xx = 0 ,

c is a constant,

u (x, 0) = f (x) ,

u t (x, 0) = g(x) .

(2.57)

Recall that the characteristics are the lines x ± ct = const. Taking these charac­ teristics as new coordinates ξ = x + ct ,

η = x − ct ,

we rewrite the wave equation in the form u ξη = 0 .

(2.58)

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30 | 2 Scalar second-order PDEs The general solutions of this equation read u = F (ξ ) + G(η ) , where F and G are arbitrary functions of their arguments; hence in the variables x and t we arrive at u (x, t) = F (x + ct) + G(x − ct) . (2.59) From this, we obtain F (x) =

cf (x) + g(x) , 2c

G (x) =

cf (x) − g(x) , 2c

(2.60)

which implies f (x) 1 + F (x) = 2 2c

x

g(z)dz ,

f (x) 1 G(x) = − 2 2c

0

x

g(z)dz .

(2.61)

0

Thus we conclude that the solution u (x, t) is given by the mean value relation f (x + ct) + f (x − ct) 1 u (x, t) = + 2 2c

x + ct

g(z)dz .

(2.62)

x − ct

This representation can be extended to the 3D case (see [73] and [33]). Let us con­ sider the wave equation for a function u (x, t), x ∈ R3 : u tt − c2 Δu = 0 , u (x, 0) = f (x) ,

u t (x, 0) = g(x) .

(2.63)

In [73], it is shown that the solution to (2.63) has the following integral representation: ⎛ ⎞   1 ∂ ⎜ 1 ⎟ g(y)dS y + f (y)dS y ⎠ . (2.64) u (x, t) = ⎝ 4πc2 t ∂t 4πc2 t | y − x |= ct

| y − x |= ct

Indeed, let us consider the spherical mean of the function u (x, t)  1 N u (x, t; r) = u (x + rξ, t)dS ξ . 4π | ξ |=1

As shown in Section 2.2.1, the function N u (x, t; r) solves the Darboux problem (2.26); hence,

∂2 ∂2 2 ∂ 2 Nu = c + Nu ∂t2 ∂r2 r ∂r (2.65) ∂ N u (x, 0; r) = N f (x, r) , N u (x, 0; r) = N g (x, r) , ∂t

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2.5 Wave equation |

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where N f and N g are spherical means of the functions f and g, respectively. This im­ plies that ∂2 ∂2 (rN u ) = c2 2 (rN u ) , (2.66) 2 ∂t ∂r which in turn means that rN u considered as a function of r and t solves the one-di­ mensional wave equation with the conditions rN u (x, 0; r) = rN f (x, r),

∂ (rN u (x, t; r))|t=0 = rN g (x, r) . ∂t

(2.67)

ˆ u (x, t; r) = rN u (x, t; r) we have by (2.65) Using the notation N ˆ tt − c2 N ˆ rr = 0 , N with the general solution described above for the one-dimensional case. Conse­ quently, from (2.66) and (2.67) we obtain N u (x, t; r) =

' 1 & (ct + r)N f (x, ct + r) − (ct − r)N f (x, ct − r) 2r ct+ r 1 zN g (x, z)dz . + 2cr

(2.68)

ct − r

Now, as r → 0, we arrive at u (x, t) = tN g (x, ct) +

 ∂  tN f (x, ct) ∂t

which proves (2.64).

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3 High-order elliptic equations 3.1 Balayage operator In this section, we give a general scheme of construction of the spherical mean value relations for high-order elliptic equations (see [157] and [158]). Let X be a locally compact Hausdorf space and let C0 (X ) be a space of continuous functions f : X → R with a compact support. We denote by R(X ) the space of Radon measures μ : C0 (X ) → R. ¯ ) we denote Let G ⊂ Rn be a bounded domain with the boundary ∂G = Γ. By C k (G α ¯ functions u (x) such that the derivatives D u exist in G for the space of continuous on G ¯ Let C∞ all |α | ≤ k and admit continuous continuation to G. 0 ( G ) be a space of infinitely differentiable functions with a compact support in G. Introduce a vector measure + , R(Rn ) , sup μ = sup μ α ⊂ G . μ = (μ α )| α|≤k ∈ | α |≤ m −1

| α |≤ k

Let us consider an arbitrary elliptic operator in Rn of order 2m with real-valued coef­ ficients a α ∈ C∞ (Rn ):  a α (x)D α (3.1) L(x, D) = | α |≤2m

n and m differential operators B j (x, D) with coefficients from C∞ 0 (R ) such that {the order of B j } = m j ≤ 2m − 1. Let us consider the boundary value problem

L(x, D)u = 0 , B j (x, D)u = g j ,

x∈G, j = 1, 2, . . . , m ,

(3.2)

x ∈ ∂G .

We will assume that functions g j , operators B j , and boundary Γ satisfy the condi­ tions which guarantee that the problem (3.2) has a unique solution (concerning these conditions, see, e.g. [189]). We also assume that the generalized maximum principle is satisfied: m  s        B j (x, D)u (x)Γ  l−m j |Q k (x, D)u (x)| ≤ c , (3.3) sup x ∈ G k =1

C

j =1

(Γ)

where Q k are differential operators of order ≤ l with smooth coefficients (they are re­ lated on Γ to the operators B j (x, D); see [189]). In particular, for the Dirichlet problem L(x, D)u = 0 , x ∈ G ,     D α u  = D α v , |α | ≤ m − 1 , Γ

it reads sup

Γ



x ∈ G | α |≤ m −1

|D α u (x)| ≤ c sup Γ

v ∈ C m−1 (Rn ) 

|D α u | .

| α |≤ m −1

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3.1 Balayage operator

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| 33

We denote by Π the balayage operator [189] for the problem (3.2): Π: μ → ν ,

¯, sup μ ⊂ G

sup ν ⊂ Γ .

Let us introduce a functional space ( ) ¯ ) = h = (D α u )| α|≤m−1 : u ∈ C m−1 (G ¯ ) ∩ C2m (G), L(x, D)u = 0 in G . H (G

The following balayage principle is true under the above assumptions and (3.3): for an arbitrary measure μ = (μ α )| α|≤m−1 , sup μ ⊆ G there exists a measure ν = (ν α )| α|≤m−1 , sup ν ⊆ Γ such that ¯) . for all h ∈ H (G

μ (h) = ν(h) ,

(3.4)

Let B(x0 , r) be a ball and let ΠB(x0 , r) be the balayage operator for the problem Lu = 0 in the ball B(x0 , r) ⊂ G where r is sufficiently small. Let σ rx0 be a measure uniformly distributed on the sphere S(x0 , r):  u (y)dS(y) , σ rx0 (u ) = c(r) S ( x 0 ,r )

where c(r) is the normalizing factor. Throughout this chapter, we shall derive mainly spherical mean value relations generated by the measure σ rx0 (u ). Let δ x be a unit mass concentrated at the point x and let δ x0 ,α0 be a vector measure whose components are equal to zero for α ≠ α 0 , while they are equal to δ x0 if α = α 0 . Now, let us take a sphere S(x0 , r) ⊂ G. Then, under the same assumptions as stip­ ulated above when formulating (3.4), there exists a vector measure ν j = (ν j β )| β|≤l−m j , sup ν j β ⊂ S(x0 , r) such that the solutions to the problem (3.2) satisfy the system Q k u(x) =

s 



jβ

Q j β u (x )dν x (x ) .

(3.5)

j =1 | β|≤ l − m j

Indeed from (3.4) it follows that s  

Q k (x, D)u (x)dμ k (x) =

m 

ν j (B j (x, D)u (x)) .

j =1

k =1 ¯ G

Choosing here the measures μ k equal to δ x0 ,k = ΠB(x0 , r)δ x0 ,k and the representation [189]   D β v(x)dν j β (x) , ν j (v) = | β|≤ l − m j Γ

we get (3.5). This is a general statement which ensures that under the formulated assumptions it is possible to derive spherical mean value relations of the types (3.5). Concrete rela­ tions will be derived in this chapter for a polyharmonic and metaharmonic equations.

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34 | 3 High-order elliptic equations

3.2 Biharmonic equation For the high-order PDEs it is possible to derive spherical mean value relations of differ­ ent type. For instance, in [139] (see also [124]) a series of spherical mean value relations for polyharmonic equation is derived which relates the spherical mean and the powers of the Laplacian at a fixed point. However, our objective is to construct a mean value relation that uniquely characterizes the relevant boundary value problem. We will see that a vector generalization of the converse spherical mean value theorem presented above for the harmonic functions can be formulated which provides a system of inte­ gral equations equivalent to the original boundary value problem under discussion. In this section we deal with the biharmonic equation ΔΔu (x) = 0 ,

x ∈ G,

(3.6)

which governs (in 2D) the bending of a thin elastic plate. We will consider two cases: (a) simply supported boundary of the plane    1 − σ ∂2 u     = g1 , u  = g0 , Δu − (3.7) Γ ρ ∂n Γ and (b) rigid fixing of the boundary   u  = g0 , Γ

 ∂u   = g1 . ∂n Γ

(3.8)

Here σ (=const) is Poisson’s ratio of the plate, n is the exterior normal vector to the boundary Γ, and ρ is the curvature of the boundary.

3.2.1 Direct spherical mean value relation In this section we treat the case (a) for the plates with ρ = ∞, so that the second boundary condition in (3.7) is Δu |Γ = g1 .   We seek the regular solution u ∈ C4 (G) C2 (G Γ) to the biharmonic equation Δ2 u ( x) = 0 ,

x ∈ G ⊂ Rm .

(3.9)

Let us recall (see Chapter 2) that, for any analytic function v defined in Rm , for any x and r such that S(x, r) ⊂ G, one has r2 Δv(x) N r v(x) = v(x) + 2m   ∞  r2k k Δ v(x) . + 2k k!m(m + 2) · · · (m + 2k − 2) k =2

(3.10)

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3.2 Biharmonic equation | 35

Using expansion (3.10) we get the following mean value relation: r2 N r (Δu )(x) , 2m Δu (x) = N r (Δu )(x) . u(x) = N r u(x) −

(3.11) (3.12)

Of course, it is possible to derive this mean value relation using the Gauss formula, as in the case of the Laplace equation (see Chapter 2). However, we give in Section 3.2.2 a different derivation which provides the spherical mean value relation for an arbitrary point inside the sphere. This actually means that for this problem, the generalized Poisson formula for a ball is obtained. Now we give the converse mean value relation that presents an integral formula­ tion of the boundary value problem (3.9), (3.7) (we formulate it for an arbitrary bihar­ monic function in G ⊂ Rm ). Proposition 3.1 (Integral formulation). Suppose the problem (3.9), (3.7) is uniquely  solvable. Assume that v1 , v2 belong to {C(G Γ); v1 |Γ = g0 , v2 |Γ = g1 } and for each ¯ for which the following mean value relation x ∈ G there exists a sphere S(x, r) ⊂ G holds: v1 = N r v1 −

r2 N r v2 , 2m

v2 = N r v2 . Then Δu = v2 and the function v1 is the unique solution to the problem (3.9), (3.7). Proof. Let us take the solution to (3.9), (3.7). It satisfies (3.11), (3.12); thus u − v1 satisfies r2 N r (Δu − v2 ) . 2m Using the results from the second section, we get that Δv2 (x) = 0 in G ; since Δu is harmonic and Δu |Γ = v2 |Γ = g1 , we have Δu ≡ v2 in G ; thus, u − v1 = N r (u − v1 ) −

u − v1 = N r (u − v1 ) , which implies that the function u − v1 is harmonic in G. Since u |Γ = v1 |Γ = g0 , we get u ≡ v1 in G.

3.2.2 Generalized Poisson formula Let us consider a 2D domain and let S(x0 , r) = ∂K (x0 , r) be a circle of radius r centered at x0 . We use the notation dS = rdω2 = ω2 rdσ rx0 ,

dx = tdtdω2 ,

ω2 = 2π .

Let α be the angle between the vecxtors x − x0 and y − x0 , where y is the vector lying on the sphere and let t = |x − x0 |.

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36 | 3 High-order elliptic equations Theorem 3.1. The regular solution to the biharmonic equation satisfies the following spherical mean value relation:  u (y)dS y r2 − t2 u(x) = rω2 |x − y|2 S ( x 0 ,r )     r2 − t2 r cos2 α |x − x0 | sin α + (3.13) arctg rω2 |x − x0 | sin α r − |x − x0 | cos α S ( x 0 ,r )  1 r cos α |x − y|2 − − Δu (y)dS y , ln 2 2t r2  Δu (y)dS y r2 − t2 . (3.14) Δu (x) = rω2 |x − y|2 S ( x 0 ,r )

Proof. The biharmonic equation has the following general representation (e.g. see [200]): (3.15) u (x) = (|x − x0 |2 − r2 )v(x) + w(x) , where v(x), w(x) are harmonic functions. For the boundary points we have w ( y ) = u ( y ) := g 0 ( y ) ,

y ∈ S ( x 0 , r) .

The Poisson formula reads [33] w( x ) =

r2 − t2 rω2

 S ( x 0 ,r )

g0 (y) dS y , ρ2

(3.16)

where ρ = |x − y|. It remains to find the function v. From (3.15) we get ) ( Δu (x) = Δ (|x − x0 |2 − r2 )v(x) + Δw(x)    ∂v(x) ∂  |x − x0 |2 − r2 = v(x)Δ |x − x0 |2 − r2 + 2 ∂x1 ∂x1  ∂v ∂  2 2 |x − x0 | − r +2 ∂x2 ∂x2 + (|x − x0 |2 − r2 )Δv(x) + Δw(x) .

Let x = (x1 , x2 ), x0 = (x01 , x02 ). It is easy to show that ∂v(x) ∂v(x) + 4(x2 − x02 ) Δu (x) = 4v(x) + 4(x1 − x01 ) ∂x1 ∂x2 $ % x1 − x01 ∂v(x) x2 − x02 ∂v(x) = 4v(x) + 4r + . r ∂x1 r ∂x2

(3.17)

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3.2 Biharmonic equation |

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For x on the surface of the sphere x1 − x01 ∂v(x) x2 − x02 ∂v(x) ∂2 v(x) = + , ∂n r ∂x1 r ∂x2 where n is the normal vector exterior to the surface S(x, r). Hence we get from (3.17) for the boundary points: g1 (y) := Δu (y) = 4v(y) + 4r

∂v(y) . ∂n

Thus we conclude that the function v(x) is the unique solution of the following bound­ ary value problem: Δv(x) = 0 ,

x ∈ K ( x 0 , r) ,

1 g1 (y) ∂v + v(y) = , ∂n r 4r

y ∈ S ( x 0 , r) .

(3.18) (3.19)

To solve the problem (3.18), (3.19) we use the polar coordinates x = ρ exp(iφ). Then we assume that there is an expansion v(x) = v(ρeiφ ) =

∞  ρ n (A n cos(nφ) + B n sin(nφ)) r + A0 · , n −1 ( n + 1) r 2 n =1

where A n , B n are the Fourier coefficients of the expansion of the function −g1 (ρeiφ )/4r: 1 An = − 4πr

2π

g1 (reiθ ) cos(nθ) dθ ,

n = 0, 1, . . . ,

g1 (reiθ ) sin(nθ) dθ ,

n = 0, 1, . . . .

0

1 Bn = − 4πr

2π 0

Using 1 A n cos(nφ) + B n sin(nφ) = − 4πr

2π

g1 (reiθ ) cos[n(θ − φ)] dθ , 0

A0 = −

1 4πr

2π

g1 (reiθ ) dθ , 0

we get v(ρeiφ ) = −

1 4πr

2π

g1 (reiθ ) 0

Let a = ρ /r and let R(a, θ) =

2π  ρ n cos[n(θ − φ)] dθ . r n (n + 1) n =1

∞  a n cos(nθ) . n+1 n =1

(3.20)

(3.21)

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38 | 3 High-order elliptic equations The function

∞

R1 (t, θ) =

1  n + t cos(nθ) 2 n =1

is uniformly convergent for t2 < 1 and R1 (t, θ) =

1 1 − t2 . 2 1 − 2t cos θ + t2

The uniform convergence for t < 1 implies that it is possible to carry out the termwise integration that gives a 1 1 (3.22) R1 (t, θ)dt − . R(a, θ) = a 2 0

Let us evaluate the integral I=

1 a

a

R1 (t, θ)dt =

1 2a

0

a 0

1 − t2 dt . 1 − 2t cos θ + t2

(3.23)

Using the known integrals [46] 1 2a

a 0

. a − cos θ dt 1 = arctan − arctan (− cot θ ) , 1 − 2t cos θ + t2 2a sin θ sin θ

and 1 2a

a 0

t2 dt 1 cos θ = + ln |1 − 2a cos θ + a2 | 1 − 2t cos θ + t2 2 2a . a − cos θ cos2 θ − 1/2 arctan − arctan(− cot θ) − a sin θ sin θ

we find   cos2 θ a − cos θ I= arctan − arctan(− cot θ) a sin θ sin θ 1 cos θ − − ln |1 − 2a cos θ + a2 |. 2 2a

Let α = arctan

a − cos θ , sin θ

β = arctan(− cot θ) ,

then

a − cos θ , sin θ Simple transformations yield tan α =

tan β = − cot θ .

α − β = arctan {a sin θ/(1 − a cos θ)} .

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3.2 Biharmonic equation | 39

Hence, I=

cos2 θ 1 cos θ a sin θ − − arctan ln |1 − 2a cos θ + a2 | . a sin θ 1 − a cos θ 2 2a

We rewrite (3.20), using (3.21)–(3.23), 

2π

a sin(θ − φ) cos2 (θ − φ) arctan a sin(θ − φ) 1 − a cos(θ − φ) 0  1 cos(θ − φ) − − ln |1 − 2a cos(θ − φ) + a2 | dθ . 2 2a

1 v(ρe ) = − 4π iφ

iθ

g1 (ρe )

We recall that a = ρ /r, hence 

2π

ρ sin(θ − φ) r cos2 (θ − φ) arctan ρ sin(θ − φ) r − ρ cos(θ − φ) 0  1 r cos(θ − φ) r2 − 2rρ cos(θ − φ) + ρ 2 ln − − dθ . 2 2ρ r2

1 v(ρe ) = − 4π iφ

iθ

g1 (ρe )

From (3.15), (3.16), and from the equalities ρ = |x − x0 | ,

|x − y|2 = r2 + ρ 2 − 2rρ cos(θ − φ) ,

we get the desired relation (3.13). The theorem is proved. Corollary 3.1. At the central point of the circle K (x0 , r), the biharmonic function satis­ fies the spherical mean value relation   1 r u (y)dS y − Δu (y)dS y , (3.24) u(x0 ) = rω2 4ω2 S ( x 0 ,r ) S ( x 0 ,r )  1 Δu (y)dS y . Δu (x0 ) = rω2 S ( x 0 ,r )

Note that this relation was obtained above (see formula (3.12)) through series expan­ sion.

3.2.3 Rigid fixing of the boundary The generalized Poisson formula can also be obtained for biharmonic functions with rigid fixing on the boundary, i.e. for the solutions to the boundary value problem (3.8), (3.9).

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40 | 3 High-order elliptic equations Theorem 3.2. The biharmonic function satisfies at arbitrary point x ∈ K (x0 , r) ⊂ G the following spherical mean value relation:  g1 (y)dS y (r2 − |x − x0 |2 )2 u(x) = − (3.25) 2r2 ω2 |x − y|2 S ( x 0 ,r )

( r − |x − x0 | ) + 2r3 ω2 2

2 2



S ( x 0 ,r )

(2r2 − 2r|x − x0 | cos α )g0 (y)dS y , |x − y|4

where α is the angle between the vectors x − x0 and y − x0 . Proof. Again, we use representation (3.15). First, from the boundary condition we have w( y) = u ( y) = g0 ( y) ,

y ∈ S ( x 0 , r) .

The Poisson formula yields w( x ) =

r2 − t2 rω2



g0 (y)dS y , ρ2

(3.26)

where ρ = |x − y|, t = |x − x0 |. From (3.15) it follows that ∂u ∂v ∂w = 2tv + (t2 − r2 ) + , ∂t ∂t ∂t and ∂/∂t = −∂/∂n as y → y ∈ S(x0 , r). Hence, ∂w −2rv(y) + ( y) = g1 ( y) , y ∈ S ( x 0 , r) . ∂n Let t ∂w h(x) = v(x) + 2 . 2r ∂t From (3.27), it follows that for y ∈ S(x0 , r) h(y) = v(y) −

(3.27)

1 ∂w g1 (y) =− , 2r ∂n 2r

and hence for the function h(x) we can write  r2 − t2 h(x) = − 2 2r ω2

S ( x 0 ,r )

g1 (y)dS y . ρ2

On the other hand, ∂w 2t =− ∂t rω2

 S ( x 0 ,r )

g0 (y)dS y 2(r2 − t2 ) − 2 ρ rω2

 S ( x 0 ,r )

(t − r cos α )g0 (y)dS y ρ4

which proves the theorem.

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3.2 Biharmonic equation |

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Corollary 3.2. At the center of S(x0 , r), the mean value relation (3.25) takes the form   1 1 u(x0 ) = − g1 (y)dS y + g0 (y)dS y . 2ω2 rω2 S ( x 0 ,r )

S ( x 0 ,r )

Note that the mean value relation (3.26) can be differentiated with respect to x. Let x = (x1 , x2 ), x0 = (x01 , x02 ), y = (y1 , y2 ). Then   g1 (y)dS y g1 (y)dS y 4 2 2(r2 − t2 )2 Δu (x) = (r − 2t2 ) − rω2 ρ2 r2 ω2 ρ4 S ( x 0 ,r ) S ( x 0 ,r )  + k (x1 , x01 ) A(x1 , x01 , y1 )g1 (y)dS y S ( x 0 ,r )



+ k (x2 , x02 )

A(x1 , x02 , y2 )g1 (y)dS y S ( x 0 ,r )

8 − 2 (r2 − 2t2 ) r ω2 ( r 2 − t 2 )2 + r3 ω2

k (x1 , x01 ) − r −

k (x2 , x02 ) r



 S ( x 0 ,r )

r − t cos α g0 (y)dS y r4

B(y)g0 (y)dS y S ( x 0 ,r )



A(x1 , x01 , y1 ) g0 (y)dS y S ( x 0 ,r )



A(x2 , x01 , y2 ) g0 (y)dS y , S ( x 0 ,r )

where t = |x − x0 |, ρ = |x − y|, k ( z1 , z2 ) =

2 (r2 − t2 )(z1 − z2 ), rω2

A(p, q, z) = (2(q − p) r2 − 8(r2 − rt cos α )(p − z))/ρ 6 , and B(y) = [16r(r − t cos α ) − 8r2 + 5((x1 − y1 )(x1 − x01 ) + (x2 − y2 )(x2 − x02 ))]/ρ 6 .

Corollary 3.3. At the central point, 2 Δu (x0 ) = 2 r ω2



g1 (y)dS y . S ( x 0 ,r )

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42 | 3 High-order elliptic equations Let us now consider the boundary value problem x ∈ K ( x 0 , r) ⊂ R2 ,

ΔΔu (x) = 0 ,

u (y) = g0 (y) ,

(3.28)

1 − ν ∂u (y) Δu (y) + = g1 (y) , ρ ∂n

y ∈ S ( x 0 , r) .

(3.29)

From the above two theorems the following mean value relation for the problem (3.28), (3.29) is true:     1 r αr −1 u(x0 ) = g0 (y)dS y − 1+ g1 (y)dS y , rω2 4ω2 2 S ( x 0 ,r )

   1 αr −1 Δu (x0 ) = 1+ rω2 2

S ( x 0 ,r )

g1 (y)dS y ,

S ( x 0 ,r )

where α = (1 − ν)/ρ.

3.2.4 Nonhomogeneous biharmonic equation Let us consider a nonhomogeneous biharmonic equation Δ2 u = g . For this equation, we now derive mean value relations in the circle K (x, r). To do this we use the equality [160] Nr u =

m  2j j  r Δ u(x) + B rx (v m Δ m+1 u ) , 2 2 ( j! ) j =0

(3.30)

where B rx (u ) is the averaging operator of u (x) over the circle K (x, r), i.e.  udσ rx . B rx (u ) = K ( x,r )

The functions v m are determined from the recurrence formulae v0 (ρ ) =

1 r ln , 2π ρ

r

v j +1 =

ρv j (ρ ) ln 0

ρ dρ . r

Hence using Δ2 u = g we obtain from (3.30) for m = 0   g r . Δu (x) = N r (Δu ) + B rx − ln 2π r1

(3.31)

(3.32)

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3.2 Biharmonic equation | 43

Integrating (3.31) by parts we find r

v1 =

ρv0 (ρ ) ln r1

=

1 2π



ρ dρ ρ1

r2 + r21 r2 − r21 r ln − 4 r1 4

.

From this we get, using (3.30) for m = 1, the identity r2 u (x) = N r (u ) − N r (Δu ) 4  r 1 r2 − r21 r2 + B rx − 1 ln g . 2π 4 4 r1

(3.33)

Hence, formulae (3.32) and (3.33) present a generalized mean value relation in the case when the right-hand side is not equal to zero. For plates with rigid fixing of the boundary, an analogous mean value relation can be derived. Theorem 3.3. An arbitrary regular solution to the equation Δu = g satisfies the mean value relation   r ∂u + B rx (q1 g) , (3.34) u(x) = N r (u) − N r 2 ∂r   2 ∂u + B rx (q2 g) , (3.35) Δu (x) = N r r ∂r where

  r2 + r21 r21 r − ln , 8 4 r1  

1 r 1 r2 − + 12 , ln q2 ( r1 ) = − 2π r1 2 2r

1 q1 ( r1 ) = 2π

and the integration B rx of the functions q1 (r1 )g(r1 ) and q2 (r1 )g(r1 ) is carried out over the circle K (x, r). Proof. By the second Green formula 

B rx (Δu ) = 2πr · N r

∂u ∂r



.

Substituting this expression into the Poisson formula for Δu (x), $ % 1 1 r r2 − r21 Δu (x) = 2 B rx (Δu ) − r2 ln(r/r1 ) − g , Bx πr 2π 2

(3.36)

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44 | 3 High-order elliptic equations we obtain (3.35). Now substituting (3.36) into (3.30) for m = 1 we obtain by (3.35) r2 Δu (x) − B rx (v1 g) 4 $ %   r ∂u 1 r r2 − r21 2 + r ln(r/r1 ) − g B = N r (u) − N r 2 ∂r 8π x 2

u(x) = N r (u) −

− B rx (v1 g)

r = N r (u) − N r 2



 ∂u + B rx (q1 g) . ∂r

This proves (3.34). Remark 3.1. Note that from the two mean value relations obtained above we can de­ duce a mean value relation for the case of simply supported boundary of the plate. Indeed, combining these relations we find that an arbitrary regular solution to Δ2 u = g satisfies the mean value relation   ∂u r2 2 + u(x) = N r (u) − N r Δu − α B r ( q g) , 4 − 2αr ∂r 2 − αr x 1 and

  ∂u 2 2 + Δu (x) = N r Δu − α B r ( q g) , 2 − αr ∂r 2 − αr x 2 where α is an arbitrary constant and

1 r2 − r21 r2 αr q1 = − ln(r/r1 ) − q1 , 2π 4 4 2 1 αr q 2 = − ln(r/r1 ) q2 . 2π 2

3.3 Fourth-order equation governing the bending of a plate Let us consider the following fourth-order equation with constant coefficients: ΔΔu (x) + a1 Δu (x) + a2 u (x) = 0 ,

x ∈ K ( x 0 , r) ,

(3.37)

y ∈ S ( x 0 , r) .

(3.38)

with the boundary conditions u (y) = g0 (y) ,

∂u (y) = g1 (y) , ∂n

In (3.37) a1 , a2 are constants; hence it can be rewritten in the form (Δ − k 1 )(Δ − k 2 )u (x) = 0 .

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3.3 Fourth-order equation governing the bending of a plate | 45

In what follows it is assumed that k 1 ≠ k 2 and Rek i > λ0 , i = 1, 2, where λ0 is the prin­ cipal eigenvalue of the Laplace operator. The case k 1 = k 2 is treated in Section 3.4.1. In [62], the spherical mean value relation was obtained for the boundary condi­ tions u (y) = g0 (y) , Δu (y) = g1 (y) . (3.39) Theorem 3.4. The solution to (3.37) satisfies the following spherical mean value rela­ tion: u(x0 ) =

c11 (r) 2π

2π

u (x0 + reiθ )dθ +

2π

c12 (r) 2π

Δu (x0 + reiθ )dθ ,

0

c21 (r) Δu (x0 ) = 2π

0

2π

c22 (r) u (x0 + re )dθ + 2π

2π

iθ

0

Δu (x0 + reiθ )dθ, 0

where c11 (r) = d11 (I0 )/δ ,

c12 (r) = d12 (I0 )/δ ,

c21 (r) = d21 (I0 )/δ ,

c22 (r) = d22 (I0 )/δ ,

and δ = (k 21 − k 22 )I0 (k 1 r)I0 (k 2 r) , d11 (I n ) = k 21 I n (k 1 r) − k 22 I n (k 2 r) , d12 (I n ) = −(I n (k 1 r) − I n (k 2 r)) , d21 (I n ) = k 21 k 22 (I n (k 1 r) − k 21 I n (k 2 r)) , d22 (I n ) = −(k 22 I n (k 1 r) − k 21 I n (k 2 r)) , and I n is the modified Bessel function (I n (z) = J n (iz)i−n ), where J z is the standard Bessel function (see the power expansion (2.6) of this function in Section 2.1.1). Theorem 3.5. The solution (in K (x0 , r) ⊂ R2 ) to (3.37) satisfies the mean value relation   b 11 (r) b 12 (r) ∂u u(x0 ) = u (y)dS y + (y)dS y , (3.40) rω2 rω2 ∂n S ( x 0 ,r ) S ( x 0 ,r )   b 21 (r) b 22 (r) ∂u u (y)dS y + (y)dS y , (3.41) Δu (x0 ) = rω2 rω2 ∂n S ( x 0 ,r )

S ( x 0 ,r )

where b 11 (r) = (d11 (I0 ) − d11 (I2 ))/δ1 ,

b 12 (r) = d12 (I0 )/δ1 ,

b 21 (r) = (d21 (I0 ) − d21 (I2 ))/δ1 ,

b 22 (r) = d22 (I0 )/δ1 ,

δ1 = δ + S ,

(3.42)

S = k 21 I0 (k 2 r)I2 (k 1 r) − k 22 I0 (k 1 r)I2 (k 2 r) .

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46 | 3 High-order elliptic equations Proof. The general solution to (3.37) at a point x with polar coordinates (r, θ) can be written as u (r, θ) = A0 I0 (k 1 r) + B0 I0 (k 2 r) +

∞ 

(A n I n (k 1 r) + B n I n (k 2 r)) sin nθ

n =1

+

∞ 

(3.43)

(C n I n (k 1 r) + D n I n (k 2 r)) cos nθ ,

n =1

where A n , B n , C n , D n (n ≥ 0) are the arbitrary constants. We differentiate this repre­ sentation with respect to r (which corresponds to the differentiation along the radius of the circle) and use the known formulae 2I n = I n−1 + I n+1 ,

I0 = I1 .

This leads to ∂u (r, θ) = A0 k 1 I1 (k 1 r) + B0 k 2 I1 (k 2 r) ∂r ∞  k1 A n (I n−1(k 1 r) + I n+1(k 1 r)) + 2 n =1 . k2 +B n (I n−1 (k 2 r) + I n+1 (k 2 r)) sin nθ 2 ∞  k1 + C n (I n−1 (k 1 r) + I n+1 (k 1 r)) 2 n =1 . k2 +D n (I n−1 (k 2 r) + I n+1 (k 2 r)) cos nθ . 2

(3.44)

Integrating (3.43) and (3.44) over (0 ≤ θ ≤ 2π) yields 1 2π

2π

u (r, θ)dθ = A0 I0 (k 1 r) + B0 I0 (k 2 r) ,

(3.45)

0

1 2π

2π 0

∂u (r, θ)dθ = A0 k 1 I1 (k 1 r) + B0 k 2 I1 (k 2 r) . ∂n

We multiply the last equality by 2/r and use the known relation 2I n = x(I n−1 − I n+1) for n = 1 (e.g. see [46]). This gives Δu (x0 ) = A0 k 21 (I0 (k 1 r) − I2 (k 1 r)) + B0 k 22 (I0 (k 2 r) − I2 (k 2 r)) .

(3.46)

We also used here the relation Δu (x0 ) =

1 2 2π r

2π 0

∂u (r, θ)dθ . ∂n

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Let us consider relations (3.45) and (3.46) at the point x0 , the center of the circle. Note that I0 (0) = 1 and I2 (0) = 0. Therefore, u(x0 ) = A0 + B0 , Δu (x0 ) = k 21 A0 + k 22 B0 . Let 1 S1 = 2π

2π

u (r, θ)dθ , 0

1 2 S2 = 2π r

2π 0

∂u (r, θ)dθ . ∂r

We resolve (3.45) and (3.46) with respect to the unknown constant A0

S1 I 0 ( k 2 r) 1 A0 = det S2 k 22 (I0 (k 2 r) − I2 (k 2 r)) δ1 = (S1 k 22 (I0 (k 2 r) − I2 (k 2 r)) − S2 I0 (k 2 r))/δ1 ,

and B0

I 0 ( k 1 r) 1 B0 = det 2 k 1 (I0 (k 1 r) − I2 (k 1 r)) δ1

S1 S2

= (S2 I0 (k 1 r) − k 21 S1 (I0 (k 1 r) − I2 (k 1 r)))/δ1 ,

where



I 0 ( k 1 r) δ1 = det 2 k 1 (I0 (k 1 r) − I2 (k 1 r))

I 0 ( k 2 r) k 22 (I0 (k 2 r) − I2 (k 2 r))

= δ − k 22 I0 (k 1 r)I2 (k 2 r) + I0 (k 2 r)I2 (k 1 r) = δ + S .

Corollary 3.4. The solution to (3.37) satisfies the mean value relation (in the circle K (x0 , r) ⊂ R2 )   ∂u e11 (r) e12 (r) u (y)dS y + (Δu (y) + α (y))dS y , u(x0 ) = rω2 rω2 ∂n S ( x 0 ,r ) S ( x 0 ,r )   ∂u e21 (r) e22 (r) u (y)dS y + (Δu (y) + α (y))dS y , Δu (x0 ) = rω2 rω2 ∂n S ( x 0 ,r )

S ( x 0 ,r )

where αr αr )d11 (I0 ) + d11 (I2 ))/δ2 , 2 2 e12 (r) = d12 (I0 )/δ2 , αr αr e21 (r) = ((1 + d21 )(I0 ) + d21 (I2 ))/δ2 , 2 2 e22 (r) = d22 (I0 )/δ2 , αr αr δ 2 = (1 + ) δ + S , 2 2 and the coefficients δ, s, d ij are defined in (3.42), α is an arbitrary constant. e11 (r) = ((1 +

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48 | 3 High-order elliptic equations

3.4 Metaharmonic equations We introduce the metaharmonic operator in Rm , L ( Δ ) = ( Δ − λ n ) p n ( Δ − λ n − 1 ) p n −1 · · · ( Δ − λ 1 ) p 1 , where p1 ≤ p2 ≤ · · · ≤ p n are positive integers, λ1 , λ2 , . . . , λ n are real or complex numbers; it is assumed that Reλ i , the real parts of the constants λ i , i = 1, 2, . . . , n are larger than the principal eigenvalue of the Laplace operator. The integer number ν = p1 + p2 + · · · + p n is the order of the metaharmonic operator.

3.4.1 Polyharmonic equation We first derive a spherical mean value relation for the polyharmonic equation Δ p v(x) = 0 in Rm . To this end we apply representation (3.10). Since Δ p u = 0, this representation reads p −1   m  r 2i Δ i v(x0 ) + N r v(x) , v( x0 ) = −Γ( ) 2 i =0 2 i!Γ(i + m/2) where x0 is the center of the sphere S(x0 , r) in Rm and N r is the spherical mean over this sphere. Successive use of this representation yields the following triangular sys­ tem: Δ p −1 v ( x 0 ) = N r Δ p −1 v , Δ p −2 v ( x 0 ) = − c 1 Δ p −1 v ( x 0 ) + N r Δ p −2 v , .. . etc., the last equation coinciding with (3.10): v(x0 ) = −c1 Δv(x0 ) − c2 Δ2 v(x0 ) − · · · − c p−1 Δ p−1 v(x0 ) + N r v . Here ci =

Γ(m/2)(r/2)2i . i!Γ(i + m/2)

We rewrite this triangular system in the form p −1

Δ j v(x0 ) = −



Δ i v(x0 ) + N r Δ j v ,

j = 0, 1, . . . , p − 1 ,

i = j +1

where Δ0 v ≡ v.

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3.4 Metaharmonic equations | 49

Substituting in this system the first equation into the second one, then the second into the third one, etc., we obtain Δ p −1 v ( x 0 ) = N r Δ p −1 v , Δ p −2 v ( x 0 ) = N r Δ p −2 v − s 1 N r Δ p −1 v , .. . etc., with the last equation v(x0 ) = N r v − s1 N r Δv − · · · − s p−1 N r Δ p−1 v . Here the coefficients s i are obtained by the recurrent formula s1 = c 1 ,

si = ci −

i −1 

c j s i−j ,

i > 1.

j =1

Thus we obtained the desired spherical mean value relation Δ j v(x0 ) = N r Δ j v −

p −1

si Nr Δi v ,

j = 0, 1, . . . , p − 1

(3.47)

i = j +1

which can be written in a matrix form v = N r [ K ( r )v ] , where v = (v, Δv, . . . , Δ p−1 v) and K (r) is the triangular matrix ⎛

1 ⎜ ⎜ 0 ⎜ K ( r) = ⎜ ⎜· · · ⎜ ⎝· · · ···

−s1 1 ··· ··· ···

−s2 −s1 ··· ··· ···

··· ··· ··· 1 0

⎞ − s p −1 ⎟ − s p −2 ⎟ ⎟ ··· ⎟ ⎟. ⎟ −s1 ⎠ 1

Let us consider the following boundary value problem for the polyharmonic equa­ tion in a bounded domain G: Δ p v(x) = 0,

x ∈ G,

v| Γ ( y ) = g ( y ) ,

¯) , v ∈ C2p (G) ∩ C2p−2 (G

(3.48)

y ∈ Γ,

where v = (v, Δv, . . . , Δ p−1 v), g = (g1 , . . . , g p ) = (v, Δv, . . . , Δ p−1 v)|Γ . Theorem 3.6. Assume that the boundary value problem (3.48) is uniquely solvable for ¯ ) satisfies the each continuous g. Then if a function v from the class C2p (G) ∩ C2p−2 (G spherical mean value relation (3.47) at all points x ∈ G for at least one sphere S(x, r) ⊂ G then u is the unique solution to (3.48).

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50 | 3 High-order elliptic equations The proof repeats the arguments given in Theorem 2.5 and Proposition 3.1. Remark 3.2. As mentioned in Chapter 1, it is possible to derive different spherical mean value relations for one and the same equation. For instance, the polyharmonic functions (Δ p v = 0) also satisfy the following spherical mean value relation at an arbitrary point x ∈ B(x0 , r) [189]: $

%  p  p−1 ∂ p− j r m −2 v(x) = P(x, r) g j (y) do(y) , ∂ ( r2 ) p− j |x − y |m j−1 j =1 S ( x 0 ,r )

where P(x, r) =

(−1)p−1 (r2 − |x − x0 |2 )p (p − 1)r m−1 ω m

and

∂ j −1 v    , j = 1, . . . , p . ∂n j−1 Γ It is however not clear if this relation characterizes the relevant boundary value prob­ lem for the polyharmonic equation. The same is true for another mean value relation for the polyharmonic equation obtained in [124] gj =

p −1

v(x) =



c(m, p, q)κ rq,x (v) ,

q =0

where the means κ rq,x(v) are obtained recursively starting from the uniform measure σ rx (v) over the sphere S(x, r): κ rq,x(v) =

m rm

r

t m−1 κ rq−1,x(v) dt ,

q = 1, 2 . . . ,

0

and the constants c(m, p, q) are known (e.g. see [124]).

3.4.2 General case Let us now consider the general case of the metaharmonic equation. We start with the equation (Δ − λ)p u = 0, such that the constant λ (real or complex) satisfies the condition: Reλ > λ0 where λ0 is the principal eigenvalue of the Laplace equation. Next we show that the general case of the metaharmonic equation is derived from this particular case. Theorem 3.7. The solution of the equation ( Δ − λ) p u = 0

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51

satisfies the mean value relation v(x0 ) = N r (Av) .

(3.49)

Here v = (u, (Δ − λ)u, . . . , (Δ − λ)p−1 u )T , 1 A = ( I + β 1 H + · · · + β p −1 H p −1 ) w − m ( r, λ) ,

where the entries β 1 , . . . , β p−1 are determined from the recursive relation β 1 = −α 1 ,

β 2 = −α 2 − β 1 α 1 ,

... ,

β k = −α k − α k−1 β 1 − · · · − α 1 β k−1 , (3.50)

I is a p × p identity matrix, H is a matrix whose nonzero entries have the form h i,i+1 = 1. In (3.50) α i = 𝛾i /w m (r, λ) , i = 1, . . . , p − 1 , where 𝛾k =

1 ∂k wm r2k w m+2k (r, λ) ( r, λ ) = . k k k! ∂λ k! 2 m(m + 2) · · · (m + 2k − 2)

Proof. Expanding in (2.8) the function W m (irλ1/2 ) in a power series in (Δ − λ), we obtain, using the relation (Δ − λ)p u = 0, (e.g. see [139]) N r (u) =

p 

∂ k−1 w m (r, λ) 1 ( Δ − λ ) k −1 u ( x 0 ) . k −1 ( k − 1 ) ! ∂λ k =1

(3.51)

Using relations similar to (3.51) for the functions (Δ − λ)u, (Δ − λ)2 u, . . . and (Δ − λ)p−1 u, we obtain a system of equations which can be written in a matrix form N xr 0 (v) = Rv(x0 ) ,

(3.52)

p

where R = {r ij }i,j=1 is a matrix with diagonal entries 𝛾0 = r ii = w m (r, λ) ,

i = 1, 2, . . . , p

and nonzero entries 1 ∂w m 1 r2 (r, λ) = w m+2 (r, λ) , 1! ∂λ 1! 2m 1 ∂2 w m 1 r4 w m+4 (r, λ) 𝛾2 = r i,i+2 = (r, λ) = , 2 2! ∂λ 2! 22 m(m + 2) .. . 𝛾1 = r i,i+1 =

etc., 𝛾k = r i,i+k =

1 ∂k wm 1 r2k w m+2k (r, λ) ( r, λ ) = , k! ∂λ k k! 2k m(m + 2) · · · (m + 2k − 2)

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52 | 3 High-order elliptic equations where i, k = 1, 2, . . . , p − 1. Consequently, using the specific structure of R, this matrix can be written in the form R = ( I + α 1 H + α 2 H 2 + · · · + α p −1 H p −1 ) w m , 𝛾 α i = i , i = 1, . . . , p − 1 , wm where the matrix H is defined in the theorem. It is clear that the matrix A = R−1 has the same structure, i.e. 1 R −1 = ( I + β 1 H + β 2 H 2 + · · · + β p −1 H p −1 ) w − m .

From R−1 R = I we obtain (using the condition H k = 0 for k ≥ p) the recurrent rela­ tions (3.50) for the entries β 1 , . . . , β p−1 . Hence, the matrix A = R−1 is easily constructed explicitly and we have from (3.52) v(x0 ) = N r (Av) which coincides with (3.49). Consider the following boundary value problem for the metaharmonic equation in a domain G ⊂ Rm : Lu (x) = 0 ,

x ∈ G,

u |∂G = g1 ,

(3.53)

Δu |Γ = g2 , . . . , Δ

ν −1

u |Γ = g ν ,

(3.54)

where ν is the order of the metaharmonic operator L. Here and in the following dis­ cussion v|Γ = g means that v( x ) → g ,

as

x → ξ ∈ Γ,

(x ∈ G) .

It is not difficult to pass from (3.53) and (3.54) to the equation Lu (x) = 0 ,

x ∈ G,

with the boundary conditions u |Γ = ψ1 ,

( Δ − λ 1 ) u | Γ = ψ 2 , . . . , ( Δ − λ 1 ) p 1 −1 u | Γ = ψ p 1 , . . . ,

( Δ − λ n ) p n − 1 ( Δ − λ n − 1 ) p n −1 · · · ( Δ − λ 1 ) p 1 u | Γ = ψ ν ,

where ψ i are linearly dependent on g1 , g2 , . . . , g ν . We will seek the solution to (3.53) and (3.54) in the form u = u1 + u2 + · · · + u n

(3.55)

where u i satisfies the equation (Δ − λ i )pi u = 0 ,

i = 1, 2, . . . , n .

(3.56)

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From (3.53) and (3.54) it is obvious that Lu = 0. To obtain a unique representation of the solution u (x) in the form of (3.55), it is necessary to derive the appropriate bound­ ary conditions for (3.56). We introduce the notation (l)

( Δ − λ i ) l −1 u i = v i , (l)

v i |Γ = ψ il ,

i = 1, 2, . . . , n,

l = 1, 2, . . . , p i

and consider vectors of dimension ν:   ( p ) (1) (p ) (p ) (1) (2) (2) V = v1 , v1 , . . . , v1 1 , v2 , v2 , . . . , v2 2 , . . . , v(n1) , v(n2) , . . . , v n n ,   ψ = ψ11 , ψ12 , . . . , ψ1p1 , ψ21 , ψ22 , . . . , ψ2p2 , . . . , ψ n1 , ψ n2 , . . . , ψ np n , Ψ = (Ψ1 , . . . , Ψ ν ) . Let μ 1 = λ1 ,

μ 2 = λ1 ,

μ p 1 +1 = λ 2 ,

...,

μ p1 = λ1 ;

...,

μ p1 + p2 = λ2 ;

μ p1 +p2 +...,+p n−1 +1 = λ n ,

μν = λn

...,

and define a vector U = (U1 , . . . , U ν ) with U1 = u and Uj =

j −1 /

(Δ − μ i )u ,

j = 2, . . . , ν .

i =1

We derive a linear relation between the vector V and U in the form BV = U ,

(3.57)

i.e. the problem is to find the matrix B. To do this, we apply to (3.56) the operator 0k i =1 ( Δ − μ i ) for k = 0, 1, . . . , ν − 1 (k = 0 corresponds to the identity operator). Hence, (1) (1) U1 = v1 + v2 + · · · + v(n1) (k = 0) and from (3.56), ( l ) (1)

(l ) (p1 )

( l ) (2)

U l = b 11 v1 + b 12 v1 + · · · + b 1p1 v1 (l)

(l)

( l ) (1)

( l ) (2)

(l ) (p2 )

+ b 21 v2 + b 22 v2 + · · · + b 2p2 v2 (p n )

l) + · · · + b n1 v(n1) + b n2 v(n2) + · · · + b (np v n n

,

l = 1, 2, . . . , ν .

The lth row of the matrix B has the form   (l) (l) (l) (l) (l) (l) (l) (l) b 11 , b 12 , . . . , b 1p1 , b 21 b 22 , . . . , b 2p2 , . . . , b n1 , b n2 , . . . , b (npl)n . Applying the operator (Δ − μ i )[(Δ − λ i ) + (λ i − μ i )]

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54 | 3 High-order elliptic equations in (3.57) to the ith row we obtain the following recurrence formula: ( l +1 )

b il

(l)

= (λ i − μ l )b il ,

i = 1, . . . , n ;

( i +1 )

b ik

(l)

(l)

= (λ i − μ l )b ik + b ik−1 ,

(3.58)

k = 2, 3, . . . , p i .

As follows from (3.58), the matrix B is triangular with nonzero diagonal entries b ii = 1, i = 1, 2, . . . , p1 , b ii =

k −1 / (λ k − λ j )pj ,

p 1 + p 2 + · · · + p k −1 ≤ i ≤ p 1 + p 2 + · · · + p k .

j =1

To obtain the boundary values ψ il, we take the trace of (3.57) on the boundary and then take the inverse of the matrix B (this procedure is not difficult because B is triangular: ψ = B−1 Ψ). The original problem is thus reduced to a boundary value problem of the type ( Δ − λ) p u = 0 ,

u |Γ = ψ1 ,

x ∈ G, ( Δ − λ ) u | Γ = ψ 2 , . . . , ( Δ − λ ) p −1 u | Γ = ψ p

for which the mean value relation (3.49) was obtained above. In conclusion, we note that the generalization of the converse mean value relation for the problem (3.52), (3.54) is completely the same as in the case of the polyharmonic equation (keeping in mind Proposition 2.2 of Section 2.2.

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4 Triangular systems of elliptic equations 4.1 One-component diffusion system Let us start with the following system (which will generate nonisotropic distribution on the sphere, see Chapter 8). We seek the regular solution to the boundary value problem in a bounded domain G ⊂ R3 with a boundary Γ : ⎧ ∂θ ⎪ ⎪ Δu (x) + 𝛾 ∂x (x) = 0 , ⎪ 1 ⎨ (4.1) Δθ(x) = 0 , ⎪ ⎪ ⎪ ⎩u | = φ , θ| = φ . Γ

1

Γ

2

¯ be an arbitrary sphere and let s be a unit vector in R3 with compo­ Let S(x, r) ⊂ G nents s i , i = 1, 2, 3. We denote by Ω the unit sphere: Ω = {s : |s| = 1}. Theorem 4.1. For each regular solution to (4.1), the following mean value relation holds: ⎧  ⎨ u (x) = N r u (x) + 𝛾r 8π Ω s1 θ ( x + rs) dΩ( s) , (4.2) ⎩ θ(x) = N θ(x) . r

Proof. Since Δ2 u (x) = 0, from (3.11) (see Chapter 3) we get u(x) = N r u(x) − Since

∂θ ∂x 1 (·)

r2 r2 ∂θ (x) . Δu (x) = N r u (x) + 𝛾 6 6 ∂x1

(4.3)

is harmonic, by the volume mean value relation 3 ∂θ (x) = ∂x1 4πr3

 B ( x,r )

∂θ (y)dy , ∂x1

so that, integrating by parts, one gets  ∂θ 3 3 (x) = s1 θ(x + rs)dΩ(s) = N r (s1 θ) . ∂x1 4πr r

(4.4)

Ω

From (4.3) and (4.4), we get (4.2). We now prove that the converse mean value relation holds. Proposition 4.1 (Integral formulation for the system). Suppose that (4.1) is uniquely solvable for each continuous φ1 and φ2 ; fix such functions and assume that functions ¯ with u˜ |Γ = φ1 , θ˜ |Γ = φ2 satisfy (4.2) at each point ˜ (x) and θ˜ (x) continuous in G u  x ∈ G for at least one sphere S(x, r x ) ⊂ G Γ. Then the functions u˜ and θ˜ solve the problem (4.1).

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56 | 4 Triangular systems of elliptic equations Proof. From Theorem 2.5 we get that θ˜ is harmonic in G; thus it coincides with θ of (4.1). In view of (4.2), it holds that  𝛾r ˜ )(x) = N r (u − u ˜ )(x) + (u − u s1 (θ − θ˜ )(x + rs)dΩ(s) , 8π Ω

˜ is harmonic in G (again we apply Theorem 2.5). But u |Γ = which implies that u − u ˜ ˜ u |Γ = φ1 ; thus u ≡ u in G. Using this proposition we can write down a system of integral equations equivalent to the problem (4.1). Let v = (u, θ)T . Let us consider the following integral equation constructed analogous to (2.16): v = Kε v + f ε , where the matrix-integral operator Kε is defined by the matrix-kernel ⎞ ⎧⎛ 𝛾rs i ⎪ ⎪ δ x (y) δ x (y) ⎪ 2 ⎨⎝ ⎠ , x ∈ G \ Γε k ε (x, y) = ⎪ 0 δ x (y) ⎪ ⎪ ⎩0, x ∈ Γε , and

⎞ ⎧⎛ ⎪ u(x) ⎪ ⎪ ⎨⎝ ⎠, f ε (x) = ⎪ θ(x) ⎪ ⎪ ⎩0 ,

x ∈ G \ Γε x ∈ Γε .

Theorem 4.2. The spectral radius of the operator Kε is less than 1, if ε > 0. Proof. Following the proof scheme of Theorem 2.6 we find   𝛾nd∗ Knε L ∞ ≤ [1 − ν(ε]n−1 1 + 2 which proves the theorem.

4.2 Two-component diffusion system For fixed integers i and j equal to 1, 2, or 3, we consider the following boundary value problem in a domain G ⊂ R3 : Δu (x) + 𝛾

∂2 θ (x) = 0 , ∂x i ∂x j (4.5)

Δθ(x) = 0 , u |Γ = φ1 ,

θ|Γ = φ2 .

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Theorem 4.3. Each regular solution to (4.5) satisfies the following mean value relation:  . δ ij 5 θ(x + rs)dΩ(s) , 𝛾 c si sj − u(x) = N r u(x) + 8π 3 (4.6) Ω Δθ(x) = N r θ(x) , where δ ij is the Kronecker symbol. Proof. From Lemma 4.1 which follows, we prove that the regular harmonic functions satisfy the relation . δ ij ∂2 θ 15 (x) = s s − θ(x + rs)dΩ(s) . (4.7) i j ∂x i ∂x j 4πr2 3 Ω

Substituting (4.7) into the mean value relation for the biharmonic function u (x), one gets r2 𝛾 ∂ 2 θ r2 u (x) = N r u (x) − Δu (x) = N r u (x) + (x) , 6 6 ∂x i ∂x j and thus (4.6). Lemma 4.1. Each analytical function θ defined in a bounded domain G ⊂ R3 satisfies the relations (i, j = 1, 2, 3):  ∞  b r2k F k (θ) 1 (a + bs2i )θ(x + rs)dΩ(s) = (a + )θ(x) + , (4.8) 4π 3 (2k + 1)!(2k + 3) k =1 Ω

where F k (θ) = a(2k + 3)Δ k θ(x) + bΔ k θ(x) + 2kbΔ k−1 and 1 4π



s i s j θ(x + rs)dΩ = Ω

∂2 θ (x) , ∂x2i

∞ 

2kr2k ∂2 Δ k −1 θ ( x ) (2k + 1)!(2k + 3) ∂x i ∂x j k =1

(4.9)

for i ≠ j. Proof. We integrate the expansion θ(x + rs) = θ(x) + r

3 3  ∂θ r2  ∂ 2 θ (x) s k + (x) s k s j + · · · ∂x k 2! j,k=1 ∂x k ∂x j k =1

over the sphere S(x, r) with the weights (a + bs2i ) and s i s j . Then the factor of the term r2k /(2k )! (all odd terms are, obviously, equal to zero) is  ∂2k C(2α 1 , 2α 2 , 2α 3 )B α 2α1 2α2 2α3 , ∂x1 ∂x2 ∂x3 A αk

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58 | 4 Triangular systems of elliptic equations where Bα =

1 4π



2α

2α

2α

s1 1 s2 2 s3 3 dΩ(s) Ω

and the set of indices is defined as A αk = {α : α 1 + α 2 + α 3 = k }. The structure of the coefficients C(2α 1 , 2α 2 , 2α 3 ) is clear (see Chapter 5). Using the relations Bα = and Δk =

(2α 1 − 1)!! (2α 2 − 1)!! (2α 3 − 1)!! [2(α 1 + α 2 + α 3 ) + 1]!! 

C(α1 , α2 , α3 )

A αk

∂2k 2α 2α 2α ∂x1 1 ∂x2 2 ∂x3 3

,

where α i = 0, 1, . . . , i = 1, 2, 3, we obtain (4.8) and (4.9). Note that for harmonic functions, relation (4.7) follows. Proposition 4.2 (Integral formulation). Suppose that the problem (4.5) is uniquely solv­  able for each continuous φ1 , φ2 . Fix two such functions. Let u and θ in C(G Γ) satisfy the mean value relation (4.6) at each point x ∈ G at least for one sphere S(x, r x ) ⊂ G; then they solve the problem (4.5). The proof repeats the considerations of Proposition 4.1.

4.3 Coupled biharmonic–harmonic equation The results of the preceding section can be generalized for the case when the operator Δ k acts on u. Let us treat the case k = 2. We fix an integer i equal to 1, 2, or 3 and consider the following boundary value problem: Δ2 u ( x) + 𝛾

∂θ (x) = 0 , ∂x i

x ∈ G ⊂ R3 , (4.10)

Δθ(x) = 0 , u |Γ = φ1 ,

Δu |Γ = φ2 ,

θ|Γ = φ3 .

Theorem 4.4. The regular solutions to (4.10) satisfy the mean value relation r2 7r3 N r Δu (x) − 𝛾N r (s i θ)(x) , 6 120 𝛾r Δu (x) = N r Δu (x) + N r (s i θ)(x) , 2 θ(x) = N r θ(x) . u(x) = N r u(x) −

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Proof. Since Δ3 u (x) = 0, from (3.10) we find that r2 r4 2 Δu (x) − Δ u(x) , 6 120 2 r Δu (x) = N r (Δu )(x) − Δ2 u (x) , 6 θ(x) = N r θ(x) . u(x) = N r u(x) −

Thus, r2 7r4 ∂θ (x) , N r (Δu )(x) − 𝛾 6 360 ∂x i 𝛾r2 ∂θ Δu (x) = N r (Δu )(x) + (x) , 6 ∂x i u(x) = N r u(x) −

θ(x) = N r θ(x) , which, in view of (4.4), gives the desired result. Proposition 4.3 (Integral formulation). Suppose that the problem (4.10) is uniquely solvable for all continuous functions φ1 , φ2 , φ3 .  Let φ1 , φ2 , φ3 be fixed. Let v := (v1 , v2 , v3 )T , v i ∈ C(G Γ) satisfy the mean value relation v(x) = N r (Av) , ⎛

where

1

⎜ ⎜ A = ⎜0 ⎝ 0

r2

−6

7𝛾

− 120 r3 s i

1

𝛾r 2

0

1

⎞ ⎟ ⎟ ⎟ ⎠

¯ Also suppose that for each x ∈ G, at least for one sphere S(x, r x ) ⊂ G. v1 |Γ = φ1 ,

v2 |Γ = φ2 ,

v3 |Γ = φ3 .

Then v solves the problem (4.10). The proof repeats the arguments given in Proposition 4.1. In conclusion we give the following extension. Let us consider the problem: Δ2 u ( x) + 𝛾

∂2 θ (x) = 0 , ∂x i ∂x j

x ∈ G ⊂ R3 , (4.11)

Δθ(x) = 0 , u |Γ = φ1 ,

Δu |Γ = φ2 ,

θ|Γ = φ3 .

The above integral formulation is true here with the matrix ⎞ ⎛ 2 δ 7𝛾r 2 1 − r6 − 24 (s i s j − 3ij ) ⎟ ⎜ ⎟ ⎜ δ ij 5𝛾 ⎟. A = ⎜0 1 ( s s − ) i j 2 3 ⎠ ⎝ 0 0 1

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5 Systems of elasticity theory 5.1 Lamé equation 5.1.1 Direct spherical mean value theorem Suppose a homogeneous isotropic medium G ⊂ Rn with a boundary Γ is given, whose state in the absence of body forces is governed by the classical static equation, the Lamé equation: Δu(x) + α grad div u(x) = 0 , x ∈ G , (5.1) where u(x) = (u 1 (x1 , . . . , x n ), . . . , u n (x1 , . . . , x n )) is a vector of displacements, whose components are real-valued regular functions. The elastic constant α α=

λ+μ 1 = μ 1 − 2σ

is expressed through the Lamé constants of elasticity λ and μ, and σ, the Poisson ratio (for −1 < σ < 1/2 we have 1/3 < α). Here we will employ the summation convention, for example, n  ∂2 u i . u i,jj = ∂x2j j =1 Hence, equation (5.1) can be written in the following form: u i,jj + αu j,ji = 0 ,

i, j = 1, . . . , n .

(5.2)

The first boundary value problem for the Lamé equation consists in finding a vector ¯ ) satisfying the boundary condition function u ∈ C2 (G) ∩ C(G u(y) = g(y) ,

y ∈ Γ,

(5.3)

where g ∈ C(Γ) is a given vector function. We first give the direct spherical mean value relation carried out in [40]. In addition to the mean value theorems for the biharmonic functions proved above, the following spherical mean value relation is deduced. Lemma 5.1. For any function v(x1 , . . . , x n ) biharmonic in G ⊂ Rn the following relation is valid for arbitrary ball B(x, r) ⊂ G: ⎡ ⎤   n ⎢n + 2 1 ⎥ ⎣ n v dV − n−1 v dS⎦ , (5.4) v(0) = 2ω n R R B (0,R )

S (0,R )

where ω n = 2π n/2 /Γ(n/2) is the area of the surface of the unit sphere in Rn .

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Proof. The function Δv is harmonic; therefore   n n ΔvdV = Δv(0) = ωn rn ωn rn B (0,r )

S (0,r )

Thus rΔv(0) =

n ωn



∂v n dS = ∂r ωn r



∂v dΩ . ∂r

Ω

∂v dΩ , ∂r

(5.5)

Ω

and the integration from r = 0 to r = ρ > 0 (interchanging the order of integration on the right-hand side) yields ⎤ ⎡  ρ2 n ⎢ ⎥ Δv(0) = ⎣ v dΩ − ω n v(0)⎦ . 2 ωn Ω

Replacing Δv(0) by its value from (5.5) we get ⎤ ⎡   ∂v ⎥ ⎢ ρn dΩ = 2ρ n−1 ⎣ v dΩ − ω n v(0)⎦ . ∂ρ Ω

By adding nρ

 n −1

Ω

v dΩ to both sides we come to ⎡ ⎤   ∂[ρ n v] ⎢ ⎥ dΩ = ρ n−1 ⎣(n + 2) v dΩ − 2ω n v(0)⎦ . ∂ρ Ω

Ω

(5.6)

Ω

Integrating (5.6) with respect to ρ from ρ = 0 to ρ = R yields   ωn Rn n R v dΩ = (n + 2) v dV − 2 v(0) , n B (0,R )

Ω

from which the desired result follows. Theorem 5.1. The regular solutions to the system (5.2) satisfy the following spherical mean value relation:   xi xj n (2 − α )δ ij + (n + 2)α 2 u j dΩ , (5.7) u i (0) = 2(n + α )ω n R Ω

i = 1, . . . , n, δ ij is the Kronecker symbol. Proof. It is easy to show that the displacement components u i satisfying (5.2) are bi­ harmonic functions and the divergence of u is harmonic. Indeed, (5.2) yields u i,jji + αu j,jii = 0 ,

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62 | 5 Systems of elasticity theory or (1 + α )Δu j,j = 0 .

Next, applying the operator Δ to (5.2) we get 0 = Δ[u i,jj + αu j,ji] = Δ2 u i + α (Δu j,j),i , or Δ2 u i = 0 .

(5.8)

We now use the fact that u i are biharmonic and that the mean value theorem of Lemma 5.1 is true to derive the spherical mean value relation for the displacements u i . We integrate by parts the Lamé equation  0= r2 [u i,jj + αu j,ji]dV (5.9) B (0,R )



=

-

R

2

S (0,R )

xj xi u i,j + αu j,j R R



.

dS − 2

[x j u i,j + αx i u j,j] dV , B (0,R )

where x i /R = s i is the ith component of the direction cosines of the vector s. The surface integral in (5.9) is zero since   . xj xi 2 2 R u i,j + αu j,j dS = R [u i,jj + αu j,ji] dV = 0 . R R S (0,R )

B (0,R )

Integrating the remaining integral in (5.9) by parts yields   . xi xj Ru i + α (n + α )u i dV . 0= u j dS − R S (0,R )

B (0,R )

Therefore, (since dS = R n−1 dΩ)   -  . xi xj Rn u i dV = δ ij + α 2 u j dΩ . n+α R B (0,R )

(5.10)

Ω

Applying the mean value theorem for biharmonic functions (5.4) to the displace­ ment u i , and then omitting the integral over B(0, R) we obtain the desired result. Note that the direct spherical mean value relation (5.7) can be derived from the Betty formula (e.g. see [114]). This formula in R3 has the form   (v · Au − u · Av)dV = (v · t(u) − u · t(v))dS . (5.11) G

∂G

Here A = Δ∗ and t(v) is the stress vector: t (v ) =

3 

τ ik cos(n, x i )ek ,

i,k =1

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where τ ik are the components of the stress tensor   ∂u i ∂u k + λδ ik div u , + τ ik = μ ∂x k ∂x i and ek is the ort of the axis x k . Assume that the function u is twice continuously differentiable in G and belongs ¯ ). Let us cutoff a sphere S(x, ε) ⊂ G, ε being small enough. We apply the Betty to C(G formula (5.11) to the domain G \ S(x, ε) and take the limit as ε → 0. Taking into account the relations   vi · AudV = vi · Au dV , lim ε →0

G \ S ( x,ε )

G



ε →0



vi · t(u)dS = 0 ,

lim

ε →0

S ( x,ε )



we obtain u i (x) =

u · t(vi )dS = −u i (x)

lim

S ( x,ε )



vi · AudV − G

{u · t(v i ) − v i · t(u)}dS .

(5.12)

∂G

Here vi = (v i1 , v i2 , v i3 )T , v ik being the components of the tensor of fundamental solu­ tions V :   1 δ ik (x i − y i )(x k − y k ) (3 − 4σ ) + . v ik = 16πμ (1 − σ ) r r3 Multiplying (5.12) by ei and summing over i we find   u(x) = V · AudV − {T (V ) · u − V · t(u)}dS , G

(5.13)

∂G

where T (V ) is a tensor whose ith row coincides with t(vi ). In our case, G is the ball B(x, R); therefore, . 1 − 2σ 3 (x i − y i )(x k − y k ) δ ik + . T ik (V ) = 8π(σ − 1)R2 1 − 2σ R2 ˆ by Let us define a tensor V ˆv ik =

  1 δ ik (x i − y i )(x k − y k ) (3 − 4σ ) + . 16πμ (1 − σ ) R R3

ˆ for r = R. Applying the Betty formula for V ˆ we get Note that V = V   ˆ · t(u)dS V · t(u)dS = V S ( x,R )

S ( x,R )





ˆ ) · udS + T (V

= S ( x,R )

ˆ · Au − u · AV ˆ }dV . {V B ( x,R )

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64 | 5 Systems of elasticity theory Substituting this representation into (5.13) we come to the following mean value relation:   ˆ ) · AudV − ˆ · udV u(x) = (V + V AV B ( x,R )





−



B ( x,R )

(5.14)

ˆ ) · udS . T (V ) − T (V

S ( x,R )

Here ˆ )ik = (AV

2λ + 3μ δ ik , 8πμ (1 − σ )R3

and ˆ ))ik = ( T (V

1 {μR2 δ ik + (4λ + 3μ )(x i − y i )(x k − y k )} . 16πμ (1 − σ )R4

In the case when u is the solution to the Lamé equation we have Au = 0; hence  ˆ · udV AV u(x) = − B ( x,R )



(5.15)

ˆ )} · udS , {T (V ) − T (V

− S ( x,R )

or, more explicitly, u i (x) =

2λ + 3μ 8πR3 μ (σ − 1)



u i dV B ( x,R )

1 − 2 16πR (σ − 1)

   λ (x i − y i )(x k − y k ) (3 − 4σ )δ ik + 9 + 4 u k dS . μ R2

S ( x,R )

Note that if we change the volume integral in (5.15) with the representation (5.10) we obtain the spherical mean value relation (5.7).

5.1.2 Converse spherical mean value theorem As in the case of harmonic functions, it is not difficult to show that the weak converse spherical mean value relation holds also for the regular solutions to the Lamé equa­ tion (e.g. see [39]). It can be derived, for instance, from the series expansion of the spherical mean with respect to a vector measure (see expansion (5.21)). However the strong converse spherical mean value relation can be proved by using the approach [39] and the converse mean value relation of Theorem 2.5 (see [161]).

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Theorem 5.2 (Integral formulation). Suppose that the boundary Γ of G ⊂ Rn is regu­ lar so that the Dirichlet problem (5.1), (5.3) is uniquely solvable for arbitrary continuous boundary functions g1 , . . . , g n , and the solutions have third continuous derivatives in G ∪ Γ. Assume that a vector function u ∈ C3 (G ∪ Γ), (u|Γ = g) satisfies the mean value ¯ Then u(·) solves the relation (5.7) for each x ∈ G at least for one sphere S(x, r x ) ⊂ G. Dirichlet problem (5.1), (5.3). ¯ and set for an arbitrary point x ∈ B(y, r) Proof. Take an arbitrary sphere S(y, r) ⊂ G, 2−α (x j − y j )(u i,j(x) − u j,i(x)) 2(n + α ) (n + 1)α + 2 + (x i − y i )u j,j(x) . 2(n + α )

Θ i ( x ) := u i ( x ) +

(5.16)

Note that the Lamé equation can be rewritten in the form: ΔΘ(x) = 0. Indeed, we first differentiate the Lamé equation. Using the properties Δu j,j = 0,

Δ(u i,j − u j,i) = 0

we find by differentiating the Lamé equation that u i,jj + This yields

(n + 1)α + 2 2−α (u i,jj − u j,ij) + u j,ji = 0 . n+α n+α

n+2 α (n + 2) ∂ div u = 0 , Δu i + n+α n + α ∂x i

which coincide with the Lamé equation, since n + α > 0. Obviously, these transfor­ mations can be converted, i.e. starting from the equation ΔΘ(x) = 0 we come back along the same transformations to the Lamé equation. Thus, the Lamé equation is equivalent to the equation ΔΘ(x) = 0. Now, we start from the spherical mean value relation given in the theorem formu­ lation and integrate by parts the right-hand side of relation (5.7) on the sphere S(y, r). This leads, as we will prove below, to   1 n u i (x) + a(x j − y j )(u i,j (x) − u j,i(x)) (5.17) u i (y) = ωn rn B ( y,r )  +(a + b )(x i − y i )u j,j(x) dx , where a=

(2 − α ) n , 2(n + α )

b=

n(n + 2)α . 2(n + α )

This relation implies by Theorem 2.5 that the functions Θ i in (5.16) are harmonic in G. Thus, we get ΔΘ(y) = 0 for all y ∈ G which implies that at y, one has that Δu i (y) + α

∂ div u(y) = 0 , ∂x i

i = 1, . . . , n .

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66 | 5 Systems of elasticity theory Since y was chosen arbitrarily in G, this completes the proof. It remains only to trans­ form the right-hand side of relation (5.7) to the right-hand side of (5.17). However, we carry out the transformations backward, i.e. we deduce the right-hand side of (5.7) from the right-hand side of (5.17). The transformations can then be easily traced back. Thus, we integrate by parts the right-hand side of (5.17) using the formula    ∂f (x) ∂g(x) g(x) dx = f (z)g(z)n i dS z − f (x) dx . ∂x i ∂x i B ( y,r )

S ( y,r )

B ( y,r )

This leads to the following representations:    a (x j − y j )u i,j (x)dx = ar u i (x)dS x − an u i (x) dx , B ( y,r )

S ( y,r )





(x j − y j )u j,i (x)dx = a

a B ( y,r )

and

S ( y,r )

B ( y,r )

(x j − y j )(x i − y i ) u j (x)dS x − a r

 (a + b)

 (x i − y i ) div u(x)dx = (a + b )

B ( y,r )

S ( y,r )



u i (x) dx , B ( y,r )

(x i − y i )(x j − y j ) u j (x)dS x r



− (a + b)

u i (x) dx . B ( y,r )

We now evaluate the right-hand side of (5.17) using these representations. The volume integrals disappear, since an + b = n. Thus, we finally get ⎧ ⎫ ⎪ ⎪   ⎨ ⎬ (x i − y i )(x j − y j ) 1 a u dS + b u dS , (5.18) u i (y) = i x j x n − 1 2 ⎪ ⎪ ωn r r ⎩ ⎭ S ( y,r )

S ( y,r )

which is the desired spherical mean value relation coinciding with (5.7). As mentioned above, these transformations can be traced backward to deduce (5.17) from (5.7), which proves the theorem.

5.2 Pseudovibration elastic equation Let us define the operator Δ∗ by Δ∗ = μΔ + (λ + μ )grad div . Consider the pseudovibration elastic equation Δ∗ u(x) − νu(x) = 0 ,

x ∈ G ⊂ Rn ,

(5.19)

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where u = (u 1 (x), . . . , u n (x)) is a real-valued regular vector function; ν is a complex parameter. Now we obtain the mean value relation for this equation. In accordance with the scheme described in Section 2.1.1, we introduce an averaging measure over a sphere S(x, r):  n 7 8 1 aδ + bs s dS ( s ) , dη1 (s) = ij i j ω n r n −1 i,j =1 where a and b are some constants, s i is the ith direction cosine of the vector s ∈ Ω(x, 1), δ ij is the Kronecker symbol, ω n is the surface area of Ω(x, 1). The corresponding operator averaging the vector function u over the sphere S(x, r) with respect to the measure dη1 will be denoted by N 1 u(x):  u(x + rs)dη1 (s) , (5.20) N 1 u(x) = S ( x,r )

where the integral of the vector u(x + rs) is understood to be a vector whose compo­ nents are integrals of the corresponding elements. The integrand in (5.20) is regarded as a product of the matrix dη1 (s) by the vector u(x + rs). We now present some auxiliary results which will be used in obtaining the main result, namely, Theorem 5.7 on the mean value relation for (5.19) (see [167]). Theorem 5.3. For a real-valued vector function u = (u 1 (x), . . . , u n (x))T , x ∈ G ⊂ Rn , analytic in G, the following expansion:   ∞  b r2k (n − 2)!! u(x) + N 1 u(x) = a + n (2k )!!(n + 2k )!! k =1   × (n + 2k )aΔ k u(x) + bΔ k u(x) + 2kbΔ k−1 grad div u(x)

(5.21)

holds if the series on the right-hand side converges. Proof. For simplicity, consider the case n = 3. The proof for arbitrary dimension is analogous. A scalar case was discussed briefly in Section 4.2, Lemma 4.1. Consider the first relation of the system (5.21). By definition of N 1 (see (5.20)) we have   1  1 N u 1 (x) = (a + bs21 )u 1 (x + rs) dS (5.22) 4πr2 S ( x,r )   b b + s s u ( x + rs ) dS + s1 s3 u 3 (x + rs) dS . 1 2 2 4πr2 4πr2 S ( x,r )

S ( x,r )

Substituting the Taylor series expansion of the analytic function u 1 u 1 (x + rs) = u 1 (x) + r

3 3  ∂u 1 (x) r2  ∂ 2 u 1 ( x ) sk + sk sj + · · · . ∂x k 2! j,k=1 ∂x k ∂x j k =1

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68 | 5 Systems of elasticity theory into the first term of (5.22), then integrating this series, term by term, we find  1 (a + bs21 )u 1 (x + rs)dS 4πr2 S ( x,r )

=

∞  r2k  ∂2k u 1 (x) C(2α 1 , 2α 2 , 2α 3 ) 2α1 2α2 2α3 (2k )! A k ∂x1 ∂x2 ∂x3 k =0 α  1 2α 2α 2α × (a + bs21 )s1 1 s2 2 s3 3 dS , 4πr2

(5.23)

S ( x,r )

where the set of indices is defined as A kα = {(α 1 , α 2 , α 3 ) : α 1 + α 2 + α 3 = k }, and C(l, n, m) =

( l + n + m )! . l!n!m!

Taking into account the relations  (2α 1 − 1)!!(2α 2 − 1)!!(2α 3 − 1)!! 2α 2α 2α , s1 1 s2 2 s3 3 dΩ = 4π ((2α 1 + 2α 2 + 2α 3 ) + 1)!! Ω ( x,1)

and Δk =



∂2k

C(α1 , α2 , α3 )

2α 2α 2α ∂x1 1 ∂x2 2 ∂x3 3

A kα

,

we obtain  1 (a + bs21 )u 1 (x + rs) dS 4π S ( x,r )

  ∞    b r2k = a+ u1 (x) + a(2k + 3)Δ k u 1 (x) + bBu 1 (x) , 3 (2k + 1)!(2k + 3) k =1

(5.24) where Bu 1 (x) =



C(2α 1 , 2α 2 , 2α 3 )(2α 1 + 1)

A kα

∂2k u 1 (x) 2α 2α 1 2α ∂x1 ∂x2 2 ∂x3 3

.

We transform the operator B. To this end, we introduce an auxiliary operator F depen­ dent on parameter 𝛾 F𝛾u1 (x) =



C(2α 1 , 2α 2 , 2α 3 )𝛾(2α1 +1)

A kα

such that

  B = F𝛾 

𝛾=1

But



∂2k u 1 (x) 2α

2α 3

,

.

∂2 ∂2 ∂2 F𝛾 u1 (x) = 𝛾 𝛾 2 + 2 + ∂x1 ∂x2 ∂x23 2

2α

∂x1 1 ∂x2 2 ∂x3

k

u1 (x) ;

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5.2 Pseudovibration elastic equation | 69

therefore, to get the desired expression of B, it is sufficient to differentiate the last expression with respect to 𝛾 at the point 𝛾 = 1: Bu 1 (x) = Δ k u 1 (x) + 2kΔ k−1

∂2 u1 (x) . ∂x21

Similar expansions for the second and third integral terms in (5.22) can also be derived:    ∞  ∂2 b k −1 s1 s2 u 2 (x + rs) dS = b M k (r) 2k Δ u2 (x) , 4πr2 ∂x1 ∂x2 k =1 S ( x,r )

b 4πr2



 ∂2 k −1 s1 s3 u 3 (x + rs) dS = b M k (r) 2k Δ u3 (x) , ∂x1 ∂x3 k =1 ∞ 

S ( x,r )



where M k (r) = r2k /[(2k + 1)!(2k + 3)]. Then expression (5.22) transforms to 

∞   N 1 u 1 (x) =

r2k (2k + 1)!(2k + 3) k =0   ∂ div u . × a(2k + 3)Δ k u 1 (x) + bΔ k u 1 (x) + 2kbΔ k−1 ∂x1

(5.25)

Repeating calculations (5.22)–(5.25) for the second and third components of the aver­ aging operator N 1 u, we obtain the desired result (5.21) for the case u ∈ R3 N 1 u(x) =

 r2k a(2k + 3)Δ k u(x) ( 2k + 1 ) ! ( 2k + 3 ) k =1    b +bΔ k u(x) + 2kbΔ k−1 grad div u(x) + a + u(x) . 3 ∞ 

Corollary 5.1. For the solution to equation (5.19) the following expansion holds: N 1 u(x) − u(x) =

∞ 

r2k (n − 2)!! (2k )!!(n + 2k )!! k =1   kn(n + 2)ν k−1 k × Δ u(x) − n(k − 1)Δ u(x) . λ + μ (n + 1)

(5.26)

Proof. Setting β = (n + 2)(λ + μ )/2[λ + μ (n + 1)], a = 1 − β, b = nβ in (5.21), for the solution to equation (5.19) we obtain (5.26). Corollary 5.2. The mean value relation for the Lamé equation Δ∗ u(x) = 0 follows immediately from (5.26) N 1 u(x) = u(x) .

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70 | 5 Systems of elasticity theory We need certain properties of the solutions to equation (5.19). Let us formulate the following theorem (see [96], III.2.4). Theorem 5.4. The general solution to (5.19) can be represented in the form u(x) = up (x)+ us (x), where up and us are regular vector functions determined by the equations (Δ − k 2p )up = 0 ,

rot up = 0 ,

k 2s )us

s

(Δ −

= 0,

(5.27)

div u = 0 ,

9

 where k p = ν/(λ + 2μ ), k s = ν/μ, and

up =

(Δ − k 2s )u , k 2p − k 2s

us =

(Δ − k 2p )u

k 2s − k 2p

,

u ∈ C ∞ (G).

(5.28)

Using this decomposition of u(x) into the potential (up (x)) and solenoidal us (x) com­ ponents, we now obtain the following properties. Lemma 5.2. The solution to (5.19) satisfies the relations s 2n p Δ n u(x) = k 2n s u (x) + k p u (x) , n

Δ div u(x) =

k 2n p

div u(x) .

(5.29) (5.30)

Proof. The repeated operator Δ∗ is represented as

n −1 ) ν νu − μΔu ( νn u = μn Δn u + (λ + 2μ )n − μ n . λ + 2μ λ+μ Hence, n −1 $ %

n −1 $ % ν λ + 2μ νu ν μ νu Δu − Δu − . − Δ u= μ λ+μ λ + 2μ λ + 2μ λ+μ μ n

Recalling that (k 2s − k 2p )−1 =

μ (λ + 2μ ) ν(λ + μ )

we obtain (see Theorem 5.4) Δ n u = k 2n s

(Δ − k 2p )u

k 2s − k 2p

+ k 2n p

(Δ − k 2s )u , k 2p − k 2s

thus, (5.29) is proved in view of Theorem 5.4. Property (5.30) is directly implied by the last formula (it is sufficient to apply the operator div to both sides of (5.29) and make use of (5.27)). Theorem 5.5. For any analytical functions up , us (case R3 ) satisfying equations (5.27) and (5.28) the following relation holds: Nr u =

sinh (rk p ) p sinh (rk s ) s u (x) + u (x) . rk p rk s

(5.31)

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Proof. In Section 2.1.1, we used the expansion of the averaging operator N r v corre­ sponding to the uniform measure dΩ. Let us complete this definition for the case of vector function v: ( N r v) i = N r ( v i ) =

∞ 

r2n Δ n v i (x) , (2n + 1)! n =0

i = 1, 2, 3 .

Then, by definition and on the basis of (5.29) we have N r u(x) =

  r2n s 2n p k 2n u ( x ) + k u ( x ) p (2n + 1)! s n =0

=

sinh (rk p ) p sinh (rk s ) s u (x) + u (x) . rk s rk p

∞ 

Theorem 5.6. For the solution to equation (5.19) (case R3 ) the following auxiliary mean value relation holds: N 1 u(x) = W1 us (x) + W2 up (x), (5.32) where

  cosh (rk s ) sinh (rk s ) sinh (rk s ) +b − , rk s (rk s )2 (rk s )3   sinh (rk p ) cosh (rk p ) sinh (rk p ) W2 = ( a + b ) − 2b − , rk p (rk p )2 (rk p )3

W1 = a

a = 1 − b,

b = 3β,

β=

5(λ + μ ) . 2(λ + 4μ )

Proof. Using expansion (5.26) and properties (5.29), (5.30) (in R3 ), we obtain N 1 u(x) − u(x) =

∞ 

r2n (2n + 1)!(2n + 3) n =1   s 2n p × (2n + 3 + 2nβ )k 2n . s u − 3nβk p u

(5.33)

Summing up the series in the right-hand side of this equality yields (5.32). Theorem 5.7 (The mean value relation). Assume that the parameter ν satisfies the in­ equalities k p r < π and k s r < π for each S(x, r) ⊂ G. Then the general solution to (5.19) in B(x, r) ⊂ R3 satisfies the following mean value relation: 7 8 sinh ( rk ) sinh ( rk s ) − rk p p N 1 u ( W2 − W1 ) N r u + rk s (5.34) u(x) = sinh ( rk ) sinh ( rk s ) W2 − rk p p W1 rk s or in a standard integral-matrix form u(x) =

1 4π

 K(x, s)u(x + rs)dΩ , Ω

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72 | 5 Systems of elasticity theory where the entries of the matrix K are given by

−1 sinh rk p sinh rk s k ij = W2 − W1 rk s rk p  $ % sinh rk p sinh rk s × W2 − W1 + a − δ ij rk s rk p $ %  sinh rk p sinh rk s +b − si sj . rk s rk p

(5.35)

Proof. Let us consider the system of equations (5.31), (5.32), and u(x) = up (x)+ us (x). Solving this system for u, we get the desired relation. Remark 5.1. This statement can be extended to the general case of n dimensions. Let ν be a complex number, the parameter of the equation Δ∗ u(x) + νu(x) = 0 ,

x ∈ G,

 satisfying the conditions Rek s r < π and Rek p r < π where k s = ν/μ and k p = 9 ν/(λ + 2μ ) for any S(x, r) ⊂ G. The spherical mean value relation under these assumptions has the form (see [123])   1 u i (x) = A + Bs i s j u j (x + rs)dΩ , i = 1, 2, . . . , n , (5.36) Cω n Ω

where A = τ α (ξ ) − (n − 1)τ α+1(ξ ) − τ α+1 (η) , B = τ α (η) − τ α (ξ ) , and C = 2α Γ(α + 1){τ α (η)[τ α (ξ ) − (n − 1)τ α+1 (ξ )] − τ α (ξ )τ α+1 (η)} . Here ξ = k p r, η = k s r, and τ α (z) = z−α J α (z). Note that the function τ α (z) is related to W α (z) through W α (z) = Γ(α /2)2α/2−1 τ α/2−1(z) and it is therefore not difficult to rewrite the relevant spherical mean value relation in terms of the functions w n and w n+2 . Indeed, the solution to (5.19) satisfies the same relation (5.36) with n−1 1 w n +2 ( ξ ) − w n +2 ( η ) , n n B = w n (η) − w n (ξ ) ,   n−1 1 C = w n (η) w n (ξ ) − w n +2 ( ξ ) − w n +2 ( η ) w n ( ξ ) . n n

A = w n (ξ ) −

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5.3 Thermoelastic equation |

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5.3 Thermoelastic equation In this section we deal with a homogeneous isotropic medium G, whose state is char­ acterized by the classical thermoelastic equation (there are no body forces and no ther­ mal sources): Δu(x) + α grad div u(x) − 𝛾 grad θ(x) = 0 ,

(5.37) 3

Δθ(x) = 0 ,

x∈G⊂R ,

(5.38)

where u = (u 1 (x), . . . , u 3 (x)), u i , θ are real-valued regular functions, 𝛾 = (2μ + 3λ)c, c is the coefficient of linear heat expansion [96]. We now obtain a mean value relation for (5.37), (5.38) which generalizes the result (5.34) (see [166]). Theorem 5.8. The general solution to the system (5.37), (5.38) satisfies the following mean value relation:  3r𝛾 sθ(x + rs) dΩ(s) , u(x) = N 1 u(x) − ω 3 (3 + α ) Ω ( x,1)

θ(x) = N r θ

(5.39)

where s is the direction cosine vector. Proof. For the harmonic function θ(x), we have θ(x) = N B θ ,

(5.40)

where N B θ is the volume integral of θ(x) over the ball B(x, r) [33]. Thus, for the har­ monic function ∂θ/∂x k , relation (5.40) takes the form   3 ∂θ(y) 3 xk ∂ θ(x) = dy = θ(x + rs) dΩ , k = 1, 2, 3 , ∂x k 4πr3 ∂x k 4πr r B ( x,r )

that is ∂ 3 θ(x) = ∂x k 4πr

Ω ( x,1)



s k θ(x + rs) dΩ ,

k = 1, 2, 3 .

(5.41)

Ω ( x,1)

By Δ2 u i = 0 and from (5.21) and (5.41) we obtain ∞ r2 (Δ + α grad div)u(x)  r2n + 3+α (2n + 1)!(2n + 3) n =2   n −1 15n𝛾Δ grad θ(x) n − 3(n − 1)Δ u(x) × 3+α  𝛾r2 grad θ(x) 3𝛾r = sθ(x + rs) dΩ(s), = 3+α 4π(3 + α )

N 1 u(x) − u(x) =

Ω ( x,1)

and (5.39) is proved.

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6 The generalized Poisson formula for the Lamé equation 6.1 Plane elasticity In this section, we deal with the first elasticity boundary value problem for a disk. So let us suppose a homogeneous isotropic medium G ⊂ R2 is given, for which Δ∗ u = 0 ,

(6.1)

where Δ∗ ≡ μΔ + (λ + μ )grad div is the Lamé operator; u = (u 1 (x), u 2 (x)) is a vector of displacements whose compo­ nents are real-valued regular functions.

6.1.1 Poisson formula for the displacements in rectangular coordinates Let us suppose that we have an elastic disk S(x0 , R) of radius R centered at x0 , lying in ¯ Let x (with the polar coordinates (ρ, φ )) be any interior point of the disk K (x0 , R), G. (K (x0 , R) = {x : |x0 − x| ≤ R}), and let y be a point situated on the circle S(x0 , R), whose coordinates are (R, θ), where θ = φ + α, and let z be defined by z = y − x, Z = |z|; ψ is the angle between x and z. Let us introduce the notation s1 = cos ψ ,

s2 = sin ψ ,

sˆ1 = cos φ ,

ˆs2 = sin φ ,

φ = ψ + φ ,

where ˆs i and s i (i = 1, 2) are the direction cosines of the vector z, 9 k = ρ /R, J = 1 − k 2 sin2 ψ . The next relations follow from the above definitions immediately Z 2 = R2 + ρ 2 − 2Rρ cos α , hence

(6.2)

9 Z = −ρ cos ψ + R2 − ρ 2 sin2 ψ ,

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and Z k = Z /R = −ks1 + J , cos α = Z /R cos ψ + ρ /R = Z k s1 + k , sin α = Z /R sin ψ = Z k s2 , ˆs1 = (R cos θ − ρ cos φ )/Z , ˆs2 = (R sin θ − ρ sin φ )/Z .

(6.3) (6.4)

(6.5)

To prove the main result we first represent some weighted spherical means in the form of a series of operators. Theorem 6.1. For any real-valued biharmonic analytical function u (x) in the domain G, the following representations hold for an arbitrary S (x0 , r) ⊂ G: 1 ω2 R

 S ( x 0 ,R )

ˆs21 u (y) R2 + 2ρ 2 cos2 φ ∂u (x) (2 + cos 2φ )ρ dS = u (x) + 4 2 2 3 Z 2(R − ρ ) ∂n 4 ( R 2 − ρ 2 )2 ∂u (x) sin 2φ Δu (x) + ∂t 4(R2 − ρ 2 )2 16(R2 − ρ 2 ) ∞  1 (−1)p ρ p−4 ∂ p−4 + L u ( x ) + L1 Δu (x) (6.6) 1 8( R 2 − ρ 2 ) 24 (p − 1)! ∂n p−4 p =4

−

where

∂2 ∂2 ∂2 cos2 φ + 2 sin2 φ − sin 2φ , 2 ∂n ∂t ∂n∂t

L1 = while 1 ω2 R

 S ( x 0 ,R )

ˆs1 ˆs2 u (y) dS Z4

sin 2φ ∂u (x) ρ sin 2φ ∂u (x) ρ cos 2φ + + 2 ∂n 4(R2 − ρ 2 )2 ∂t 4(R2 − ρ 2 )2 ∞ p p − 4  (−1) ρ 1 ∂ p −4 + L u ( x ) + L2 Δu (x) , 2 8( R 2 − ρ 2 ) 24 (p − 1)! ∂n p−4 p =4

= u(x)

where L2 =

1 2



∂2 ∂2 − 2 2 ∂n ∂t

sin 2φ +

(6.7)

∂2 cos 2φ , ∂n∂t

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 S ( x 0 ,R )

u(y) dS Z4 (6.8)

( R2 + ρ 2 ) u ( x) ∂u (x) Δu (x) ρ = + + , ( R 2 − ρ 2 )3 ∂n (R2 − ρ 2 )2 4(R2 − ρ 2 )  s1 u ( y) 1 dS ω2 R Z3 S ( x 0 ,R )

∞  ρu (x) ∂u (x) (−1)p+1 ρ p−3 ∂ p−2 Δu (x) 1 + = + , ( R 2 − ρ 2 )2 ∂n 2(R2 − ρ 2 ) p=3 23 (p − 1)! ∂n p−2  s2 u ( y) 1 dS ω2 R Z3

S ( x 0 ,R )

(6.9)

(6.10)

∞  ∂u (x) (−1)p+1 ρ p−3 ∂ p−4 ∂2 Δu (x) 1 . + = 2 2 ∂t 2(R − ρ ) p=3 23 (p − 1)! ∂n p−4 ∂n∂t

Proof. It suffices to prove any of formulae (6.6)–(6.10), say, formula (6.6). Let us locate the origin of coordinates (x1 , x2 ) at the point x0 , and introduce the coordinates (n, t) with the origin at x and turned with respect to the axes (x1 , x2 ) through the angle φ . Then the Taylor series expansion of the function u at any point x ∈ K (0, R) is ∂u (x) ∂u (x) Zs1 + Zs2 ∂n ∂t p Zp  α ∂p u p− α + ··· + C sα s + ··· . p! α=0 p ∂n α ∂t p−α 1 2

u (y) = u (x + Zs) = u (x) +

(6.11)

Let us multiply both sides of (6.11) by the weight ˆs21 /Z 4 and integrate the result over the circle S(0, R). Hence, due to the expansion Zp =

p 

j

p− j j

C p R p (−k 2 )p−j s1

J

j =0

we get 1 ω2 R

 S (0,R )

ˆs21 u(x) u (y) dS = 4 Z ω2 ×

p −3 j =0

 S (0,R ) j

C p −3

p ˆs21 1  α ∂ p u(x) dS + · · · + C Z4 ω2 α=0 p ∂n α ∂t p−α

R p −3 (−k 2 )p−j−3 p!



s2 α + p − j −3 p − α j ˆ s2 J 1

s1 S (0,R )

Z

dS + · · · . (6.12)

All transformations of series (6.11) (integration term-to-term, summation order change) are possible by virtue of the hypotheses of the theorem.

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The Jacobian of the transformation of a circle element dS to an angle element is equal to Z /J. Since the function J is positive and symmetric relative to the point π in the interval [0, 2π], the integrals in (6.12) differ from zero if the powers of s i , (i = 1, 2) are even. Consequently, in (6.12), only J 2j (j = 1, 2, . . .) should be considered. Hence J 2j can be decomposed using the binomial formula. To make the presentation clear we treat separately the sums involving the even and odd terms. The even terms of the series in (6.12) are (for p > 3) ⎧ j p ∞ ⎨  R2p−3  2j−1 2p−2j−4  l ∂2p u 2 j−l C2p−3 k C j (−k ) ⎩ C2α 2p 2α ω (2p)! j ∂n ∂t2p−2α p =3 2 α =0 l =0 2π 2p−2j +2α −4 2p−2α +2j −2l × (s21 cos2 φ + s22 sin2 φ )s1 s2 dψ

(6.13)

0 p −1

−



+1 C2α 2p

α =0

2p

∂ u sin 2φ ∂n2α+1 ∂t2p−2α−1

2π

2p−2j +2α −2 2p−2α +2j −2l s2

s1 0

⎫ ⎪ ⎬

dψ⎪ . ⎭

The remaining integrals can be taken from tables and are expressed in terms of Euler integrals, 2π 1 (2n − 1)!!(2m − 1)!! 2m s2n . 1 s2 dψ = ω2 (2n + 2m)!! 0

Let us roll up the sum over α in (6.13). To this end, we introduce the independent pa­ rameters β and 𝛾 such that (2n − 2m − 1)!! ∂ n  −(2m+1)/2     (−2)−n = β . β =1 (2m − 1)!! ∂β n The same property holds for the parameter 𝛾. Then (6.13) takes the form j ∞  R2p−3  2j−1 2p−2j−4  l (2p − 1)!! C2p−3 k C j (−k 2 )j−l (−2)p−1−l ( 2p ) ! ( 4p − 2l − 2)!! p =3 j l =0 ⎧ p ⎨ ∂2p u ×⎩ C αp 2α 2p−2α (A cos2 φ + B sin2 φ ) ∂n ∂t α =0 ⎫ p −1 ⎬ 2p  ∂ u − C αp−1 2α+1 2p−2α−1 2pC sin 2φ ⎭ , ∂n ∂t α =0

where

 ( p − j − 1 )  ( j − l )    𝛾−(2p−2α+1)/2 , A = β −(2α+1)/2 β 𝛾 𝛾= β=1  ( p − j − 2)  ( j − l + 1 )    𝛾−(2p−2α+1)/2 , B = β −(2α+1)/2 β 𝛾 𝛾= β=1  ( p − j − 2)  ( j − l )    𝛾−(2p−2α+1)/2 . C = β −(2α+3)/2 β

𝛾

𝛾= β=1

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78 | 6 The generalized Poisson formula for the Lamé equation Changing the order of differentiation with respect to the parameters and the summa­ tion over α, we can then rewrite the sum over α as j ∞  R2p−3  2j−1 2p−2j−4  l (2p − 1)!! C2p−3 k C j (−k 2 )j−l (−2)p−1−l ( 2p ) ! ( 4p − 2l − 2)!! p =3 j l =0

⎛ ⎞ p − j −1 j−l ∂ ∂ 1 p × ⎝ p−j−1 j−l ⎝ 9 L u ⎠ cos2 φ ∂β ∂𝛾 β𝛾 ⎛

⎛ ⎞ ∂ p − j −2 ∂ j − l +1 ⎝ 1 p ⎠ 2 9 L u sin φ + ∂β p−j−2 ∂𝛾j−l+1 β𝛾

∂ p − j −2 ∂ j − l − p p − j −2 j − l ∂β ∂𝛾 where L=



2 1 p −1 ∂ u L (β𝛾)3/2 ∂n∂t



   sin 2φ  

, β=𝛾=1

1 ∂2 1 ∂2 + . β ∂n2 𝛾 ∂t2

It is easy to see that L is the Laplace operator if β = 𝛾 = 1. Hence, if after the dif­ ferentiation with respect to β and 𝛾, the operator L acts on the function u more than twice, then this term vanishes. Therefore, only the terms with l = 0 do not vanish. Among these, the nonzero terms appear only when the parametric derivatives of L are taken, since otherwise there are terms involving L q with q > 1 (recall that Δ2 u = 0). Now,

Ln  ∂2n−2m−2 ∂2m ∂ n − m −1 ∂ m   = n!(−1)n−1 2n−2m−2 2m , n − m − 1 m l k β=𝛾=1 ∂β ∂𝛾 β𝛾 ∂n ∂t

m m 2l 2m  ∂2 ∂2m l l ∂ m − l +1 m ∂ Δu Δu = Δ − Δu = C (− 1 ) Δ u = (− 1 ) . m ∂t2m ∂n2 ∂n2l ∂n2m l =0

These relations are proved by direct differentiation. The property [( p −1)/2] 2j +1 C p = 2 p −1 j =0

was used above. In order to get the desired result (6.6) it remains to add the odd terms transformed by the same procedure and the first terms with p < 3 to the expression for even p. The first term values are given by the next formulae, which are proved using a simple

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technique: 1 2πR 1 2πR 1 2πR 1 2πR 1 2πR 1 2πR 1 2πR 1 2πR



1 R2 + ρ 2 dS = 2 , 4 Z ( R − ρ 2 )3

S (0,R )



s1 ρ dS = 2 , Z3 ( R − ρ 2 )2

S (0,R )



s2i 1 dS = , Z2 2(R2 − ρ 2 )

S (0,R )



s21 R2 + 2ρ 2 dS = , 4 Z 2(R2 − ρ 2 )3

S (0,R )





s31 3ρ dS = , Z3 4 ( R 2 − ρ 2 )2

S (0,R )

S (0,R )



S (0,R )

(6.14)

s22 R2 dS = , Z4 2(R2 − ρ 2 )3

S (0,R )



i = 1, 2 ,

s1 s22 ρ dS = , 3 2 Z 4 ( R − ρ 2 )2 s2i s2j Z2

dS =

1 + 2δ ij , 8( R 2 − ρ 2 )

i = 1, 2 ,

j = 1, 2 .

Thus the required result (6.6) is deduced. Formulae (6.7)–(6.10) are proved in a similar way. The following auxiliary result is used in Theorem 6.2. Lemma 6.1. For the solution of the Lamé equation u the properties   ∂u 1 ∂u 2 = 0, Δ + ∂n ∂t   ∂u 1 ∂u 2 =0 − Δ ∂t ∂n hold. Proof. It is known [96] that 

 ∂u 1 ∂u 2 = 0, − ∂x2 ∂x1   ∂u 1 ∂u 2 + = 0. Δ ∂x1 ∂x2

Δ

Rewriting these equations in the coordinates (n, t) and taking the corresponding lin­ ear combinations we get the result.

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80 | 6 The generalized Poisson formula for the Lamé equation Theorem 6.2. For any analytical in the domain G solution of equation (6.1) and for ev­ ¯ the following auxiliary mean value relation holds at an ery circle K (x0 , R), lying in G arbitrary point x, x ∈ K (x0 , R): 1 ω2 R

 S ( x 0 ,R )

  δ ij u j (y) ρ ˆs i ˆs j − dS − 2 Z4 2(R2 − ρ 2 )ω2 R = −Δu i (x)

where



A = a ij



cos (2φ + ψ) = i,j =1,2 sin (2φ + ψ)

Proof. Let us consider the integral  1 I= ω2

S ( x 0 ,R )

 S ( x 0 ,R )

a ij u j (y) dS Z3

1 + 2/α , 16(R2 − ρ 2 )

i, j = 1, 2 ,

sin (2φ + ψ) − cos (2φ + ψ)

(6.15)

  δ ij u j (y) ˆs1 ˆs j − dS . 2 Z4

By Theorem 6.1, this integral can be expressed as a series of differential operators. Since the function u i is biharmonic and the first equation of the Lamé system in the coordinates (n, t) is

2 2 2 2 ∂ 2 ∂ ∂ Δu 1 + α cos φ + sin φ − sin 2φ u1 ∂n2 ∂t2 ∂n∂t

2 1 ∂2 ∂2 ∂ + +α − cos 2φ u2 = 0 , sin 2φ 2 ∂n2 ∂t2 ∂n∂t then the integral I is transformed to I=

  ρ2 1 + 2/α cos 2φ u 1 + sin 2φ u 2 − Δu 1 2(R2 − ρ 2 )3 16(R2 − ρ 2 )     ρ cos 2φ ∂u 1 ∂u 2 ρ sin 2φ ∂u 2 ∂u 1 + + − . + 4(R2 − ρ 2 )2 ∂n ∂t 4(R2 − ρ 2 )2 ∂n ∂t

(6.16)

The same transformations are made with the integrals   1 s1 u 1 + s2 u 2 s1 u 2 − s2 u 1 1 dS, dS . 3 ω2 Z ω2 Z3 S ( x 0 ,R )

S ( x 0 ,R )

On the basis of Theorem 6.1, these integrals are expanded and then Lemma 6.1 is ap­ plied:    1 s1 u 1 + s2 u 2 ρ 1 ∂u 1 ∂u 2 dS = u + + , 1 ω2 Z3 ( R 2 − ρ 2 )2 2(R2 − ρ 2 ) ∂n ∂t S ( x 0 ,R )    1 s1 u 2 − s2 u 1 ρ 1 ∂u 2 ∂u 1 dS = u + − . (6.17) 2 ω2 Z3 ( R 2 − ρ 2 )2 2(R2 − ρ 2 ) ∂n ∂t S ( x 0 ,R )

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Extracting the expressions in the braces in (6.17) and substituting them into (6.16) we obtain the first relation (i = 1) of the system (6.15). The second relation (i = 2) is proved analogously. Theorem 6.3. For any analytical in G solution u(x) of the Lamé equation and for any circle S(x0 , R) ⊂ G at an arbitrary interior point of K (x0 , R), the following mean value relations hold: u i (x) =

2π   2  u j (θ) 1 − k2  2 δ ij + ˆs i ˆs j dθ 2π j=1 σ Z 2k 0

+

1−k 2πσ

2

2π 2   j =1 0

(6.18)

b ij u j (θ) dθ Z 2k

i = 1, 2, where 

b ij



2 cos2 θ − k cos (θ + φ ) = sin 2θ + k sin (θ + φ )

sin 2θ + k sin (θ + φ ) 2 sin2 θ − k cos (θ + φ )

,

ˆs i , Z k are defined in formulae (6.2), (6.3), and σ = 1 +

2μ . λ+μ

Proof. Let i = 1, (the case i = 2 is deduced in a similar manner). We introduce the notation u i (y) = g i (y) ,

Δu i (y) = g1i (y) ,

i = 1, 2 ,

y ∈ S ( x0 , R) .

It is known [200] that any biharmonic function can be represented through two har­ monic functions u i ( x) = ( R2 − ρ 2 ) v i ( x) + w i ( x) , (6.19) where the functions v i and w i are the solutions to the following problems: ⎧ ⎨ Δw i (x) = 0 , x ∈ K ( x0 , R) , ⎩w (y) = g (y) , y ∈ S ( x0 , R) i i and

⎧ ⎨Δv i (x) = 0 ,  ⎩ ∂v i (y) + 1 v   ∂r r i

x ∈ K ( x0 , R) r=R

=

g 1i ( y ) − 4R

,

y ∈ S ( x0 , R)

The Poisson formula yields the solution to the problem (6.20):  (1 − k 2 ) R g i (y) dS . wi = 2π Z2

.

(6.20)

(6.21)

(6.22)

S ( x 0 ,R )

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82 | 6 The generalized Poisson formula for the Lamé equation As to the problem (6.21) – we treat it with the help of the separation of variables [138]. v i (ρ, φ ) =

∞    ρ i ρn A in cos nφ + B in sin φ , A0 + n − 1 2 R (n + 1) n =1

A in = −

2π

1 4πR

g1i (y) cos nθ dθ,

A0i = −

1 4πR

0

B in = −

2π

g1i (y) dθ , 0

2π

1 4πR

g1i (y) sin nθ dθ . 0

Then 1 v i (ρ, φ ) = − 4π

2π

g1i (y) 0

 ∞   kn 1 dθ . cos n(θ − φ ) + n+1 2 n =1

The change of integration and summation order is possible since for k < 1 the series is uniformly convergent with respect to α = θ − φ. It is not difficult to calculate its sum [63] 2π k 1 1 − t2 v i (ρ, φ ) = − g1i (y) dt dθ. 16πk 1 − 2t cos α + t2 0

0

In the region [0; 2π]×[0; k ] the integrated functions are continuous, so we can change the order of integration ⎡ ⎤ k 2π 1 g ( y ) 1 ⎢ ⎥ i v i (ρ, φ ) = − dθ⎦ dt . ⎣(1 − t2 ) 16πk 1 − 2t cos α + t2 0

0

Now in the square brackets is the Poisson integral formula for the harmonic function Δu i (indeed, Δ(Δu i ) = 0); therefore 1 v i (ρ, φ ) = − 8kR

ρ

Δu i (t1 , φ ) dt1 .

0

But for any point x ∈ K (x0 , R), x = (t1 , φ ), the value of Δu i (x) is given by Theo­ rem 6.2. Hence, we can write for i = 1: 2 v1 (ρ, φ ) = πkσ R2

2π

k 2

(1 − t ) 0

1 − πkσ R2 1 + πkσ R2

0

ˆs1 ˆs2 (ˆs21 − 0.5) u1 + 4 u2 Z 4t Zt

dθ dt

k 2π  1  t 4 cos ( θ + φ )u 1 + sin ( θ + φ ) u 2 dθ dt Zt 0

0

k

2π

t 0

2 0

 1  cos 2φ u 1 + sin 2φ u 2 dθ dt , Z 4t

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where Z 2t = 1 − 2t cos α + t2 . Changing the order of integration we can then calculate the integrals over t on the basis of tables [63]. Adding the expression obtained for v i (x) with the weight (1 − k 2 ) and for w i (x) from (6.22), we get the desired relation. Corollary 6.1. If k = 0 (i.e. x = x0 is the center of the circle K (x0 , R)), then relations (6.19) become the mean value relations for the Lamé equation.

6.1.2 Poisson formula for displacements in polar coordinates Let us consider the same plane elasticity problem in the disk D(0, R): μΔu(x) + (λ + μ ) grad div u(x) = 0 ,

x ∈ D(0, R) ,

u(y) = g(y) ,

y ∈ S(0, R) ,

(6.23)

where u = (u 1 , u 2 )T is the displacement column vector which is prescribed on the boundary as a column vector g = (g1 , g2 )T , λ and μ are the elasticity constants. Let x = reiθ be a point in the disk Disk(0, R), and let ρ = Rr . The Poisson integral formula for the solution to the Lamé equation (6.23), presented in the previous section can be rewritten as follows: 2π

u(reiθ ) =

K (ρ; θ − φ)B(ρ; θ, φ)g(Reiφ )dφ ,

(6.24)

0

where K (ρ; θ − φ) is the Poisson kernel of the Laplace equation K (ρ; θ − φ) =

∞ 1 − ρ2 1 1  k 1 = + ρ cos[k (θ − φ)] , 2π 1 − 2ρ cos(θ − φ) + ρ 2 2π π k=1

and the matrix B has the form B =I+

λ+μ λ + 3μ



Q11 Q21

Q12 Q22

,

(6.25)

with the entries given explicitly by cos(2φ) − 2ρ cos(θ + φ) + ρ 2 cos(2θ) , 1 + ρ 2 − 2ρ cos(θ − φ) sin(2φ) − 2ρ sin(θ + φ) + ρ 2 sin(2θ) , Q12 = sin(2φ) − ρ sin(θ + φ) + 1 + ρ 2 − 2ρ cos(θ − φ) Q11 = cos(2φ) − ρ cos(θ + φ) +

and Q22 = −Q11 , Q21 = Q12 , I being an identity matrix.

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84 | 6 The generalized Poisson formula for the Lamé equation This form of the Poisson-type integral formula is simple and convenient to use in numerical simulations. It is seen that in contrast to the Laplace equation case, the matrix kernel has lost the nice property of depending only on the difference of the angles θ and φ. However, it is possible to find a linear transformation which leads to a Poisson integral formula with a matrix kernel depending only on the difference θ − φ. It turns out that this can be done by a proper transformation of the vector u = (u 1 , u 2 )T to polar coordinates. So let us turn to the expansion of our displacement vector u in polar coordinates u = u r er + u θ eθ , where er , eθ are unit vectors in directions r and θ, respectively. Then, the vectors (u 1 , u 2 )T and (u r , u θ )T are related through the rotation





u 1 (r, θ) cos θ − sin θ u r (r, θ) , = sin θ cos θ u 2 (r, θ) u θ (r, θ) and conversely,



u r (r, θ) T u 1 ( r, θ ) = Rθ u θ (r, θ) u 2 (r, θ)

where we use the notation for the rotation matrix

cos θ − sin θ Rθ = , sin θ cos θ and RTθ means the transpose to Rθ . Theorem 6.4 (Poisson formula in polar coordinates). The displacements in polar co­ ordinates, (u ρ , u θ ), satisfy the following integral relation:

2π u r (r, θ) L11 (ρ; θ − φ) = u θ (r, θ) L21 (ρ; θ − φ) 0

L12 (ρ; θ − φ) L22 (ρ; θ − φ)



g r (Reiφ ) dφ , g θ (Reiφ )

(6.26)

where the entries L ij (ρ; θ − φ), i, j = 1, 2, are explicitly given below. Proof. The Poisson integral formula (6.24) can be obviously rewritten as follows:

2π u r (r, θ) 1 + βQ11 T = Rθ K (ρ; θ − φ) u θ (r, θ) βQ12 0

βQ12 g r (Reiφ ) Rφ dφ 1 − βQ11 g θ (Reiφ )

λ+ μ λ +3μ .

where β = After obvious transformations we come to the desired form of the Poisson integral formula



2π u r (r, θ) G11 G12 g r (Reiφ ) 1 = K (ρ; θ − φ) dφ u θ (r, θ) G21 G22 g θ (Reiφ ) λ + 3μ 0

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6.1 Plane elasticity

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where the entries of the new matrix kernel G = G(θ − φ) are G11 = 2(λ + 2μ ) cos(θ − φ) − (λ + μ )ρ cos(θ − φ) − 2ρ + ρ 2 cos(θ − φ) , 1 + ρ 2 − 2ρ cos(θ − φ) (1 − ρ 2 ) sin(θ − φ) , G12 = 2μ sin(θ − φ) − (λ + μ ) 1 + ρ 2 − 2ρ cos(θ − φ) (1 − ρ 2 ) sin(θ − φ) , G21 = −2(λ + 2μ ) sin(θ − φ) − (λ + μ ) 1 + ρ 2 − 2ρ cos(θ − φ) cos(θ − φ) − 2ρ + ρ 2 cos(θ − φ) . G22 = 2μ cos(θ − φ) + (λ + μ )ρ − (λ + μ ) 1 + ρ 2 − 2ρ cos(θ − φ) + (λ + μ)

We expand the matrix kernel in the Fourier series. In the expansion below, we use the following formulae simply obtained via differentiations: ∞  1 − ρ2 = 1 + 2 ρ k cos(kθ) , 1 + ρ 2 − 2ρ cos θ k =1 ∞  ρ sin θ = ρ k sin(kθ) , 1 + ρ 2 − 2ρ cos θ k=1 ∞ ρ (cos θ − 2ρ + ρ 2 cos θ)  k = kρ cos(kθ) , (1 + ρ 2 − 2ρ cos θ)2 k =1 ∞  ρ sin θ(1 − ρ 2 ) = kρ k sin(kθ) . (1 + ρ 2 − 2ρ cos θ)2 k =1

Introducing the notation L(ρ; θ − φ) =

1 K (ρ; θ − φ)G(ρ; θ − φ) λ + 3μ

and expanding in the series yields L11 =

∞ ρ 1  + λ11 (ρ, k )ρ k cos[k (θ − φ)] , 2π π k =1

L12 =

∞ 1  λ12 (ρ, k )ρ k sin[k (θ − φ)] , π k =1

L21 =

∞ 1  λ21 (ρ, k )ρ k sin[k (θ − φ)] , π k =1

L22 =

∞ ρ 1  + λ22 (ρ, k )ρ k cos[k (θ − φ)] , 2π π k =1

(6.27)

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86 | 6 The generalized Poisson formula for the Lamé equation where we use the notation

% $ 1 2(λ + 2μ ) k (λ + μ )(1 − ρ 2 ) λ11 (ρ, k ) = 2μρ + + , 2(λ + 3μ ) ρ ρ % $ 1 2μ k (λ + μ )(1 − ρ 2 ) − 2μρ − , λ12 (ρ, k ) = 2(λ + 3μ ) ρ ρ % $ 1 2(λ + 2μ ) k (λ + μ )(1 − ρ 2 ) 2(λ + 2μ )ρ − − , λ21 (ρ, k ) = 2(λ + 3μ ) ρ ρ % $ 1 2μ k (λ + μ )(1 − ρ 2 ) 2(λ + 2μ )ρ + − . λ22 (ρ, k ) = 2(λ + 3μ ) ρ ρ

Thus the desired Poisson integral formula reads

2π u r (r, θ) L11 (ρ; θ − φ) = u θ (r, θ) L21 (ρ; θ − φ) 0

L12 (ρ; θ − φ) L22 (ρ; θ − φ)



g r (Reiφ ) dφ . g θ (Reiφ )

This formula will be repeatedly used in the subsequent sections for the construction of numerical algorithms.

6.2 Generalized spatial Poisson formula for the Lamé equation We turn now to the 3D case (see [191]). Suppose for simplicity that we have a ball B(0, R) of radius R centered at the origin of coordinates. Let M be an arbitrary point with the spherical coordinates (ρ, θ, φ) inside the ball and let P be a point on the sphere S = ∂B(0, R). We denote by W the distance |M − P| and let ˆs = (ˆs1 , ˆs2 , ˆs3 ) be the direction cosines of the vector P − M. We need also a triple of coordinate axes (n1 , n2 , n3 ) cen­ tered at the point M and oriented with respect to the axes (x1 , x2 , x3 ) so that n1 coin­ cides with the vector M − O. In these new coordinates, the direction cosines of the vector P − M are s = (s1 , s2 , s3 ) and those of the vector P − O are α = (α 1 , α 2 , α 3 ). From geometrical considerations, we write down the relations W 2 = R2 + ρ 2 − 2Rρα 1 , 2

2

2

R = W + ρ + 2Wρs1 ,

(6.28) (6.29)

hence W = R(−ks1 + J k ) , where

9 J k = 1 − k 2 + k 2 s21 ,

k = ρ/R .

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The vectors s and ˆs are related by s = AT ˆs , where

⎛

sin θ cos φ ⎜ A = ⎝ sin θ sin φ cos θ

cos θ cos φ cos θ sin φ − sin θ

(6.30) ⎞ − sin φ ⎟ cos φ ⎠ , 0

(6.31)

and s1 = W −1 (Rα 1 − ρ ) ,

s2 = W −1 Rα 2 ,

s3 = W −1 Rα 3 .

(6.32)

To prove the main theorem we need the integrals of the solution over the averaging measure represented as a series of operators. We first prove Theorem 6.5. For any analytic function u which is biharmonic in a ball the following expansions hold. First,  s i u(P) 2ρδ i1 ∂u (M ) 1 1 dS = u (M ) + 4πR2 W4 3R(R2 − ρ 2 )2 ∂n i 3R(R2 − ρ 2 ) S (6.33) ∞  (−1)p−3 (2ρ )p−3 ∂ ∂ p−3 Δu (M ) + , p −3 R 2(2p − 1)!! ∂n i ∂n1 p =3 next 1 4πR2

 S

s2i u (P) 5R2 − (1 − 8δ i1 )ρ 2 ∂u (M ) 2ρ + 4ρδ i1 dS = u ( M ) + W5 15R(R2 − ρ 2 )3 ∂n1 15R(R2 − ρ 2 )2 +

∞  Δu (M ) (−1)p (2ρ )p−4 ∂2 ∂ p−4 Δu (M ) + , p −4 2 2 30R(R − ρ ) p=4 R 2(2p − 1)!! ∂n2i ∂n1

(6.34) and 1 4πR2

 S

s i s j u(P) ∂u (M ) 2ρ (1 − δ i2 ) ∂2 u(M ) 1 dS = + W5 ∂n i 15R(R2 − ρ 2 )2 ∂n i ∂n j 15R(R2 − ρ 2 ) ∞  (−1)p (2ρ )p−4 ∂2 ∂ p−4 Δu (M ) . + p −4 R 2(2p − 1)!! ∂n i ∂n j ∂n1 p =4

(6.35)

In the above formulae, i = 1, 2, 3, and j > i. Finally,  u(P) 3R2 + ρ 2 ∂u (M ) 2ρ Δu (M ) 1 dS = u (M ) + + , 2 5 4πR W 3R(R2 − ρ 2 )3 ∂n1 2R(R2 − ρ 2 )2 6R(R2 − ρ 2 ) S

(6.36) where δ ij is Kronecker’s symbol.

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88 | 6 The generalized Poisson formula for the Lamé equation Proof. It suffices to prove any of formulae (6.33)–(6.36), say, formula (6.34). Let us ex­ pand the function u (P) by its Taylor series expansion in the coordinates (n1 , n2 , n3 ): u (P) = u (M + Ws) = u (M ) + Wp + p!



3  ∂u (M ) si W + · · · ∂n i i =1

∂ p u(M ) c c c ( c 1 , c 2 , c 3 ) c1 c2 c3 s1 1 s2 2 s3 3 + · · · ∂n ∂n ∂n 1 2 3 c1 + c2 + c3 = p

where (c1 , c2 , c3 ) =

(6.37)

( c 1 + c 2 + c 3 )! . c1 !c2 !c3 !

Let us multiply both sides of (6.37) by the weight s21 /W 5 and integrate them over the sphere S(0, R). To this end, we pass on to integration with respect to the spherical coordinates (W, θ , φ ) at the point M. Note that the vector s can be represented as s1 = cos φ sin θ ,

s2 = sin φ sin θ ,

s3 = cos θ .

The Jacobian of the transition is given by the following lemma. Lemma 6.2. A surface element dS at a point P of the sphere S is given by dS =

W 2 sin θ dφ dθ . Jk

The proof follows from the known formulae for the change of variables [90]. Hence, (6.37) can be written as  2 ∞   s i u(P) 1 ∂ p u(M ) 1 dS = I + (c1 , c2 , c3 ) c1 c2 c3 0 2 5 4πR W p! c1 +c2 +c3 =p ∂n1 ∂n2 ∂n3 p =4 S

×

p −3 j =0

j C p−3 R p−3 (−k )p−3−j

π 2π

p −1 − j + c 1 c 2 c 3 j −1 s2 s3 J k sin θ

s1

dθ dφ .

0 0

(6.38) Here we used the expansion W p = R p (−ks1 + J k )p =

p 

p− j j Jk

j

R p C p (−k )p−j s1

.

j =0

The symbol I0 stands for the sum of first terms with p ≤ 3. Now it is convenient to partition the series over p into sums over even and odd terms and consider each case separately. We start with the case p = 2p . It is easy to verify that the integrals of the form π 2π

p − j + c 1 −1 c 2 c 3 j −1 s2 s3 J k sin θ

s1

dθ dφ

0 0

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6.2 Generalized spatial Poisson formula for the Lamé equation |

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89

differ from zero only if all the powers of the directional cosines s i are even. Hence if p = 2p , then for the integrals different from zero we find c i = 2c i ,

j = 2j + 1

i = 1, 2, 3 ,

and we have an even power of J k , which can then be multiplied out. Note that

2j



J k = (1 − k 2 + k 2 s21 )j =

j 

2( j − l )



C lj (1 − k 2 )l (k 2 )j −l s1

.

l =0

Therefore, we can integrate the remaining cosines s i by the formula [63] 1 4π

π 2π

2m

2m

2m 3

s1 1 s2 2 s3

sin θ dθ dφ =

0 0

(2m1 − 1)!!(2m2 − 1)!!(2m3 − 1)!! . (2m1 + 2m2 + 2m3 + 1)!!

Let us consider the inner sum in (6.38), i.e. the sum as above, takes the form (2p − 1)!! (4p − 2l − 1)!!



#

c 1 + c 2 + c 3 = p . This sum transformed



(c 1 , c 2 , c 3 )

c 1 + c 2 + c 3 = p

∂2p u 2c 1

2c 3

2c 2

∂n1 ∂n2 ∂n3

(2p − 2l + 2c 1 − 3)!! . (2c 1 − 1)!!

(6.39)

Let us introduce in (6.39) a parameter β such that

(2p − 2l + 2c 1 − 3)!! ∂ p − l −1 = (−2)p −l−1 p −l−1 (2c1 − 1)!! ∂β





1

.



β (2c1 +1)/2

β =1

Placing the differentiation with respect to the parameter β outside the sign of summa­ tion over c i , we can then contract the sum in (6.39) to get

(2p − 1)!! ∂ p − l −1 (2p − 2l − 1)!! ∂β p −l−1  ⎫ ⎧ 

p ⎪ ⎪  ⎬ ⎨ 1 2 2 2 1 ∂ ∂ ∂ p − l −1  9  ×⎪ u ⎪ (−2) 2 + 2 + 2  ∂n2 ∂n3 ⎭ ⎩ β β ∂n1 

(6.40) . β =1

Now let us consider the differentiation of the expression in braces with respect to β. In both cases, we may change the order of differentiation, because the sum c i is finite and β is an independent parameter. Since u is a biharmonic function, i.e.

∂2 ∂2 ∂2 2 + 2 + ∂n1 ∂n2 ∂n23

2

u = 0,

then at the point β = 1, all the terms in (6.40) with l = 0 vanish and only terms in which we take derivatives of the parenthetical factor in (6.40) will not vanish in the remaining sum. As this happens, its power reduces.

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90 | 6 The generalized Poisson formula for the Lamé equation As a result, we obtain in (6.40) the following terms:

(2p − 1)!! p −1 ∂ 2p −2 p ! 2p −2 Δu . 2 (4p − 1)!! ∂n1

It remains only to add terms transformed by the same procedure and the nonzero first terms with p < 3 to the expression for even p in order to get the desired formula. The first term can be integrated directly. The corresponding results are given as the following lemma. Lemma 6.3. The following table of integrals is true:  1 1 3R2 + ρ 2 dS = , 2 5 4πR W 3R(R2 − ρ 2 )3 S  s1 2ρ 1 dS = , 4πR2 W 4 3R(R2 − ρ 2 )2 S  2 si 1 1 dS = , i = 1, 2, 3 , 4πR2 W 3 3R(R2 − ρ 2 ) S  2 1 s1 5R2 + 7ρ 2 dS = , 4πR2 W 5 15R(R2 − ρ 2 )3 S

1 4πR2



s2j W5

dS =

S

1 4πR2

 S

1 4πR2

 S

1 4πR2



S

1 4πR2



s1 s2j

dS =

2ρ , 15R(R2 − ρ 2 )2

s4i 1 dS = , W3 5R(R2 − ρ 2 ) s2i s2j W3

S

(6.41)

s31 2ρ dS = , W4 5R(R2 − ρ 2 )2

W4 

j = 2, 3 ,

1 5R4 + 10R2 ρ 2 + ρ 4 dS = , W7 5R(R2 − ρ 2 )5

S

1 4πR2

5R2 − ρ 2 , 15R(R2 − ρ 2 )3

dS =

1 , 15R(R2 − ρ 2 )

j = 2, 3 , i = 1, 2, 3 , i = 1, 2,

j = 2, 3,

i < j,

where s i , W, and ρ are defined in (6.28)–(6.32). Proof. Let us prove the first of formulae (6.41). The proof of each subsequent integral is based on the preceding one and on (6.28)–(6.32). Let us represent the surface element of the unit sphere as dΩ = JdW k dθ .

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6.2 Generalized spatial Poisson formula for the Lamé equation | 91

Taking into account that 9 9 W k = 1 + k 2 − 2kα 1 = 1 + k 2 − 2k cos φ sin θ

we have J=

Wk W k α3 = . k sin φ kα 2

Then  Ω

dΩ =2 W k5

1 −k

1+ k

1 dW k kW k4

⎧ ⎨ ⎩



1 − cos2 θ −

1 + k 2 − W k2 2k

(−1/2)

2 ⎫ ⎬

⎭

sin θ dθ .

Let us consider the inner integral. It is known (see, e.g. [3]) that ⎧

⎫(−1/2) ⎨ 2 2⎬ 2 1 + k − W k 1 − cos2 θ − sin θ dθ ⎩ ⎭ 2k ⎛⎧ ⎞

⎫(−1/2) 2 2⎬ ⎨ 2 1 + k − W ⎜ ⎟ k = arcsin ⎝ 1 − cos θ⎠ . ⎭ ⎩ 2k Because the plane (M, O, P) coincides with (n1 , 0, n3 ), we have at the ends of the in­ tervals of integration, : :

2

2 ; ; ; ; 1 + k 2 − W k2 1 + k 2 − W k2 < cos θ = 1 − , − cos θ = 0, and B nl =

(2n + 1)(n − l)! 2π(n + l)!

2π π

φ1 (θ , φ )P(nl) (cos θ ) sin lφ sin θ dθ dφ .

0 0

The values of the function v are not changed if we introduce a parameter t and integrate (6.52) as follows: v(ρ, θ, φ) = −

1 4k 3/2

k

∞ 

t 1 /2

0

t n Y n dt .

(6.53)

n =0

The functional series in (6.53) can be integrated term by term, since we have an absolutely and uniformly convergent series (t < 1 always holds). The resultant series is easy to contract [6], so we get v(ρ, θ, φ) = −

R 16πρ 3/2

ρ



2π π

t1

0

φ1 (R, θ , φ )

0 0

R2 − t21 sin θ dθ dφ dt1 . W t31

(6.54)

Adding properly weighted expressions for v and w in (6.51) and (6.54), we arrive at the desired relation. Corollary 6.2. If k = 0, then (6.48) reduces to a known mean value relation for bihar­ monic functions at the center of the ball: ⎧ ⎫ ⎪  ⎪ ⎨ ⎬ 2  1 R φ φ . u (0, θ, φ) = 0 dΩ − 1 dΩ ⎪ ⎪ 4π ⎩ 6 ⎭ Ω

Ω

To prove this, it is sufficient to take the limit in (6.48) as k → 0. Theorem 6.8. The components of the displacement vector in the Lamé equation satisfy the following mean value relation at an arbitrary interior point in the ball:  1 k ji (x, y)u i (y) dS(y) , (6.55) u j (x) = 4πR2 S

δ ij R(R2 − ρ 2 ) k ji (x, y) = W3



R(R2 − ρ 2 )ρ (1−3σ )/(2σ ) + σ

ρ

Φ 1 (t1 ) + Φ 2 (t1 ) + Φ 3 (t1 ) (1− σ )/(2σ )

0

t1

dt1 ,

where Φ 1 (t1 ) =

δ ij , 2W t31

Φ2 (t1 ) = −3t1

(a i1 ˆs j + a j1 ˆs i ) − δ ij [a k1 ˆs k ] , W t41

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96 | 6 The generalized Poisson formula for the Lamé equation and Φ 3 (t1 ) =

3(5ˆs i ˆs j − 2δ ij )(R2 − t21 ) . 2W t51

Here we used the notation [a k1 ˆs k ] =

#3

k =1

a k1 ˆs k .

Proof. The displacement vector u is termwise biharmonic; therefore we can take ad­ vantage of the previous theorem. Let us consider (6.54). The inner integral is the Pois­ son kernel in R3 for the harmonic function Δu i , (since Δ(Δu i ) = 0): 1 vj (ρ, θ, φ) = − 3/2 4ρ

ρ

 Δu j (t1 , θ, φ) t1 dt1 .

(6.56)

0

If we substitute the values of Δu j (t1 , θ, φ) from (6.45) into (6.56) and solve the resultant equation for vj , we prove the theorem. Thus, for j = 2 (other versions are proved in a similar manner) formula (6.45) is Δu 2 (t1 , θ, φ) = −

2u 2 (t1 , θ, φ) + f1 (t1 , θ, φ) , (R2 − t21 )σ

where the sum of all the integrals is designated by the function f1 . Now we substitute the last mentioned expression in (6.56) and obtain 1 v2 (ρ, θ, φ) = − 3/2 4ρ

ρ



t1 f1 (t1 , θ, φ) dt1 +

0

ρ √

1 2ρ 3/2 σ

0

t1 u 2 dt1 . R2 − t21

By virtue of (6.49) and (6.51), the second term can be transformed, namely 1 v2 (ρ, θ, φ) = − 3/2 4ρ

ρ



t1 f1 (t1 , θ, φ) dt1 +

0

+

R 8ρ 3/2 σ πR2

ρ  0 S

1 2ρ 3/2 σ

ρ



t1 v2 (t1 , θ, φ) dt1

0

√

u 2 t1 dt1 dS . W t31

Let us introduce the notation 1 f = −f1 + 2πσ R

 S

then ρ

3 /2

1 v2 (ρ, θ, φ) − 2σ

ρ 0



u2 dS , W t31

1 t1 v2 (t1 , θ, φ) dt1 = 4

ρ



t1 f (t1 ) dt1 .

0

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6.2 Generalized spatial Poisson formula for the Lamé equation | 97

After we differentiate both sides of this equality with respect to ρ and collect the similar terms, we obtain   1 3 1 1 f (ρ, θ, φ) . v2 (ρ, θ, φ) + − v2 (ρ, θ, φ) = ρ 2 2σ 4ρ This is an ordinary differential equation. Its general solution is 1 v2 (ρ, θ, φ) = ρ (1−3σ )/(2σ ) 4

ρ

f (t1 , θ, φ) (1− σ )/(2σ )

0

t1

dt1 .

Integrating in new variables t1 = tR we can rewrite v2 (ρ, θ, φ) =

1 (1−3σ )/(2σ ) k 4

k 0

f (tR, θ, φ) dt . t(1−σ )/(2σ )



By definition, σ = 1 + 2/α and α = 1 + λ/μ; hence, σ > 0, t < 1, and the integral has no singularities, except when k = 0. At k = 0 it has a removable singularity to be considered below. We have thus expressed vi (ρ, θ, φ) and wi (ρ, θ, φ) in terms of their values on the sphere, and the theorem is proved, because u i = (R2 − ρ 2 )vi + wi . To verify (6.49), let us compare it with the known value for u i (0, θ, φ). Corollary 6.3. In the limit, as k → 0, the above mean value relation becomes a known relation for the system of the Lamé equations at the center of the ball:   3 u i (0) = (2 − α )u i + 5αs i s j u j dΩ . 8π(3 + α ) Ω

Proof. When taking the limit ρ → 0 in (6.55), we have to check those integrands whose numerators contain t1 to the power (σ − 1)/(2σ ). The other functions contain t1 to higher powers, and therefore their integrals are equal to zero in the limit, as ρ → 0. Consider the integral k

1 k (3σ −1)/(2σ )

0





t(σ −1)/(2σ ) dt (1 + t2 − 2t cos α )3/2

(6.57)

and rewrite its kernel in the form



tC3 t(σ −1)/(2σ ) = t(σ −1)/(2σ ) + t(σ −1)/(2σ ) , (1 + t2 − 2t cos α )−3/2 + 1 W t3 where C3 =

3 

C3i t i−1 (t − 2 cos α )i .

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98 | 6 The generalized Poisson formula for the Lamé equation Then the integral in (6.57) is



k (1−3σ )/(2σ )

k









t(σ −1)/(2σ ) + t(σ −1)/(2σ )

0

tC3 dt W t−3 + 1

2σ = + k (1−3σ )/(2σ ) 3σ − 1

k 0





t(3σ −1)/(2σ ) C3 dt . W t−3 + 1

The other two similar integrals can be transformed in a similar way: k

(1−3σ )/(2σ )

k 0





k (1−3σ )/(2σ )

k 0





t(σ −1)/(2σ ) 2σ dt = + k (1−3σ )/(2σ ) 5 3σ − 1 Wt t

( σ −1)/(2σ )

W t7

where Cq =

q 

dt =

2σ + k (1−3σ )/(2σ ) 3σ − 1

C iq t i−1 (t − 2 cos α )i ,

k 0

k 0









t(3σ −1)/(2σ ) C5 dt , W t−5 + 1 t(3σ −1)/(2σ ) C7 dt , W t−7 + 1

q = 5, 7 .

i =1

Now let us take the limit in (6.55) as k → 0. We obtain  3 u j (0, θ, φ) = (5ˆs i ˆs j + (σ − 2)δ ij )u i dΩ . (3σ − 1)ω3 Ω



Putting σ = 1 + 2/α, we obtain the desired results.

6.3 An alternative derivation of the Poisson formula In Sections 6.1 and 6.2, we have derived the generalized Poisson formula for the Lamé equation starting from the functional power expansion. In this section, we show that the proof given in [96] and [101] leads to the same mean value relation. Let us consider the first boundary value problem for the Lamé equation in the ball Δ∗ u(x) = 0 , u(y) = f (y) ,

x ∈ B(0, R) , y ∈ S(0, R) .

(6.58)

A general representation of the solution to (6.58) is already known in the form of a power series (see, e.g. [101]). Under some general assumptions about the function f (for instance, it should be assumed that f can be expanded into a Fourier series in # i spherical functions: f i (y) = ∞ n =0 A n ( y )) this representation has the form  n ∞ ∞   r ∂d n+2 i An + ( r2 − R2 ) D n +2 , i = 1, 2, 3 , (6.59) u i (x) = R ∂x i n =0 n =0

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6.3 An alternative derivation of the Poisson formula | 99

where x = (r, θ, φ), λ+μ Dn = − , 2[λ(n − 1) + μ (3n − 2)]

  n  3  ∂ r dn = A in , ∂x R i i =1

and A in are the standard spherical functions of order n, i.e.  2n + 1 A in (θ, φ) = P n (cos 𝛾)f i (θ, φ)dS y , 4πR2 S (0,R )

where P n (ξ ) is the Legendre polynomial of order n and 𝛾 is the angle between the vectors x and y: 3 1  xk yk . cos 𝛾 = rR k=1 The derivation of the expansion given by Thomson is reported in [101]. A different approach to get (6.59) based on potential theory is given in [96]. Using this representa­ tion, it is possible to obtain the spherical mean value relation. This was done in [96]. Let us formulate the relevant statement given in [96]. Theorem 6.9. Under the assumptions f i ∈ C(S(0, R)), the unique regular solution to the problem (6.58) satisfies the spherical mean value relation  k ij (x, y)f j (y)dS(y) , (6.60) u i (x) = S (0,R )

where 1 k ij = 4πR 1 

R2 − ρ 2 ∂2 Φ(x, y) δ ij + β 1 ( R2 − ρ 2 ) 3 |x − y| ∂x i ∂x j



,

3ρt cos 𝛾 R2 − ρ 2 t2 1 − − (R2 − 2Rρt cos 𝛾 + ρ 2 t2 )3/2 R R2

Φ(x, y) = 0

β1 =



λ+μ , 2(λ + 3μ )

α1 =



dt , t 1+ α 1

λ + 2μ . λ + 3μ

Here 𝛾 is the angle between the vectors x and y. Proof. The double-layer potential representation reads  T ∗ (x − y, n(y))g(y)dS(y) , u(x) =

x ∈ B(0, R) ,

(6.61)

S (0,R )

where T∗ kj ( x

1 − y, n(y)) = 4π A=



∂ 1 ∂r ∂r ∂ 1 δ kj A + 3B − AM kj ∂n(y) r ∂x k ∂x j ∂n(y) r

2μ , λ + 2μ

B=



∂ , n( y) ∂y

 1 , r

2(λ + μ ) , λ + 2μ

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100 | 6 The generalized Poisson formula for the Lamé equation and 3 9  ∂ ∂ ≡ , ρ = x21 + x22 + x23 , r2 = (x i − y i )2 , ∂n(x) ∂ρ i =1

  ∂ ∂ ∂ 1 ∂ ∂ − n k (x) = xj − xk . M kj , n( x) = n j ( x) ∂x ∂x k ∂x j ρ ∂x k ∂x j

The unknown density g satisfies the boundary integral equation  −g(z) + T ∗ (z − y, n(y))g(y)dS(y) = f (z) .

(6.62)

S (0,R )

The density g is constructed in the form of a series in standard spherical functions [96] and from (6.61) and (6.62) it follows that - m . ∞  m ∞   ρ λ+μ ρ A m + ( R2 − ρ 2 ) grad div Am , u(x) = R 2[m(λ + 3μ ) − λ − 2μ ] R m =0 m =2 (6.63) where A m = (A1m , A2m , A3m ). Standard contracting procedure applied to series (6.63) yields the desired result. An analogous approach is used in [11] to solve the same problem in 2D. Remark 6.1. Note that if we take the second derivatives of the function Φ(x, y) in the kernel of (6.60) we will come to the spherical mean value relation which we obtained in Theorem 6.8. Indeed, ∂2 Φ = ∂x i ∂x j

1  (R2 − ρ 2 t2 )ˆs i ˆs j 2δ ij 15 − 3 Q5 Q 0

6(x i ˆs j + x j ˆs i ) 3(R2 − ρ 2 t2 )δ ij − − Q4 Q5



dt t α 1 −1

,

where Q = (R2 − 2Rρt cos 𝛾+ ρ 2 t2 )1/2 . From this we come to (6.8) if we take the relation tx k = y k − Qˆs k into account and use the change of variables t1 = ρt. The same technique can be used to derive a spherical mean value relation for the sec­ ond boundary value problem Δ∗ u(x) = 0 , Tu(y) = f (y) ,

x ∈ B(0, R) , y ∈ S(0, R) ,

(6.64)

where the operator T is defined by T kj = λn k (y)

∂ ∂ ∂ + μn j (y) + μδ kj . ∂y j ∂y k ∂n(y)

For details, see [96].

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6.3 An alternative derivation of the Poisson formula | 101

Theorem 6.10. The unique (to within two arbitrary constants C and C1 ) regular solution to the problem (6.64) satisfies the following spherical mean value relations:  u(x) = K(x, y)f (y)dS(y) + C + |xC1 | , S (0,R )

where C, C1 are some arbitrary constants and  1 Tu(x) = K 1 (x, y)f (y)dS(y) , 4πρ S (0,R )

where 

  R2 − 3ρ 2 ∂2 Φ3 ∂Φ1 ∂Φ2 + xj − 2 ∂x i ∂x j ∂x i ∂x i



 2 ∂Φ3 ∂Φ1 ∂ ∂ Φ2 ∂2 Φ1 + xi 2ρ − Φ3 + ρ2 − − 2x i ∂x j ∂x j ∂ρ ∂x i ∂x j ∂x i ∂x j

1 Kij (x, y) = 8πμ

(Φ 1 + Φ 2 )δ ij +

and Kij1 (x, y) =

2 R2 − ρ 2 2 2 ∂ Φ 4 ( x, y ) δ + ( R − ρ ) . ij |x − y|3 ∂x i ∂x j

Here 1 $

Φ1 = 0

Φ3 =

R2 − ρ 2 t2 1 − Q(t) R

1 b− 1 ImΦ 0

%

dt , t

1 $

,

Φ0 = 0

α 1 = b 0 + ib 1 =

1 $

Φ2 = 0

3ρt cos 𝛾 R2 − ρ 2 t2 1 − − Q(t) R R2

R2 − ρ 2 t2 1 − Q(t) R

9 μ + i 2λ2 + 6λμ + 3μ2

2(λ + μ )

,

%

dt , t 1+ α 1

b2 =

%

dt , t2

Φ4 = Re(b 2 Φ0 ) ,

3λ + 4μ 1 +i 9 , 2 2 2λ2 + 6λμ + 3μ2

and Q(t) = (R2 − 2Rρt cos 𝛾 + ρ 2 t2 )3/2 .

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7 Spherical means for the stress and strain tensors In this chapter, we derive spherical mean value relations using the generalized Gauss formula (see [192]). This approach was used in [40] to deduce the mean value rela­ tion for the Lamé equation. Note that this approach directly gives the mean value re­ ¯ under some smoothness assumptions. In lation for an arbitrary sphere S(x, r) ⊂ G Section 7.1, we use this method to obtain the spherical mean value relation deduced in Section 5.2 by the series expansion. In Section 7.2, a mean value theorem that relates the displacements with the stress and strain components is presented. In Section 7.3, the spherical means of the stress components are related to the values of these com­ ponents at the center of the sphere. Note that throughout this chapter, we deal with the spherical mean value theo­ rems that relate the spherical means of solutions to the solutions at the center of a ball B(0, R). Therefore, we will simply write, say, u instead of u (0). It will be also conve­ nient to use in this chapter the notation sinh x and cosh x for the functions sinh(x) and cosh(x), respectively, and the summation convention (summation over the repeated index), for instance, u i,i means div u.

7.1 Spherical means for the displacement components through the displacement vector As in Section 5.2, we consider the equation μΔu + (λ + μ ) grad div u − νu = 0 ,

(7.1)

where u = (u 1 , u 2 , u 3 ) is a real-valued regular vector function. The coefficients λ, μ, and ν are assumed to be constants. In this chapter, we suppose that ν satisfies the following assumption: ks r < π , kp r < π , (A) 9  where k s = ν/μ and k p = ν/(λ + 2μ ). We multiply (7.1) by r2 /μ and integrate the result over the ball B(R) = B(0, R), 0 < r < R:   7 8 k 2s r2 u i dV = r2 u i,jj + αu j,ji dV . B(R)

B(R)

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7.1 Spherical means for the displacements | 103

Integrating the right-hand side of this equation by parts we get   . xj xi r2 u i dV = R2 u i,j + αu j,j dS k 2s R R B(R) S(R)  7 8 −2 x j u i,j + αx i u j,j dV .

(7.2)

B(R)

The surface integral in (7.2) can be transformed as follows:   7 . 8 xj xi 2 2 R u i,j + αu j,j dS = R u i,jj + αu j,ji dV R R S B(R)  = R2 k 2s u i dV .

(7.3)

B(R)

We integrate by parts the volume integral in the right-hand side of (7.2) and use relation (7.3). This yields 

k 2s



r2 u i dV − R2 k 2s B(R)

u i dV B(R)

 -

R2 u i + α

= −2 S(R)

 . xi xj (3 + α )u i dV . u j dS + 2 R

(7.4)

B(R)

We recall now (see Section 5.2) that u, the solution to (7.1), can be represented as u = up + us ,

(7.5)

where up , us are the functions solving the following equations: (Δ − k 2p )up = 0 ,

rot up = 0 ,

k 2s )us

div u = 0 ,

(Δ −

where

= 0,

(7.6)

s

9 k p = ν/(λ + 2μ ) ,

9 k s = ν/μ ,

u i ∈ C ∞ (G) .

(7.7)

We now use the volume mean value relation for the diffusion equations of the type (Δ − κ )u = 0 which were obtained under the assumption (A) in Section 2.2. This yields p p (as mentioned above, we use short notations u i and u si instead of u i (0) and u si (0), respectively):    p 4πR3 f (k p )u i + f (k s )u si = u i dV , (7.8) B(R) p 4π(f1 (k p )u i



+

f1 (k s )u si )

r2 u i dV ,

=

(7.9)

B(R)

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104 | 7 Spherical means for the stress and strain tensors where 1 f (k) = 3 R

R

r2 0

R

r4

f1 (k) = 0

shkr chkR shkR − , dr = kr (kR)2 (kR)3

(7.10)

R3 shkr shkR . dr = f (k )(R5 + 6 2 ) − 2R5 kr k (kR)3

(7.11)

We note that (7.8) and (7.9) are obtained as the sum of the mean value relations for up and us , i.e. from  shk p R p shk s R s (7.12) u + 4π u = udΩ , 4π ks R kp R Ω

2

4

when multiplying (7.12) by r (by r in the case of (7.9)) and then integrating over r. Formulae (7.8) and (7.9) allow us to escape the volume integrals in (7.4). Indeed, &

' p k 2s f1 (k s ) − k 2s R2 f1 (k p ) − 2(3 + α )R3 f (k p ) u i & ' + k 2s f1 (k p ) − k 2s R2 f1 (k s ) − 2(3 + α )R3 f (k s ) u si  xi xj 2 =− Ru i + α u j dS . 4π R S(R)

Substituting in this equality f1 (k p ), f1 (k s ), and f (k s ), f (k p ) we get the relation ob­ tained in Section 5.2: (7.13) N (1) u = W1 us + W2 up , where W1 , W2 and N 1 u are shk s R + 3βf (k s ) , ks R shk p R 5(λ + μ ) W2 = (1 + 2β ) − 6βf (k p ) , β = , kp R 2(λ + 4μ )    xi xj 1 (1 − β )δ ij + 3β u j (y)dS(y) . N1u = 4π R R W 1 = (1 − β )

S(R)

Now all the necessary results are obtained and we can formulate the main statement. Theorem 7.1. Assume that the parameter ν is chosen so that the assumption (A) is true for all S(x, r) ⊂ G. Then the solution to (7.1) satisfies the following spherical mean value relation:   shk p R sR (W1 − W2 )N r u + shk N1u ks R − kp R . (7.14) u(x) = shk R shk s R W2 − k p pR W1 ks R for any ball B(x, R) ⊂ G. Proof. Resolving the linear system of equations (7.13), (7.12), and (7.5) we get the desired relation. The theorem is proved.

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7.2 Mean value relations for the stress and strain tensors through the displacement vector 7.2.1 Mean value relation for the strain components The main results of this section are Theorems 7.2. and 7.3. To prove them we first obtain a set of preliminary statements. Let us introduce (see, e.g. [96]), the strain tensor 1 ε ik = (u i,k + u k,i ) . 2

(7.15)

Equalities (7.15), (7.5), and (7.13) give p

ε ik = ε ik + ε sik ,    1 xi xj p aδ + b u j,k dS . W1 u si,k + W2 u i,k = ij 4πR2 R R

(7.16) (7.17)

S(R)

We multiply (7.17) by R4 and integrate over R from 0 to ρ: ρ

ρ

W1 R4 u si,k dR 0

p

W2 R4 u i,k dR =

+ 0

1 4π

 (aδ ij R2 + bx i x j )u j,k dV . B (ρ)

The same procedure of integration by parts carried out in the previous section gives

 xi xj ρ2 xk p s aδ ij + b 2 u j dS f2 (k s )u i,k + f2 (k p )u i,k = 4π ρ ρ S (ρ)  1 − (2ax k u i + bx i u k + bu i x j δ ik )dV , 4π B (ρ)

where ρ

W1 R4 dR

f2 (k s ) = 0



ρ3 shk s ρ 5 = (6a − 3b ) 2 + ρ a f (k s ) − ρ 2 (2a − b ) , ks k 3s ρ f2 (k p ) = W2 R4 dR 0



ρ3 shk s ρ 5 = 6(a + 2b ) 2 + ρ (a + b ) f (k p ) − 2(a + b )ρ 2 . kp k 3p

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106 | 7 Spherical means for the stress and strain tensors The same is true for u k,i ; therefore, p

ε sik f2 (k s ) + ε ik f2 (k p )   ρa b (u i x k + u k x i )dS + x i x j x k u j dS = 8π 4πρ S (ρ)

(2a + b ) − 8π

 B (ρ)

S (ρ)

b (x k u i + x i u k )dV − 4π

(7.18)



u j x j δ ik dV . B (ρ)

To derive the desired mean value relation it remains to pass from the volume integrals in these equalities to surface integrals and use relations (7.16) and (7.17). Thus let us evaluate the volume integrals in (7.18). Lemma 7.1. The following representations are true:    p u i x k dV = 4π f3 (k p )u i,k + f3 (k s )u si,k ,

(7.19)

B (ρ)

ρ

ρ3 f (k )r dr = 2 k



4

f3 (k) = 0

or f3 (k) =

shkρ − 3f (k ) kρ

,

 1 5 ρ f (k) − f1 (k) . 2

p

p

Proof. The functions u i solve the diffusion equations; hence the derivatives of u i,k also solve the equation p p Δu i,k − k 2p u i,k = 0 . p

Therefore, u i,k satisfy the relations p

f (k p )u i,k =

1 4πr3



p

u i,k dV . B(r)

Integrating by parts, the right-hand side yields  1 p p f (k p )u i,k = u i x k dΩ . 4πr2

(7.20)

Ω

Multiplying this relation by r and integrating over r, 0 ≤ r ≤ ρ gives 4

p u i,k

ρ

f (k p )r4 dr =

1 4π

0

or p

f3 (k p )u i,k =

1 4π



p

u i x k dV , B (ρ)



p

u i x k dV . B (ρ)

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The same arguments for u si,k lead to f3 (k s )u si,k (0) =

1 4π



u si x k dV . B (ρ)

We conclude by summing the two last equalities. Note that from (7.19), it follows that    p (u i x k + u k x i )dV = 8π f3 (k p )ε ik + f3 (k s )ε sik .

(7.21)

B (ρ)

Remark 7.1. We will need a volume integral which can be obtained by proceeding as in Lemma 7.1, where equality (7.20) is multiplied by r6 . Omitting these simple calculations we obtain the result    p r2 u i x k dV = 4π f4 (k p )u i,k + f4 (k s )u si,k , B (ρ)

where ρ

f (k )r6 dr

f4 (k) = 0

= −30

3

5

ρ ρ f (k) + 2 k4 k

or



f4 (k) = Then





chkρ shkρ shkρ −5 + 15 kρ (kρ )2 (kρ )3

(7.22)

,

 10 ρ5 2 + ρ f3 (k) − 2 2 f (k) . 2 k k

  p r2 (u i x k + u k x i )dV = 8π f4 (k p )ε ik + f4 (k s )ε sik .

(7.23)

B (ρ)

Lemma 7.2. The solution to (7.1) satisfies the following integral relation:   f3 (k p ) u i x i dV = u i x i dΩ . f (k p )ρ2 B (ρ)

(7.24)

Ω

Proof. Introduce the notations vp =

3  i =1

Then,

 B

p

xi ui ,

vs =

3 

x i u si .

i =1

  p u i x i dV = (u i x i + u si x i )dV = (v p + v s )dV . B

B

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108 | 7 Spherical means for the stress and strain tensors Differentiating v p yields 3 

Δv p = 2 div up +

p

x i Δu i

i =1

(7.25)

3 

= 2 div up +

p x i k 2p u i

= 2 div up + k 2p v p ,

i =1

and Δ2 v p = 4 div Δup +

3 

p

x i Δ2 u i

i =1

= 4 div k 2p up +

3 

p

x i k 4p u i = 4 div up k 2p + k 4 v p .

i =1

We resolve the last two equations with respect to v p . This leads to the relation (Δ − k 2p )2 v p = 0 .

Then the function (Δ − k 2p )v p satisfies the relation  1 f (k p )(Δv p − k 2p v p ) = (Δ − k 2p )v p dV . 4πr3

(7.26)

B(r)

Substituting (7.25) into (7.26) we come to the relation (taking into account that v p (0) = 0)  2 p div up dV . f (k p )Δv = 4πr2 B(r)

Applying the Gauss theorem we get f (k p )Δv p =

2 4πr2



p

u i x i dΩ =

2 4πr2

Ω



v p dΩ .

(7.27)

Ω

Multiplying this equality by r4 and integrating the result over r, 0 ≤ r ≤ ρ yields  2 v p dV . f3 (k p )Δv p = 4π B (ρ)

From this, in view of (7.27), we get   f3 (k p ) v p dV = v p dΩ . f (k p ) ρ2 B (ρ)

(7.28)

Ω

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|

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Since div us = 0, the relation of the type (7.25) for v s gives that Δv s = k 2s v s , and hence

1 4πR2



v s dV = f (k s )v(0) = 0 . B (ρ)

The contribution of v s to integral (7.24) is equal to zero and hence (7.28) gives the desired result. The lemma is proved. Coming back to (7.18), we substitute the volume integrals obtained in the last two lem­ mas. This yields  p  2ε sik (f2 (k s ) + (2a + b )f3 (k s )) + 2ε ik f2 (k p ) + (2a + b )f3 (k p )   ρ3 a ρb = (u i x k + u k x i )dΩ + x i x j x k u j dΩ 4π 2π (7.29) Ω Ω  bδ ik f3 (k p ) − u j x j dΩ . 2πρ 2 f (k p ) Ω

From (7.20), 1 4πρ 2



p

(u i x k + u k x i )dΩ = 2f (k p )ε ik + 2f (k s )ε sik .

(7.30)

Ω

Thus we have a system of linear equations (7.16), (7.29), and (7.30). Solving this system yields Theorem 7.2. Assume that the parameter ν is chosen so that the assumption (A) is true for all S(x, r) ⊂ G. The strain components are related to the displacement vector through the following mean value relation: ⎡ ⎤   f (k s ) − f (k p ) ⎢ 3 f3 (k p ) ⎥ x i x j x k u j dΩ − δ ik u j x j dΩ⎦ ε ik (0) = ⎣R 4πR2 A f (k p ) Ω Ω (7.31)  ' & 5 1 R f (k p ) − 2f3 (k s ) − 3f3 (k p ) (u i x k + u k x i )dΩ , + 8πR2 A Ω

where A = R5 f (k p )f (k s ) − 3f3 (k p )f (k s ) − 2f3 (k s )f (k p ) .

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110 | 7 Spherical means for the stress and strain tensors

7.2.2 Mean value relation for the stress components In the next statement, we present the mean value relation for the stress components through the displacement vector. Theorem 7.3. Assume that the parameter ν is chosen so that the assumption (A) is true for all S(x, r) ⊂ G. The stress components τ ij at the center of the sphere are related to the integrals of the displacement vector: ⎡ ⎤     2μ f (k s ) − f (k p ) ⎢ 3 f3 (k p ) ⎥ τ ik = x i x j x k u j dΩ − δ ik u j x j dΩ⎦ ⎣R 4πR2 A f (k p ) Ω Ω  ' μ & 5 R f (k p ) − 2f3 (k s ) − 3f3 (k p ) (u i x k + u k x i )dΩ + 4πR2 A Ω  λδ ik 1 + u j x j dΩ , 4πR2 f (k p ) Ω

where the quantities A and f i (k ) are defined as in Theorem 7.2. Proof. By the Hooke law, τ ik = μ (u i,k + u k,i ) + λu j,j δ ik = 2με ik + λδ ik div u . This shows that it is required to get a mean value relation for the divergence term u i,i and substitute it in (7.31). From (7.6) it follows that (Δ − k 2p ) div u = 0 ,

hence f (k p ) div u(0) =

1 4πR3



div udV . B(R)

By the Gauss theorem f (k p ) div u(0) =

1 4πR2



u i x k dΩ , Ω

thus u i,i =

1 1 4πR2 f (k p )



u i x i dΩ .

(7.32)

Ω

Using this representation and the last theorem we get the result.

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Corollary 7.1. For the Lamé equation (ν = 0) the mean value relation of this theorem becomes ⎧ ⎪   ⎨ xi xj xk 3μ ui xk + uk xi τ ik (0) = 5 (1 − α ) + 35α u j dΩ dΩ 4π(5 + 2α )R ⎪ R R3 ⎩ Ω Ω ⎫  ⎪ ⎬ xj +(2α 2 − 4α − 5)δ ik u j dΩ . ⎪ R ⎭ Ω

It is proved by taking the limit as ν → 0 in the mean value relation of the above theo­ rem.

7.3 Mean value relations for the stress components in terms of surface tractions In this section, we deduce a spherical mean value relation between the stress compo­ nents and surface tractions τ ik x k . As in the previous section, the main result, Theorem 7.5, will follow from a set of preliminary results. We need some properties of τ ij . Theorem 7.4. For the stress components, the following representation holds: p

τ ij = τ ij + τ sij ,

(7.33)

where p

p

p

Δτ ij − k 2p τ ij = 0 ,

τ ij =

Δτ sij − k 2s τ sij = 0 ,

τ sij =

p

τ ii =

3 

p

τ ii =

i =1

(Δ − k 2s )τ ij , k 2p − k 2s (Δ − k 2p )τ ij

k 2s − k 2p

(Δ − k 2s )τ ii = τ ii , k 2p − k 2s

,

τ sii = 0 .

Proof. It is known [96] that τ ij satisfies the equilibrium equation τ ij,j − νu i = 0 ,

(7.34)

the equation ε ij =

λδ ij 1 τ mm τ ij − 2μ 2μ (3λ + 2μ )

(7.35)

and the Hooke equation τ ij = 2με ij + λu k,k δ ij .

(7.36)

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112 | 7 Spherical means for the stress and strain tensors Differentiation of (7.36) yields ε ij,kk + ε kk,ij = ε ik,jl + ε jl,ik . In view of (7.35) we get, choosing l = k, τ ij,kk + τ kk,ij −

λ (τ mm,kk δ ij + τ mm,ij δ kk ) 3λ + 2μ = τ ik,jk + τ jk,ik −

λ (τ mm,jk δ ik + τ mm,ik δ jk ) . 3λ + 2μ

Summation over k yields  k

τ ij,kk +



τ kk,ij −

k

  λδ ij  λ τ kk,ll − τ kk,ij = (τ ik,jk + τ jk,ik) . 3λ + 2μ k 3λ + 2μ k k

From this, in view of (7.34), we get that Δτ ij +

λδ ij 2(λ + μ ) τ mm,ij − Δτ mm = ν 3λ + 2μ 3λ + 2μ



∂u j ∂u i + ∂x j ∂x i

.

The right-hand side can be transformed according to (7.35); hence   λδ ij 2(λ + μ ) ν μ Δ− τ mm = 0 . Δτ ij + τ mm,ij − τ ij − 3λ + 2μ μ 3λ + 2μ ν

(7.37)

We put i = j and sum over i: 

τ mm,ii =

i

ν τ mm , λ + 2μ

(7.38)

i.e. Δτ mm = k 2p τ mm . Using (7.38) and (7.37), we get Δτ ij +

2(λ + μ ) ν τ mm,ij − 3λ + 2μ μ



τ ij +

λδ ij (λ + μ ) τ mm (3λ + 2μ )(λ + 2μ )

= 0.

(7.39)

It remains to apply the operator Δ to the expression (7.39) and use properties (7.38) and (7.39): ν ν ν ν Δ2 τ ij + · τ ij − Δτ ij − Δτ ij = 0 , λ + 2μ μ λ + 2μ μ or



ν ν Δ− Δ− τ ij = 0 λ + 2μ μ which proves the theorem.

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7.3 Mean value relations for the stress components | 113 2( λ + μ ) . 3λ +2μ

It is convenient to introduce the notation 𝛾 = (7.39) reads Δτ ij + 𝛾τ mm,ij − k 2s τ ij + δ ij

Then the compatibility equation

k 2s − 2k 2p

(7.40) 𝛾τ mm = 0 . 2 Multiplying this equality by r2 , 0 ≤ r ≤ ρ , and integrating over the ball B(ρ ) yields    δ ij 𝛾 2 2 2 2 2 r k s τ ij dV − (k s − 2k p ) r τ mm dV = r2 (Δτ ij + 𝛾τ mm,ij)dV . (7.41) 2 B (ρ)

B (ρ)

B (ρ)

Integration by parts of the last integral leads to

  xj xk 2 2 r (Δτ ij + 𝛾τ mm,ij)dV = ρ τ ij,k + 𝛾τ mm,i dS ρ ρ B (ρ) S (ρ)    xj xk −2 r τ ij,k + 𝛾τ mm,i dV . r r

(7.42)

B (ρ)

Using the Gauss theorem and equality (7.40), we transform the surface integral in (7.42) to

  xj xk 2 ρ τ ij,k + 𝛾τ mm,i dS = ρ 2 (τ ij,kk + 𝛾τ mm,ij)dV ρ ρ S (ρ)

B (ρ)



=ρ

2

k 2s τ ij

− δ ij 𝛾

B (ρ)

k 2s − 2k 2p 2

τ mm dV .

Then (7.41) can be rewritten as

  k 2s 2 2 2 r k s τ ij dV − δ ij 𝛾 − kp r2 τ mm dV 2 B (ρ)

B (ρ)

 −ρ

2

k 2s τ ij dV B (ρ)



k 2s + ρ δ ij 𝛾 − k 2p 2



2

τ mm dV B (ρ)



(τ ij,k x k + 𝛾τ mm,i x j )dV .

= −2 B (ρ)

Integrating the right-hand side by parts yields

  k2 r2 k 2s τ ij dV − δ ij 𝛾 s − k 2p r2 τ mm dV 2 B (ρ)



B (ρ)





 k 2s − k 2p − 2 τ mm dV 2 B (ρ)   xi xj 3 2 = −2ρ τ ij dΩ − 2ρ 𝛾 τ mm dΩ . ρ

τ ij dV + δ ij 𝛾 ρ 2

− (ρ 2 k 2s + 6) B (ρ)

Ω

(7.43)

Ω

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114 | 7 Spherical means for the stress and strain tensors We pass from the volume integrals to the surface integrals using τ ij . p From Theorem 7.4 it follows that τ ij and τ sij satisfy the equation Δτ tij − k 2t τ tij = 0 ,

t = p, s .

Hence, f (k t )τ tij = and f1 (k t )τ tij =



1 4πr3 1 4π

τ tij dV , B(r)



ρ 2 τ tij dV . B(r)

Then p

f (k p )τ ij + f (k s )τ sij =

1 4πr3

and

1 4π

p

f1 (k p )τ ij + f1 (k s )τ sij =



τ ij dV ,

(7.44)

ρ 2 τ ij dV ,

(7.45)

B(r)

 B(r)

where f (k ) and f1 (k ) are defined in (7.10) and (7.11). We put i = j in the last equalities and sum over i. By property (7.33), this yields  1 τ mm dV , (7.46) f (k p )τ mm = 4πr3 B(r)

and f1 (k p )τ mm =

1 4π



ρ 2 τ mm dV .

(7.47)

B(r)

Note that the representation for the volume integrals through the stress tensor compo­ nents can also be derived from the equilibrium equations. This is seen from the lemma which follows. Lemma 7.3. The following relations are true. First,   ρ τ ij dV = (τ ik x j x k + τ jk x i x k )dΩ 2 B (ρ)

Ω

 4πλδ ij k 2s f3 (k p ) k2  p + τ mm − 4π s f3 (k p )τ ij + f3 (k s )τ sij , 2(3λ + 2μ ) 2

then,





τ mm dV = ρ B (ρ)

τ il x i x l dΩ − Ω

4πν f3 (k p )τ mm , (3λ + 2μ )

(7.48)

(7.49)

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7.3 Mean value relations for the stress components | 115

next, 2 4π



r2 τ ij dV B (ρ)

=

ρ3 4π

 (τ ik x j x k + τ jk x i x k )dΩ

(7.50)

Ω 2

+

and 1 4π



  ρ λδ ij k 2s f3 (k p ) p τ mm − (4 + ρ 2 k 2s ) f3 (k p )τ ij + f3 (k s )τ sij , 3λ + 2μ

ρ3 r τ mm dV = 4π

B (ρ)





2

τ ik x i x k dΩ − τ mm Ω

ν 2+ρ 3λ + 2μ 2

f3 (k p ) .

(7.51)

Proof. We give the proof of (7.48). The same arguments work for (7.50). Formulae (7.49) and (7.51) follow from (7.48) and (7.50), respectively, when putting j = i and taking a sum over j. Now, in the equality    τ ij dV = (τ ik x j ),k dV − τ ik,k x j dV . B (ρ)

B (ρ)

B (ρ)

we take by parts the first integral in the right-hand side, and transform the second integral using the equilibrium equation (7.34). This yields    xk τ ij dV = τ ik x j dS − ν u i x j dV . ρ B (ρ)

S (ρ)

B (ρ)

Since τ ij = τ ji ,    τ ij dV = ρ (τ ik x j x k + τ jk x k x i )dΩ − ν (u i x j + u j x i )dV . 2 B (ρ)

Ω

B (ρ)

From (7.21) it follows     p 2 τ ij dV = ρ (τ ik x j x k + τ jk x k x i )dΩ − 8πν f3 (k p )ε ij + f3 (k s )ε sij . B (ρ)

Ω

From this, using (7.35), we get the desired formula (7.48). Now, from (7.44) and (7.48) we obtain the first auxiliary mean value relation for τ ij

 xj xk λk 2s δ ij 1 xi xk τ ik + τ jk dΩ + f3 (k p )τ mm 4π ρ ρ ρ ρ (3λ + 2μ )ρ 3 Ω



k 2s k 2s p = f3 (k p ) + 2f (k p ) τ ij + f3 (k s ) + 2f (k s ) τ sij . (7.52) ρ3 ρ3

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116 | 7 Spherical means for the stress and strain tensors For the function τ mm we find from (7.46) and (7.49) that

 ν ρ 3 τ il x i x l dΩ = τ mm ρ f (k p ) + f3 (k p ) . 4π 3λ + 2μ

(7.53)

Ω

Using representations (7.44)–(7.47) we get in (7.43) that   ρ𝛾 ρ3 τ ij dΩ + τ mm x i x j dΩ 4π 4π Ω Ω    p = τ ij k 2s f3 (k p ) + 3ρ 3 f (k p ) + τ sij k 2s f3 (k s ) + 3ρ 3 f (k s )

k 2s 2 3 − τ mm δ ij 𝛾 − k p f3 (k p ) − ρ f (k p ) . 2

(7.54)

Multiplying (7.54) by ρ, integrating over ρ, 0 ≤ ρ ≤ R, and in view of (7.50), we get

 p τ ij k 2s   1 τ mm x i x j dV = − 1 f1 (k p ) − 3f3 (k p ) 2 4π 𝛾 kp B(R) (7.55) 

 3k 2s k 2s + τ mm δ ij − 2 f3 (k p ) − − 1 f1 (k p ) . 2k 2p 2k 2p It remains to evaluate the last volume integral in (7.55). Lemma 7.4. The following representation of the volume integral through the surface in­ tegrals is true   ρ 1 τ kk x i x j dV = τ kl x k x i x j x l dΩ 4π 4π B (ρ)

Ω

k 2s k 2s − 2f3 (k p ) − f5 (k p ) + f4 (k p ) 2 2   s 2 − τ ij 2f3 (k s ) + 2k s f5 (k s )

k 2s λ (λ + 2μ )k 2s f4 (k p ) − f5 (k p ) , + τ mm δ ij 2(3λ + 2μ ) 2(3λ + 2μ )

p τ ij

(7.56)

where f4 (k ) is given above by formula (7.22) and f5 (k) = Note that f5 (k) =

ρ5 5 f (k) − 2 f3 (k) . k2 k

1 5 (ρ f (k ) − 5f3 (k )) , k2

and f4 (k ) = ρ 2 f3 (k ) − 2f5 (k ) .

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7.3 Mean value relations for the stress components |

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Proof. We start from the obvious equality    τ kk x i x j dV = (τ kl x k x i x j ),l dV − (τ ki x k x j + τ kj x k x i )dV B(R)

B(R)

B(R)



−

τ kl,l x k x i x j dV . B(R)

From this,







τ kk x i x j dV = R B(R)

τ kl x k x i x j dΩ − 

−

(τ ki x k x j + τ kj x k x i )dV B(R)

Ω

(7.57)

τ kl,l x k x i x j dV . B(R)

To evaluate the volume integrals we proceed as follows. Let us consider the first inte­ gral. Multiplying (7.52) by R and integrating over R, 0 ≤ R ≤ ρ yields    p 1 (τ ik x j x k + τ jk x i x k )dV = 2f3 (k p ) + k 2s f5 (k p ) τ ij 4π B (ρ)

  + 2f3 (k s ) + k 2s f5 (k s ) τ sij −

(7.58)

λk 2s δ ij f5 (k p )τ mm . 3λ + 2μ

Now let us evaluate the second volume integral in the right-hand side of (7.57). From (7.29), (7.30), and (7.32) we get p

ε ik (ρ 5 f (k p ) − 3f3 (k p )) + 2ε sik f3 (k s ) =



ρ 4π

u j x j x i x k dΩ − f3 (k p )δ ik

3 

ε ii .

i =1

Ω

Multiplying by ρ and integrating over ρ in the interval [0, R] yields p

ε ik (f4 (k p ) − 3f5 (k p )) + 2ε sik f5 (k s ) =

1 4π



u j x j x i x k dV − f5 (k p )δ ik

3 

ε ii .

i =1

B(R)

Using (7.35), we pass from the strain to the stress tensor

λf4 (k p ) p − 3f5 (k p ) δ ik τ mm τ ik (f4 (k p ) − 3f5 (k p )) + 2τ sik f5 (k s ) − 3λ + 2μ  μ = u j x j x i x k dV . 2π B(R)

From this equality and using the equilibrium equation (7.34) in the form   u j x j x i x k dV = τ jl,l x j x i x k dV , ν B(R)

B(R)

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118 | 7 Spherical means for the stress and strain tensors we obtain the desired integral  k2 1 p τ jl,l x j x i x k dV = s (f4 (k p ) − 3f5 (k p ))τ jk + k 2s f5 (k s )τ sij 4π 2 B(R)

k 2s λ − δ ij f4 (k p ) − f5 (k p ) τ mm . 2 3λ + 2μ

(7.59)

Finally, from (7.57), (7.58), and (7.59) we get (7.56). Now we substitute representation (7.56) into (7.55) to find the second auxiliary mean value relation  

 xk xl xi xj λk 2s ρ 2 1 1 + 4 f3 (k p ) τ kl dΩ + τ mm δ ij −f (k p ) + 5 4π ρ4 ρ 2(3λ + 2μ ) Ω



k 2s ρ 2 1 8 p s = τ ij 12 + f3 (k p ) − 2f (k p ) + τ ij 2f (k s ) − 5 f3 (k s ) . (7.60) ρ5 2 ρ Theorem 7.5. Assume that the parameter ν is chosen so that the assumption (A) is true for all S(x, r) ⊂ G. The stress components at the center of the sphere is related to surface tractions by  B2 − A2 τ kl x k x i x l x j dΩ τ ij (0) = 4πρ 4 (A1 B2 − A2 B1 ) Ω  B1 − A1 + [τ ik x j x k − τ jk x i x k ]dΩ 4πρ 2 (A1 B2 − A2 B1 ) Ω  C1 (B2 − A2 ) + C2 (B1 − A1 ) + δ ij τ kl x k x l dΩ , 4πρ 2 (A1 B2 − A2 B1 )C3 Ω

where



A1 = −2f (k p ) + B1 = 2f (k s ) −

k 2s 12 + ρ5 2ρ 3

f3 (k p ) ,

8 f3 (k s ) , ρ5

λk 2s 4 + 5 C 1 = −f ( k p ) + f3 ( k p ) 2(3λ + 2μ )ρ 3 ρ

,

k 2s f3 (k p ) + 2f (k p ) , ρ3 k2 B2 = 3s f3 (k s ) + 2f (k s ) , ρ λk 2s , C2 = f3 (k p ) 2(3λ + 2μ )ρ 3 ν C3 = f (k p ) + f3 (k p ) . (3λ + 2μ )ρ 3

A2 =

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7.3 Mean value relations for the stress components | 119

Proof. We conclude by resolving the linear system of equations (7.33), (7.53), (7.52), and (7.60) with respect to τ ij . Note that the mean value theorem for the Lamé equation (ν = 0) obtained in [40] is a special case of Theorem 7.5. Corollary 7.2. For the Lamé equation, (ν = 0) the mean value relation reads ⎧  ⎪ ⎨ xj xk 3 10 (1 − 𝛾) τ ik 2 dΩ τ ij (0) = 8π(5 + 2𝛾) ⎪ ρ ⎩ Ω ⎫ ⎪   ⎬ xk xi xj xl xk xl − 7𝛾δ ij τ kl 2 dΩ + 35𝛾 τ kl dΩ . ⎪ ρ ρ4 ⎭ Ω

Ω

Proof. To show that, it is sufficient to take the limit in the last expression as ν → 0.

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8 Random Walk on Spheres method 8.1 Spherical mean as a mathematical expectation The integral equations with a convergent Neumann series can be numerically solved by the Monte Carlo method using the Markov chains [52]. To make things clear, we first illustrate the situation by a simple evaluation of the spherical mean (in R3 ) by the Monte Carlo method. Let s be a random vector uniformly distributed on the sphere S(x, r). Then by def­ inition, the expectation of the random variable u (s) is equal to the spherical mean N r u:  1 E u ( s) = N r u ≡ u (y)dS(y) . 4πr2 S ( x,r )

By the law of large numbers, the expectation is approximated by the arithmetic mean: E u ( s ) ≈ U N :=

N 1  u (ˆs(i) ) , N i =1

(8.1)

where ˆs(i) are independent samples of the random vector s. √ The statistical error of this approximation is measured by the quantity σ / N, where σ 2 is the variance of u (s). This means that with probability 0.997 the absolute difference between the arithmetic mean and the exact expectation is not larger than √ 3σ / N. It is supposed of course that σ < ∞ [160]. Thus we can formulate the Monte Carlo algorithm for calculation of the spherical mean over N samples: P1 Put V := 0; i := 1; P2 Sample a random vector s uniformly distributed on the sphere S(x, r): s = x + ζr. Here ζ = (ζ1 , ζ2 , ζ3 ) is a unit isotropic vector which is simulated as follows: 2.1 ζ1 = 1 − 2α 0 ; 2.2 𝛾1 = 1 − 2α 1 ; 𝛾2 = 1 − 2α 2 ; d = 𝛾12 + 𝛾22 ; 2.3 If d > 9 1, then go to 2.2, otherwise 9 2.4 ζ2 = 𝛾1 (1 − ζ12 )/d; ζ3 = 𝛾2 (1 − ζ12 )/d; {Here α 0 , α 1 , α 2 are independent samples generated by the standard generator of pseudorandom numbers uniformly distributed on [0, 1]} P3 V := V + u (s); if i < N go to P2; P4 The arithmetic mean (8.1) is then U N = V /N.

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8.2 Iterations of the spherical mean operator

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8.2 Iterations of the spherical mean operator x,y

In this section, let us use the following notation for the spherical mean N r [u ] where the superscript indicates that the spherical mean is taken over the sphere S(x, r) and y is the variable of integration over S(x, r):  1 x,y I1 ≡ N r [u] = u (y)dS(y) , 4πr2 S ( x,r )

i.e., y = x + rs. Suppose that it is desired to calculate the second iteration of the spherical mean x,y

y,z

I2 ≡ N r1 [N r2 [u ]] . Introduce Y 2 (x), a set of Markov chains starting at the point x, with the first random state y1 uniformly distributed on S(x, r1 ) and the second state y2 uniformly distributed on S(y1 , r2 ), provided y1 is fixed. On this Markov chain we define the random variable ξ (x) = u(y2 ) . It is obvious that I2 = Eξ (x) = Eu(y2 ) , where E denotes the expectation, i.e. the average over all Markov trajectories Y2 (x). Thus we can approximately evaluate I2 ≈

N 1  (j) u(y2 ) , N j =1

(j)

where y2 , j = 1, 2 . . . , N are the second states of independently simulated Markov trajectories. Note that only the values of u at the last state y2 are involved. In the general case, to evaluate the kth iteration x,y

y ,y 2

I k ≡ N r1 1 N r21

y

· · · N r kk−1

,y k

[u] ,

we introduce a Markov chain Y k (x) starting at the point x and with the transitions x → y1 → y2 → · · · → y k simulated according to the same scheme: the point y i is uniformly distributed on the sphere S(y i−1 , r i ), i = 1, 2, . . . , k, y0 = x. Then, I k = Eu(y k ) .

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122 | 8 Random Walk on Spheres method

8.3 The Random Walk on Spheres algorithm 8.3.1 The Random Walk on Spheres process for the Dirichlet problem We now show that the spherical mean value relation and the Monte Carlo procedure of calculating the iterations of the spherical mean operator described above can be used to give a numerical method for solving the Dirichlet problem for the Laplace equation Δu (x) = 0,

x ∈ G,

u |Γ = φ .

We recall the definitions given in Chapter 2: d(x) is the distance from the point x to the boundary of the domain and Γ ε is defined as the set Γ ε = {x ∈ G : d ( x) < ε} . Let us define the Random Walk on Spheres process in the domain G. It starts at the point x and the states y1 , y2 , . . . are constructed as in the Markov chain Y k (x) de­ scribed above, where r i = d(y i−1 ) and the process stops if after, say, N ε steps, y N ε ∈ Γ ε . We denote the Random Walk on Spheres process by Y ε . This process is also called the ε-spherical process. It can be defined as a sequence of random points y i = y i −1 + d ( y i −1 ) ω i ,

i = 1, 2, . . . ,

y0 = x ,

where {ω i } is a sequence of independent unit isotropic vectors. The last random state is y N ε such that y N ε ∈ Γ ε (see Figure 8.1).

S(x,r)

G

Figure 8.1. The random ε-spherical process starting at x, the center of the first sphere S ( x, r ). In this illustration we see one trajectory which, after four steps, comes to Γ ε .

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8.3 The Random Walk on Spheres algorithm |

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The main properties of the Random Walk on Spheres process Y ε are well known (see, e.g. [160]). The following property is important since it shows that the process rapidly converges to the boundary: for bounded domains of arbitrary dimension and practically arbitrary boundaries (for details, see [160]) the random process Y ε with probability one converges to the boundary even if ε = 0 while the mean number of steps to reach Γ ε is O(ln(ε)) as ε → 0. For unbounded domains, see [169]. Using the converse spherical mean value theorem (see Theorem 2.5) and the ran­ dom estimators for the iterations I k , it is possible to construct the random estimator for the Dirichlet problem. Indeed, since the Random Walk on Spheres process reaches the set Γ ε with probability one in a finite number of steps, it follows from the converse spherical mean value theorem (see Section 2.1.2) that u(x) = Eu(y N ε ) . Choosing ε small enough, in practice one uses u (x) ≈ Eφ(y¯ N ε ) , where y¯ N ε is the point of the boundary closest to the last state y N ε ∈ Γ ε . The convergence of the Random Walk on Spheres method follows from the estima­ tion of the relevant integral operator generated by the spherical mean value relation. We present here the proof for the case of bounded domain G ⊂ R3 and domains with d∗ < ∞, where d∗ is a sphere of maximal radius which can be inscribed into the domain G. More formally, d∗ = sup d(x) , x∈G

where d(x) is the radius of the sphere S(x, d(x)) centered at the point x inscribed into G; obviously, d(x) equals the distance from the point x ∈ G to Γ, the boundary of G. Let P ε (x) be the probability that the random point uniformly distributed on a sphere S(x, r) ⊂ G hits the set Γ ε . Simple arguments lead to the uniform estimation [50] P ε (x) ≤

ε2 . 4 ( d ∗ )2

Using this estimation we get the convergence for the domains with d∗ < ∞. To this end we first write down a formal integral equation generated by the spherical mean value relation (e.g. see [50]). Let δ x (y) be the density of the uniform mass on the sphere S(x, d(x)). We define the kernel function k ε by ⎧ ⎨ δ x (y) if x ∈ G \ Γ ε , k ε (x, y) := ⎩ (8.2) 0 if x ∈ Γ ε ,

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124 | 8 Random Walk on Spheres method and define the integral operator K ε by



K ε ψ ( x ) :=

k ε (x, y)ψ(y)dy

(8.3)

G

for each ψ(·) ∈ C(G). We now fix the boundary function φ. If G is unbounded we suppose that the so­ lution to the Dirichlet problem tends to zero at infinity. Denote by u the solution to the Dirichlet problem corresponding to φ and by f ε (x) the function ⎧ ⎨0 if x ∈ G \ Γ ε , f ε (x) = ⎩ u (x) if x ∈ Γ ε . Consider the integral equation v(x) = K ε v(x) + f ε (x) .

(8.4)

Note that the mean value relation implies (8.4). The relation between equation (8.4) and the boundary value problem (2.1) and (2.2) is given by the following proposition. Proposition 8.1. Suppose that the Dirichlet problem for the Laplace equation is uniquely solvable for any continuous function φ. Then for any ε > 0, the integral equation (8.4) has a unique solution given by the Neumann series v(x) = f ε (x) +

∞ 

K εi f ε (x) .

(8.5)

i =1

This solution coincides with the solution to the original Dirichlet problem. Proof. Let ε be fixed. To show the convergence of the series f ε (x) +

N 

K εi f ε (x) ,

i =1

it is sufficient to prove the existence of 0 < λ < 1 such that for any continuous and bounded function g, K 2ε gL ∞ < λgL ∞ (this fact also implies the uniqueness of the solution to (8.5)). Let ν(ε) = ε2 /(4d∗2 ) . For x ∈ G \ Γ ε we have 



k ε (x, y) G

k ε (y, y )dy dy =

 G\Γ ε

G



=

(8.6) ⎛



⎞

⎟ ⎜ δ x (y) ⎝ δ y (y )dy ⎠ dy G

δ x (y)dy ≤ 1 − ν(ε) < 1 . G\Γ ε

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8.3 The Random Walk on Spheres algorithm |

Let v(x) := f ε (x) +

#∞

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K εi f ε (x). It is clear that v satisfies v(x) = K ε v(x) + f ε (x) .

If x belongs to Γ ε , then, for any y, k ε (x, y) = 0 and thus v(x) = f ε (x) = u (x). On the other hand, if x belongs to G \ Γ ε , then the definition of k ε implies that v satisfies the mean value relation at x with the sphere of radius d(x). We conclude by using the converse mean value theorem. The proof presented does not work if d∗ = ∞. We now consider two characteristic cases of domains with d∗ = ∞: the first is the half-space R3+ and the second is the ex­ terior of a ball. We show that in these two cases, the Random Walk on Spheres method converges. Thus let us consider the ε-spherical process starting at a point x0 in the half-space R3+ , R0 being the distance from x0 to the plane {z = 0}. Theorem 8.1. For λ < λ0 = ln 2 − 1 − ln ln 2 ≈ 0.0596 the moment E exp λN ε is finite and   lnα2 −1 exp λ R0 E exp λN ε ≤ , 2(1 − α ) ε where α < 1 is the solution of the equation α exp(−α ) =

ln 2 exp(λ) . 2

For the mean number of steps of the ε-spherical process the following estimation holds: EN ε ≤

1 ln 2 ln(R0 /ε) + + . 3 2(1 − ln 2) 2(1 − ln 2) 2(1 − ln 2)2

Proof. For arbitrary ε > 0 the exact representation of P(N ε = k ), the probability that the number of steps is equal to k, k > 1, is P(N ε = k) =

2R 2R 0 1   1 ε ε 1 − dR dR2 1 2k 2R0 R0 ε

2Rk−3

...

2Rk−2

dR k−2 ε

dR k−1 ε

ε

1

k/ −2

R k −1

j =1

1 Rj



ε 1− 2R j

.

Since ε < 2R j for all j, we get  ∞  ε exp λ  exp λ i lni {2i R0 /ε} 2R0 i=0 2 i!   ∞ i ε exp(λ)  ln 2 exp λ 1 i = [i + a] 2R0 i=0 2 i!

E exp λN ε ≤

=: f (a) ,

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126 | 8 Random Walk on Spheres method where



 1 R0 ln 2 . a := ln ε Direct calculations show that f (a) satisfies the equation

ln 2 exp λ ∂f = f (a + 1) , ∂a 2 whose solution has the form f (a) = f (0) exp(αa) , where α is the unique solution on the interval (0, 1) to the equation α exp(−α ) =

ln 2 exp λ . 2

The value f (0) is easily found: f (0) = Thus, E exp λN ε ≤

exp λ ε . (1 − α ) 2R0

   1    lnα2 −1 exp(λ) ε R0 ln 2 exp λ R0 = exp α ln . (1 − α ) 2R0 ε 2(1 − α ) ε

Note that f (0) < ∞ if ln 2 exp(λ) < 2/e, i.e., when λ < λ0 = ln 2 − 1 − ln ln 2 ≈ 0.0596. The solution to α exp(−α ) = ln 2 exp(λ/2) on (0, 1) can be found numerically, for example, for λ = 0.05 we get α = 0.86 and  0.24 R0 E exp λN ε ≤ 3.754 . ε For the mean number of steps we get EN ε ≤

 ∞  ε  ln 2 i 1 (i + 1) (i + a)i . 2R0 i=0 2 i!

Let F ( β ) := Then,

 ∞  ε  ln 2 i i+1 1 β (i + a)i . 2R0 i=0 2 i!  ∂F    EN ε ≤ . ∂β β=1

But

β F (β) = 2(1 − α )

where α exp(−α ) =

β ln 2 . 2

 ∂F    ∂β 

R0 ε



1 −1 ln 2

,

The derivative gives

= β =1



1 ln 2 ln (R0 /ε) + + . 2(1 − ln 2) 2(1 − ln 2)3 2(1 − ln 2)2

The theorem is proved.

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8.3 The Random Walk on Spheres algorithm

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The case d∗ = ∞ is a challenging problem in two respects: first, the behavior of the Random Walk on Spheres process is more complicated since there is a possibility that the process goes to infinity. Second, the standard approach [160] for constructing the random estimators for the Dirichlet problem in the bounded domain fails. It should be noted that if the problem is solved in G1 = R3 \ G, where G is a finite domain, then the Random Walk on Spheres algorithm can be constructed as follows (e.g. see [51]). We imbed the domain G into a sphere S(y, R0 ): G ⊂ S(y, R0 ); construct the Random Walk on Spheres process in the domain G1 ; outside of S(y, R0 ) use the transition to the surface of the sphere S(y, R0 ) according to the probability distribution generated by the Poisson kernel of the exterior Dirichlet problem for a ball. Actually, this is a special case of the method of iterations over subdomains described in [160]. This approach is restricted however to the case of bounded G. It reduces the problem to the standard Random Walk on Spheres process in bounded domains and says nothing about the behavior of this process at infinity. Although we study, in detail, a simple case when G is a sphere, it is believed that it provides the essential properties of the Random Walk on Spheres process in un­ bounded domains and gives an idea of how to treat more general cases. In practical problems one always tries to avoid simulation of long trajectories by different tech­ niques, for instance, using the explicit form of the Green functions as described above. However, it is of fundamental importance to investigate the Random Walk on Spheres process in the general case. In particular, the estimation of the exponential moment provides a probabilistic representation in the general case. With this representation, it is then possible to combine different strategies to improve the efficiency of the nu­ merical algorithms (see [161] and [168]). Let us consider the unbounded domain G1 ⊂ R3 which is the exterior to the sphere S(z, r) of radius r centered at a point z ∈ G1 . We also define an ε-layer Γ ε = {x ∈ G1 : r ≤ |x − z| ≤ r + ε} . The Random Walk on Spheres process in G1 starting at a point x ∈ G1 is defined as the Markov process {ξ0 , ξ1 , . . . , ξ n , . . . } such that ξ0 = x ,

ξ k +1 = ξ k + d ( ξ k ) ω k ,

k = 0, 1, . . . ,

where d(y) is the distance from the point y to the sphere S(z, r) and ω0 , ω1 , . . . , is a set of independent unit isotropic random vectors in R3 . So defined, the Random Walk on Spheres process consists of an infinite number of states {ξ0 , ξ1 , . . . , ξ k , . . . }. We define the ε-spherical process Ξ ε (x) as the Random Walk on Spheres process starting at x and which terminates if after, say, N ε steps ξ Nε ∈ Γ ε . The number of steps, N ε , is random and it is of interest to derive different sta­ tistical characteristics of it, such as the mean number EN ε , the exponential moment E exp(λN ε ), etc.

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128 | 8 Random Walk on Spheres method Theorem 8.2. Let P k be the probability that the ε-spherical process Ξ ε (x) terminates after k steps. Then P k +1 ≤

ε + 2r ε ln k {2k d(x)/ε} , 2(r + d(x)) 2d(x) 2k k!

and ε + 2r exp(λ) E exp{λN ε } ≤ 2(r + d(x)) 2(1 − α )



k = 0, 1, . . . , d(x) ε

 lnα2 −1

for λ < λ0 = ln 2 − 1 − ln ln 2 ≈ 0.0596. Here α is the unique (in the interval (0, 1)) solution to the equation ln 2 exp(λ) . α exp(−α ) = 2 Proof. Let R0 = d(x). Then P(N ε = 1) =

2πh0 R0 h0 = , 2R0 4πR20

where h0 is defined from the obvious equation (r + ε)2 − (r + h0 )2 = R20 − (R0 − h0 )2 ,

namely, ε2 + 2rε . 2(R0 + r) The probability that the ε-process has exactly two steps is h0 =

2R 0   h0 1 h1 P(N ε = 2) = 1 − dx0 , 2R0 2R0 2R1

(8.7)

h0

where the integration is carried out with respect to the measure dx0 , x0 varying be­ tween 0 and 2R0 , being the projection of the center of the second sphere on the ray z, z0 ; h1 is found in the second sphere as h0 was found in the first one: h1 =

ε2 + 2rε , 2(R1 + r)

where R1 is the radius of the second sphere in the Random Walk on Spheres process. Note that (r + R1 )2 − (r + x0 )2 = R20 − (R0 − x0 )2 ; hence, x0 =

R1 (R1 + 2r) . 2(R0 + r)

Using this relation we rewrite (8.7) 2R 0   h0 1 ε2 + 2rε dR1 . P(N ε = 2) = 1 − 2R0 2R0 2(R0 + r) 2R1

(8.8)

ε

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On the kth step we get 2R 2R 0  1     h0 1 h1 dx0 h2 dx1 1− 1− P(N ε = k) = 1 − 2R0 2R0 2R1 2R1 2R2 2R2 h0

2R k−2 

···

1− h k −2

h1

h k −1 2R k−1



dx k−2 2R k−1

2R k−1

h k −1

(8.9)

hk dx k−1 . 2R k

Using the relations ε2 + 2rε , 2(R i + r) R i (2r + R i ) = , 2(R i−1 + r)

hi = x i −1 we successively get from (8.9)

2R 0    h0 ε2 + 2rε h1 dR1 1− P(N ε = k) = 1 − 2R0 2R0 (R0 + r) 2R1 2R1 ε

2Rk−3 

··· ε

h k −2 1− 2R k−2



dR k−2 2R k−2

2Rk−2

ε

dR k−1 , 2R k−1

or, 2R 0 1     k 2R h0 ε2 + 2rε 1 dR1 dR2 P(N ε = k) = 1 − 2R0 2(R0 + r)R0 2 ε

2Rk−3

···

2Rk−2

dR k−2 ε

ε

k −2 dR k−1 / 1 R k −1 j =1 R j

ε



hj 1− 2R j

(8.10)

.

Note that expression (8.10) differs from the representation of P(N ε = k ) obtained in Theorem 8.1 for the half-space R3+ only by the factor ε + 2r . 2(R0 + r) Thus we conclude by using the results of Theorem 8.1. Note that from the results given in the above theorem, it is seen that there is a portion of trajectories (≈ 1 − (ε + 2r)/2(r + R0 )) which go to infinity. Since, however, u (x) → 0 at infinity, the conditional expectation coincides with the total expectation. From these estimations it is possible to obtain a practical scheme of implementation of the algo­ rithm for solving exterior boundary value problems for domains whose boundaries

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130 | 8 Random Walk on Spheres method have a complicated shape. For instance, to reach the probabilistic error of the order ε, it is necessary to ignore the trajectories which are out of the sphere of radius R∞ ∼ r +

r( ρ − r) , ρε

where ρ is the distance from the point where the solution is calculated to the center of the sphere S(z, r). Then u (x) ≈  φ(y¯ N ε ) where y¯ N ε ∈ Γ is the point closest to the final state y N ε ∈ Γ ε . It is interesting to note that similar arguments in a two-dimensional case lead to the estimation R∞ ∼ r(ρ /r)1/ε which implies that to reach the desired probabilistic accuracy ε it is necessary to simulate much longer trajectories compared to the 3D case. Remark 8.1. The same scheme works for equations for which the converse spherical mean value theorem can be proved and, in addition, the convergence of the iterative procedure for the relevant integral equation can be shown. For example, let us con­ sider the Dirichlet problem for the diffusion equation (λ ≥ 0) Δu (x) − λu (x) = 0,

u |Γ = φ .

The random estimator that can be used to calculate the approximate solution is √ N/ ε −1 d(y i ) λ √ . ξ = φ(y¯ N ε ) sinh[d(y i ) λ ] i =0

8.3.2 Inhomogeneous case Let us consider the Dirichlet problem for the Poisson equation Δu (x) = f (x) ,

x ∈ G,

u |Γ = 0 .

We now construct a Random Walk on Spheres process which solves this problem. We start by the integral representation  u (x0 ) = u δ (x, x0 )f (x)dx , (8.11) G

where u δ is the Green function, i.e. Δu δ (x, x0 ) = δ(x − x0 ) ,

u δ |Γ = 0 .

We seek the Green function u δ in the form u δ (x, x0 ) = E(x, x0 ) + W (x, x0 ) ,

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8.3 The Random Walk on Spheres algorithm

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where E(x, x0 ) is the fundamental solution to the Laplace equation and hence W is the solution to ΔW (x, x0 ) = 0 , W |x∈Γ = −E(x, x0 )|x∈Γ . (8.12) Thus we see that the problem is reduced to calculating the integral functional (8.11) of the solution to the Dirichlet problem for the Laplace equation (8.12). Integral (8.11) can be represented as the expectation u (x0 ) =  η(y, x0 ) , where the random estimator η(y, x0 ) is constructed as follows: η(y, x0 ) = u δ (y, x0 )f (y)/π(y) , and y is a random point sampled in G from an appropriate probability density func­ tion π(y). Since for the Green function we can also construct a random estimator, say, ξ (x0 , y) so that u δ (x0 , y) = Ey ξ (x0 , y), we come to the representation u (x0 ) = Ey ξ (x0 , y) so that η(y, x0 ) =

) f (y) ( E(y, x0 ) + ζ (y, x0 ) , π(y)

(8.13)

(8.14)

since ξ (x0 , y) = E(y, x0 ) + ζ (y, x0 ) where ζ (y, x0 ) is the standard estimator of the Random Walk on Spheres algorithm for the Dirichlet problem (8.12). Thus the double expectation (8.13) can be evaluated as follows. A point y is chosen at random in G according to the density π(y) and a δ-source is placed at y. Then the Random Walk on Spheres process starting from x0 is constructed and the estimator (8.14) is calculated. So along one trajectory a contribution from the δ-source at y is calculated. Next a new random point y1 is sampled according to the same density and the contribution from the δ-source at y = y1 is calculated, etc. The final result is obtained by averaging the contributions over a sufficiently large number of points y, y1 , . . . , y p . The algorithm described gives the solution only at one fixed point x0 . However, it is possible to construct a modification which gives the solution at a set of points. This modification is based on the symmetry of the Green function: u δ (x, x0 ) = u δ (x0 , x). Let us describe this algorithm that gives the solution at a set of points x i , i = 1, . . . , m. The δ-sources are located at the points x i , i = 1, . . . , m and fixed for all tra­ jectories. A random point y1 is sampled as above, from the density π(y). The Ran­ dom Walk on Spheres process starting from this point, y1 , is simulated and contribu­ tions from the δ-sources are calculated, i.e. the estimators η(x i , y1 ) are calculated for i = 1, . . . , m. Next a new random point, y2 , is sampled, the Random Walk on Spheres process starting from y2 is constructed and new contributions η(x i , y1 ) are calculated for i = 1, . . . , m. Again, the result is obtained by averaging the contributions over a sufficiently large number of samples y1 , . . . , y p .

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132 | 8 Random Walk on Spheres method Details of these algorithms can be found in [160]. Note that it was assumed that φ = u |Γ = 0. In the general case, when φ ≠ 0, a dif­ ferent approach can be applied. In this case we need the volume mean value relation in a ball, i.e. the Green formula for a ball:   ∂u δ (x, y) u(x) = u δ (x, x )f (x )dV (x ) . φ(y)dS(y) + ∂y S ( x,r )

B ( x,r )

In [160], a modification of the Random Walk on Spheres based on this Green formula is described. It appears that along the random point on each sphere, it is necessary to sample a random point in the ball. For details, see [160] and [50].

8.4 Biharmonic equation The vector case is treated in the same way (see [159] and [165]). Let us consider, for example, the boundary value problem for the biharmonic equation Δ2 u ( x) = 0 ,

x ∈ G ⊂ Rm

(8.15)

with the boundary conditions u |Γ = g0 ,

Δu |Γ = g1 .

(8.16)

To carry out an iterative procedure, we write down the integral equation, as we have done in Chapter 2. We can rewrite the mean value relation in the form of the following system of integral equations for the vector v = (v1 , v2 )T : v = K ε v + fε ,

(8.17)

where the matrix-kernel k ε (x, y) and the vector fε have the form ⎞ ⎧⎛ r2 ⎪ δ x (y) − 2m δ x (y) ⎪ ⎪ ⎨⎝ ⎠ , x ∈ G \ Γε k ε (x, y) = ⎪ 0 δ x (y) ⎪ ⎪ ⎩0 , x∈Γ ε

and

⎞ ⎧⎛ ⎪ ⎪ ⎪⎝ u 1 ( x ) ⎠ ⎨ , fε (x) = ⎪ u2 (x) ⎪ ⎪ ⎩0 ,

x ∈ Γε x ∈ G \ Γε .

The solution to the preceding integral equation is given by v( x ) =

∞ 

K εn fε (x) ,

n =0

where K ε is the matrix-integral operator corresponding to the kernel k ε (x, y):

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8.4 Biharmonic equation | 133

Proposition 8.2. For arbitrary d∗ > ε > 0, the function v is well defined. Proof. The following estimate holds: K εn L ∞ ≤ [1 − ν(ε)]n−1 1 +

n ( d ∗ )2 2m

.

(8.18)

Hence, ε being fixed, one can choose n large enough so that the above series converges. As in Chapter 2, we can prove that the solution to the above integral equation coincides with the solution of the boundary value problem (8.15), (8.16). The kernel of the system of integral equations is the matrix 2 1 −2mr K ( r) = . 0 1 Consequently, ξ = (ξ1 , ξ2 ) and the vector random estimator for v = (u, Δu )T is ⎧ ⎫ Nε ⎨/ ⎬ ξ (x) = ⎩ K (d(y i−1 ))⎭ v(y N ε ) , i =1

where y0 = x. This means that v( x ) = E ξ . Again, the ε-biased estimator has the form ˆξ (x) =

Nε / &

' K (d(y i−1)) g(y¯ N ε ) ,

i =1

where g = (g0 , g1 )T . Note that since the matrix K is triangular, the random estimators ξ2 and ˆξ2 for the solution to (8.15), (8.16) can be rewritten in the form ξ2 = u(X Nε ) −

N ε −1 Δu (X N ε )  d2 (X k ) , 2m k =0

N ε −1 ¯ ¯ N ε ) − g1 (X N ε ) d2 (X k ) . ξˆ2 = g0 (X 2m k=0

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134 | 8 Random Walk on Spheres method

8.5 Isotropic elastostatics governed by the Lamé equation 8.5.1 Naive generalization In Chapter 4 we treated triangular systems of elliptic equations generating triangular systems of integral equations which in turn leads to a simple analysis of the conver­ gence of the iterative process. Let us consider a system in G ⊂ Rn where all the equations are symmetrically entered: μΔu 1 (x) + (λ + μ )

∂ (div u) = 0 , ∂x1 .. .

μΔu n (x) + (λ + μ )

∂ (div u) = 0 , ∂x n

or, using the summation convention (i fixed, summation over the repeated index j) : u i,jj + αu j,ji = 0 ,

x ∈ G ⊂ Rn ,

(8.19)

where i, j = 1, 2, . . . , n, α = (λ + μ )/μ, λ, μ > 0 are constants. This is the equilibrium equation for the displacements u i (x), i = 1, 2, . . . , n , the body forces being absent and it is known as the Lamé equation; λ, μ are the Lamé constants of the elastic material. We deal in this section with the first boundary value problem for (8.19) with the boundary conditions: u|Γ = φ = (φ1 , . . . , φ n )T . We rewrite the mean value relation (5.7) (see Theorem 5.1 in Section 5.1.1) in the vector form (8.20) u = pN r u + qSr u , where p=a=

n (2 − α ) ≥ 0, 2(n + α )

q=β=

α (n + 2) ≥ 0; 2(n + α )

hence p + q = 1 and Nr is the matrix integral operator ⎛ ⎞ N r u1 0 ··· 0 ⎜ ⎟ ⎜ 0 0 ⎟ N r u2 · · · ⎜ ⎟ Nr u = ⎜ . .. ⎟ .. ⎜ .. . . ⎟ ⎝ ⎠ 0 ··· Nr un

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8.5 Isotropic elastostatics governed by the Lamé equation |

and (Sr u)i (x) =

n ωm

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135



s i s j u j (x + rs)dΩ(s) ,

i, j = 1, 2, . . . , n .

(8.21)

Ω

A naive vector Monte Carlo estimator can be constructed as a generalization of the standard isotropic Random Walk on Spheres method. We introduce the unbiased vector estimator N/ ε −1 (pI + qS i )u(X N ε ) , (8.22) ξε = i =0

and the ε-biased random estimator ˆξ ε =

N/ ε −1

¯ Nε ) , (pI + qS i )φ(X

(8.23)

i =0

where I is the n × n identity matrix and S i is the matrix-kernel of the operator (8.21) in ith sphere: ⎛ 2 ⎞ s1 s1 s2 . . . s1 s n ⎜ ⎟ ⎜ s2 s1 s22 . . . s2 s n ⎟ ⎜ ⎟ . (8.24) Sk = n ⎜ . .. ⎟ .. ⎜ .. ⎟ . . ⎝ ⎠ s m s1 · · · s2n Thus s1 , s2 , . . . , s n are the components of the unit isotropic vector in ith sphere of the Random Walk on Spheres process. From the integral formulation of the Lamé equation given in Chapter 5 it follows that u(x) = Ex ξˆε . However, estimators (8.22) and (8.23) cannot be used if ε is small enough. Indeed, the ¯ is the number of variance of ˆξ ε (x) is exponentially increasing: E(ˆξ ε )2  n N¯ , where N steps of the ε-spherical process.

8.5.2 Modification of the algorithm It is known that if ε is small, then after a certain number of steps the trajectory is concentrated near the boundary, r i being of the order of ε (see, e.g. [160] and [121]). This gives the motivation to construct the following modification of the algorithm. Let the constants C ij and D ij be such that      ∂2 u   ∂2 Δu    i  i   = C ij , sup   = D ij , (i ≠ j) sup   ∂x i ∂x j   ∂x i ∂x j  Γε Γε and set Ci =

 i≠j

C ij , D i =



D ij .

i≠j

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136 | 8 Random Walk on Spheres method Let us consider the sum of the nondiagonal terms of the operator (8.20)  1 s i s j u j (s) dΩ(s) , (j ≠ i) . mi = 4π Ω

From the expansions of Lemma 4.1 we get the three-dimensional case : mi =

r2 ∂ r4 ∂ u j,j + Δu j,j , 15 ∂x i 210 ∂x i

where the summation is taken over j ≠ i. In the 3D case, we rewrite (8.20) in the form  u i (x) = p(Nr u)i (x) + 3q s2i u i (x + rs) dΩ(s) + m i . Ω

Thus, in Γ ε we can approximate the local integral equation (8.20) by 

ˆ i (x + rs)dΩ(s) + s2i u

ˆ )i (x) + 3q ˆ i (x) = p(Nr u u

dˆ4 (x) dˆ2 (x) Ci + Di , 15 210

(8.25)

Ω

⎧ ⎨d(x) ,

where

ˆ (x) = d ⎩

0,

x ∈ Γε , x ∈ Γ ε .

Define a diagonal matrix-operator C by  3 (C v)i (x) = s2i v i (x + rs) dΩ(s) , 4π

S(x, r) ⊂ G ,

Ω

(no summation over i). By p + q = 1, we obtain that the Neumann series for (8.25) absolutely converges for an arbitrary small ε, since the norm of the second iteration of the integral operator in (8.25) has the following estimate: (pNr + qC)2 L ∞(G) < 1 − ν(ε) < 1 .

From this we can estimate the error corresponding to (8.25):

i, Γ ε

i

∞ 

2 ˆ 2 (x) ≈ max (C i , D i ) ε (pNr + qC)k d i ν( ε) x ∈ Γ ε k =1

ˆ i | ≈ max (C i , D i ) sup sup |u i − u

as ε → 0. Thus for smooth boundaries for which ν(ε) ≈ ε, we conclude that the error is of the order ε.

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8.5 Isotropic elastostatics governed by the Lamé equation |

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Remark 8.2. Note that the mean value relation u i (x) = p(Nr u)i (x) + q(C u)i (x) ,

S(x, r) ⊂ G

corresponds to the case when u i,ji = 0. This implies that we have actually three inde­ pendent diffusion equations ∂2 u i Δu i + α 2 = 0 . ∂x i Near the boundary, where r i are small, the regular solutions to these diffusion equa­ tions have the mean value property u i (x) = p(Nr u)i (x) + q(C u)i (x) + O(r4 ) , S(x, r) ⊂ G . The approximation (8.25) corresponds to the approximation of the Lamé equation near the boundary by three independent diffusion equations u i,jj + αu i,ii = 0 . Thus, the modified algorithm is the following. First, choose δ = tε where t > 1 is a constant. Evaluate (8.22), until X m ∈ Γ δ . After that, use the operator (8.25), omitting the right-hand side. We denote by CC a diagonal matrix with entries ms2i which is the kernel of the matrix integral operator C . The estimator is then N/ ε −1 ˜ξ ε = φ(X ¯ Nε ) (pI + qCC )k , k =0

where the subscript k means that the matrix is taken in the kth step of the random walk process. Note that the operator C has a probabilistic interpretation: we make with the probability p the standard isotropic step of the ε-spherical process and with probability q = 1 − p the random point X n+1 is sampled on S(X n , d(X n )) with the density ms2i . Of course, this procedure is efficient if the constants C ij and D ij are not too large. Thus, in the general case, it is not possible to improve the algorithm by this approach.

8.5.3 Nonisotropic Random Walk on Spheres We now construct a different estimator that uses the specific structure of the mean value relation. The main idea is to find a distribution on the sphere which is in some sense generated by the matrix-kernel of the mean value relation. It appears that the points on the sphere should then have the “s i ”-distribution introduced above. We begin with the case p = 0, q = 1. It is then necessary to evaluate the itera­ tions of the operator (8.21) along the spheres S(X i , d(X i )) where we have to define the points X i .

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138 | 8 Random Walk on Spheres method Let X0 = x, X1 , . . . , X m be points on the spheres S(X0 , d(X0 )) ,

S(X1 , d(X1 )) , . . . , S(X m , d(X m )) ,

respectively. We have to evaluate the products m /

Si ,

i =1

where the matrix S i is defined in (8.24). Let s(i) , i = 1, . . . , m be the unit vector that specifies the corresponding point X i on the ith sphere. We use the following representation (which is easily shown by induction): m /

S i = n m (s(1) · s(2) ) · · · (s(m−1) · s(m) ) [s(1) ⊗ s(m) ] ,

(8.26)

i =1

where

⎛

(1) ( m )

s1 s1 ⎜ (1) (m) ⎜ s2 s1 ⎜ [s(1) ⊗ s(m) ] = ⎜ .. ⎜ . ⎝ (1) ( m ) s n s1

(1) ( m )

s1 s2 (1) ( m ) s2 s2 ···

··· ··· .. .

(1) ( m ) ⎞ s1 s n (1) ( m ) ⎟ s2 s n ⎟ ⎟ . .. ⎟ ⎟ . ⎠ (1) ( m ) sn sn

To be more specific, let us first consider the two-dimensional case. The first m itera­ tions give, using (8.26):  2m cos ψ1 R( ψ1 , . . . , ψ m ) (u, s(m) )dψ1 · · · dψ m (8.27) u1 (x) = m π 2 Ωm

and

2m u2 (x) = m π



R( ψ1 , . . . , ψ m ) Ωm

sin ψ1 (u, s(m) )dψ1 · · · dψ m , 2

(8.28)

where R( ψ1 , . . . , ψ m ) =

1 2 m −1

−1 m /

cos (ψ j+1 − ψ j ) ,

j =1

(u, s(m) ) = u 1 cos (ψ m ) + u 2 sin (ψ m ) .

We now give a probabilistic interpretation to (8.27). Let us define a distribution density on the half-circle {s(1) > 0} by p 1+ ( s ) =

cos ψ , 2

−

π π ≤ψ≤ . 2 2

(8.29)

On the second circle we sample the random direction from the conditional density p 2+ ( s ) =

cos (ψ2 − ψ1 ) , 2

−

π π + ψ1 ≤ ψ2 ≤ + ψ1 , 2 2

(8.30)

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under the condition that ψ1 was sampled in the first circle from the density (8.29), etc., including the direction s(m) in the last circle. The densities p i− , i = 1, . . . , m in the half-circles {s(1) ≤ 0} are constructed analogously. In the ith circle, we sample the random direction with probability 1/2 from p i+ (s) and with probability 1/2 from p i− (s). Let N1 be the number of steps until X N1 ∈ Γ ε and let κ N1 be the number of samples that occurred in {s(1) ≤ 0}. Then (8.27) can be rewritten in the form of the expectation N taken over the ensemble of directions {s(i) }i=1 1 : u 1 (x) = Eˆξ1 (x) , where ˆξ1 (x) =

  N1 4 (−1)κ N1 [u 1 (X N1 ) cos ψ N1 + u 2 (X N1 ) sin ψ N1 ] . π

(8.31)

(8.32)

The estimator for the second component is constructed analogously: ξˆ2 (x) =

 N2 4 (−1)κ N2 [u 1 (y N2 ) cos ψ N2 + u 2 (y N2 ) sin ψ N2 ] , π N

(8.33)

N

where the process {y i }i=2 1 is obtained from {X i }i=1 1 by rotating on π/2 in the positive direction. This follows by comparing (8.27) and (8.28). The ε-biased estimators have the form   N1 4 ¯ N1 ) cos ψ N1 + φ2 (X ¯ N1 ) sin ψ N1 ] , (−1)κ N1 [φ1 (X (8.34) ξ1 (x) = π and ξ2 (x) =

 N2 4 (−1)κ N2 [φ1 (y¯ N2 ) cos ψ N2 + φ2 (y¯ N2 ) sin ψ N2 ] . π

(8.35)

Note that the variance of the estimators is still divergent as ε → 0. However, it is much less than that of the estimator (8.22).

8.5.4 Branching process We now consider the general case of the Lamé equation with p, q ≥ 0; p + q = 1. Let us define a Markov branching process η(x) starting at the point x, the center of the first circle S(x, r1 ). With probability p, the point y1 is distributed on S(x, r1 ) uniformly and with probability q = 1 − p, two points are sampled on S(x, r1 ), the first one being (1) y+ , 1 = x + r1 s where s(1) has the distribution (8.29) and the second (1) y− . 1 = x − r1 s

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140 | 8 Random Walk on Spheres method Next y+ 2 = X 1 + r2 ω , (ω isotropic) with probability p and with probability q = 1 − p (2) , y± 2 = X 1 ± r2 s − (2) has the distribution den­ where X1 is the previous state, i.e. it is y1 , y+ 1 , or y 1 and s sity (8.30) if the previous direction was not isotropic and (8.29) if it was isotropic, until the absorption in Γ ε : y k , y± k ∈ Γ ε . Hence, after each isotropic scattering the transition (8.29) is used, continuing the scattering according to (8.30), until an isotropic scatter­ ing happens again, etc. Thus, one trajectory of η(x) consists of one or two points in the first generation, 1, 2, 3, or 4 points of the second generation, etc. In the mth generation we have 1, 2, . . . , or 2m points. Note that the set of the last (absorption) states lying in Γ ε will consist of points of different generations. N Let us denote by N is the number of the last states {y k }k=is 1 ∈ Γ ε sampled isotropi­ Ns cally and by N s the number of the last states {y± k }k =1 ∈ Γ ε sampled nonisotropically. ± Let us now choose a fixed last state y k or y k and let us consider the transitions leading from x to the last state. We denote by l k the number of these transitions. Let l± k be the number of transitions made according to the scheme ( j +1 ) y± j +1 = X j ± r j +1 s

inside these l k steps. We denote the trajectory of the branching process by  ( ) ( )Ns  N η1 = η1 x ; y k k=is 1 ; y± k k =1 N

indicating the starting point, the points {y k }k=is 1 absorbed in Γ ε after an isotropic tran­ Ns sition and points {y± k }k =1 absorbed in Γ ε after a nonisotropic transition. On η1 , we now define the following random estimator: ˆζ1 (x) =

N is 

−

(−1)l k

k =1

where



 l+k +l−k   l+k +l−k  Ns   − 2 2 ( k +1 ) u 1 (y k ) + (−1)l k u(y± , k)·s π π k =1

 ± ( k +1 ) u(y± = u1 (y± k)·s k ) cos ψ k + u 2 ( y k ) sin ψ k .

By the construction, u 1 (x) = Eˆζ1 (x) for arbitrary ε > 0. The ε-biased estimator has the form  l+k +l−k   l+k +l−k  Ns   2 2 l− ( k +1 ) k φ1 (y¯ k ) + (−1) φ(y¯ ± . ζ1 (x) = (−1) k)·s π π k =1 k =1 N is 

l− k

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8.5 Isotropic elastostatics governed by the Lamé equation |

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For the second component, we construct the trajectories η2 (x) that are obtained from η1 (x) by rotation through π/2 in the first step, repeating the rest of the trajectory. Let   N Ns η2 = η2 x; {z k }k=is 1 ; {z± k } k =1 , and ˆζ2 (x) =

N is 

−

(−1)l k

k =1

 l+k +l−k   l+k +l−k  Ns   − 2 2 ( k +1 ) u 2 (z k ) + (−1)l k u(z± . k)·s π π k =1

Then u 2 (x) = Eˆζ2 (x) for arbitrary ε > 0. Remarkably, the estimators have the same form in the general n-dimensional case. For example, let us consider the 3D case. The branching process   N Ns η = η x; {y k }k=is 1 ; {y± k } k =1 is constructed according to the same scheme, using in the first sphere {s3 > 0} the density p+ (s) = s3 /π and applying the corresponding rotation beginning from the second nonisotropic scattering. The unbiased estimator in three dimensions has the form ˆζ i (x) =

N is 

−

(−1)l k

k =1

 l−k +l+k  l−k +l+k  Ns   − 3 3 ( k +1 ) u i (y k ) + (−1)l k u(y± k)·s 4 4 k =1

and the corresponding ε-biased estimator is ζ i (x) =

N is 

−

(−1)l k

k =1

 l−k +l+k   l−k +l+k  Ns   − 3 3 ( k +1 ) φ i (y¯ k ) + (−1)l k φ(y¯ ± . k)·s 4 4 k =1

8.5.5 Analytical continuation with respect to the spectral parameter There is another possibility for constructing the solution to the Lamé equation based on the analytical continuation of the solution to Δ∗ u − νu = 0

u|Γ = g

(8.36)

with respect to ν. Indeed, using the well-known spectral properties of the Lamé equa­ tion, we can apply one of the numerical procedures of analytical continuation of the solution to (8.36) to the point ν = 0. Thus it is necessary to construct a numerical pro­ cedure for the problem (8.36). We will show that for sufficiently large value of ν, the

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142 | 8 Random Walk on Spheres method Random Walk on Spheres method converges. It means that for a fixed ε > 0 we can find ν depending on ε such that the Neumann series for the integral equation gener­ ated by the relevant spherical mean value relation for the problem (8.36) converges. Let us rewrite the spherical mean value relation for (8.36) derived in Section 5.2 (see formula (5.34) in Theorem 5.7) as u(x) = Ku, or  Aδ ij + Bs i s j 1 u j (s)dS , (8.37) u(x) = 4π A𝛾1 (η) + B𝛾2 (η) S ( x,R )

where A = 𝛾1 (ξ ) − 2 + 𝛾2 (ξ ) − 𝛾2 (η) , B = 𝛾1 (η) − 𝛾1 (ξ ) , and 𝛾1 (η) =

shη , η

𝛾2 (η) =

chη shη − 3 . η3 η

Here η = k s R, ξ = k p R . We now show that for sufficiently large ν = ν(ε), the norm K L ∞ of the integral operator generated by the spherical mean value relation (8.37) is less than 1. Lemma 8.1. The following estimations are true: (a )

𝛾2 (η) < 0 ,

( b)

B > 0 if

A>0

for

η > 0,

and ξ>

 where g = 2 + λ/μ.

ln g , g−1

Proof. Indeed, since thη < η for η > 0, we find 𝛾2 (η) =

sinh η − η cosh η < 0, η3

η > 0.

Therefore, for η > 0, ξ > 0 we have A = 𝛾1 (ξ ) − 2𝛾2 (ξ ) − 𝛾2 (η) > 0 . Since η = gξ , we can rewrite B as B = 𝛾1 (η) − 𝛾1 (ξ ) =

sinh gξ sinh ξ − . gξ ξ

The inequality B > 0 implies sinh gξ − g sinh ξ > 0, or     e gξ − ge ξ + ge−ξ − e−gξ > 0 ;

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8.5 Isotropic elastostatics governed by the Lamé equation |

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therefore, e gξ − ge ξ > 0

when

ξ>

ln g , g−1

which proves (b). Theorem 8.3. For the integral operator K with the norm K L ∞ = max i,x

with the entries k ij =

3 

|k ij | ,

j =1

Aδ ij + Bs i s j A𝛾1 (η) + B𝛾2 (η)

the inequality K L ∞ < 1 holds if (λ + 2μ )ξ02 , ε2

ν>

(8.38)

where ξ0 is the solution of the equation sinh ξ −

3ξ 2 + 2ξe−ξ = 0. ξ 2 − 2ξ + 2

Proof. Indeed, K L∞ ≤

|A| + 3|B| |A| + 3|B| . ≤ |A𝛾1 (η ) + B𝛾2 (η )| |A𝛾1 (η )| − |B𝛾2 (η )|

By inequalities (a) and (b) we find K L∞ ≤

A + 3B A𝛾1 (η) + B𝛾2 (η)

for ξ > ln g/(g − 1), η > 0 . To ensure K L ∞ ≤ 1 we require A + 3B < A𝛾1 (η) + B𝛾2 (η) ; hence A[𝛾1 (η) − 1] > B[3 − 𝛾2 (η)] . Substituting A and B into this equation yields [𝛾1 (ξ ) − 2𝛾2 (ξ ) − 𝛾1 (η )][𝛾1 (η ) − 1] > [𝛾1 (η ) − 𝛾1 (ξ )][3 − 𝛾2 (η )] .

Hence, for ξ > 0 𝛾1 (η) − 1 > 𝛾1 (η) − 𝛾1 (ξ ) .

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144 | 8 Random Walk on Spheres method We find the values of ξ such that 𝛾1 (ξ ) − 2𝛾2 (ξ ) − 𝛾2 (η) > 3 − 𝛾2 (η) , or

𝛾1 (ξ ) − 2𝛾2 (ξ ) > 3 .

From this we get



sinh ξ cosh ξ sinh ξ −2 − > 3. ξ ξ2 ξ3

Simple calculations lead to (8.38).

8.6 Alternative Schwarz procedure In this section, we present a Random Walk method for solving the first boundary value problem for the Lamé equation

μΔu + (λ + μ ) grad div u = 0 ,

x ∈ G,

u(y) = g(y) ,

(8.39)

y ∈ Γ,

where G = B1 ∪ B2 is a union of two overlapping balls in 3D G = B 1 ( x 1 , r1 ) ∪ B 2 ( x 2 , r2 ) , Γ is the boundary of G. Comparing with the general case of the Random Walk on Sphere process we see that in this special case ε = 0.

R1

R2 y Θ*2

Θ*1 O1

O2

x Γ1

γ1

γ1 Γ2

Figure 8.2. Two overlapping balls of the Schwarz iteration algorithm.

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8.6 Alternative Schwarz procedure | 145

We introduce the following notation (see Figure 8.2): 𝛾1 is the part of the sphere ∂B1 lying in B2 and 𝛾2 is the part of the sphere ∂B2 lying in B1 , while Γ1 = ∂B1 \ 𝛾1 and Γ2 = ∂B2 \ 𝛾2 . We denote the curve (a circle) of intersection of the spheres ∂B1 and ∂B2 by l. It is also convenient to introduce the notation u|Γ1 = g(1) and u|Γ2 = g(2) . The generalized Poisson formula (see Theorem 6.3 in 2D and Theorem 6.8 in 3D) for an arbitrary point x ∈ B1 can be written in the matrix form as u(x) = (K1 + F1 )u , where



K1 u(x) =



C(x, y)g(y)dS(y) ,

F1 u(x) =

C(x, y)u(y)dS(y) , 𝛾1

Γ1

and the matrix C(x, y) is defined in the above mentioned theorems. The same relation is true at y ∈ B2 : u(y) = (K2 + F2 )u , where



K2 u(y) =



C(y, y )g(y )dS(y ) ,

F2 u(y) =

C(y, y )u(y )dS(y ) .

𝛾2

Γ2

Now, we can represent the solution through the iterative procedure u(x) = K1 g + F1 K2 g + F1 F2 K1 g + F1 F2 F1 K2 g + F1 F2 F1 F2 K1 g + · · · . which formally can be represented as u(x) =

∞ 

G j ( K l g) ,

(8.40)

j =1

where G j = F1 F2 F1 F2 · · · F1 F2 · · · F p , = >? @ ⎧ ⎨1 ,

l=⎩ 2,

j times

if j is even, if j is odd,

while ⎧ ⎨1 ,

p=⎩

2,

if j is even, if j is odd.

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146 | 8 Random Walk on Spheres method The multiple integrals in this sum can be evaluated by the Monte Carlo method on the corresponding Markov chain. For example, the term ⎧  ⎪ ⎨ F1 F2 F1 K2 g(x) = C(x, y1 )dS(y1 ) ⎪ C(y1 , y2 )dS(y2 ) ⎩ 𝛾1 𝛾2 ⎫  ⎪  ⎬ × C(y2 , y3 )dS(y3 ) C(y3 , y)g(y)dS(y) ⎪ ⎭ 𝛾1

Γ2

is estimated on Markov chains starting at the point x with the transitions x → y1 ∈ 𝛾1 → y2 ∈ 𝛾2 → y3 ∈ 𝛾3 → y ∈ Γ2 . However, we now describe a different iteration method, a generalized Schwarz method whose convergence was proved in [201]. Let us define the energy of deformation for any displacement vector v as . μ E (v ) = λ(div v)2 + (∇α v β + ∇β v α )2 dV . (8.41) 2 G

It is well known that the boundary value problem (8.39) is equivalent to the problem of finding a function v such that v|Γ = g and which minimizes the energy integral (8.41). The Schwarz iterative method consists in constructing a sequence of functions u(i) , i = 0, 1, . . . , such that the functions u2k are regular in B2 and B1 \ B2 , satisfy in these domains equation (8.39), they are continuous in G and ⎧ ⎨g(0) if k = 0 , u2k |Γ1 = g(1) , u2k |Γ2 = g(2) , u2k |𝛾2 = ⎩ (8.42) u2k−1 if k > 0 . Here g(0) is an arbitrary sufficiently smooth vector function whose values on the circle l coincide with those of the functions g(1) and g(2) . All functions u2k+1 are continuous in G, regular in B1 and B2 \ B1 , satisfy equation (8.39) and u2k+1 |Γ1 = g(1) ,

u2k+1|Γ2 = g(2) ,

u2k+1 |𝛾1 = u2k |𝛾1 ,

k ≥ 0.

(8.43)

The proof of convergence of uk (see [201]) is based on the estimations E(u2k ) ≤ E(u2k−1 ) and E(u2k+1 ) ≤ E(u2k ) . Finally, representing uk through the Green function and taking the limit in the Green formula one finds that limk→∞ uk solves the original problem. The same scheme also works if there are more than two overlapping balls.

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8.6 Alternative Schwarz procedure | 147

Using the generalized Poisson formulae derived above we can construct an effec­ tive numerical algorithm based on the Schwarz method. Indeed, choose a sufficiently fine grid on the spheres ∂B1 and ∂B2 and calculate successively the functions uk from the relevant Poisson formula. Note that in 3D some coefficients in the Poisson formula should be evaluated numerically in advance, for a fixed grid. In 2D all the coefficients are given explicitly.

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9 Random Walk on Fixed Spheres for Laplace and Lamé equations 9.1 Introduction In this chapter, we present the Random Walk on Fixed Spheres (RWFS) introduced in our paper [170] and further developed in [172]. We consider in details the Laplace and Lamé equations governing static elasticity problems. The approach is based on the Poisson-type integral formulae written for each disk of a domain consisting of a family of overlapping disks. The original differential boundary value problem is equiv­ alently reformulated in the form of a system of integral equations defined on the in­ tersection surfaces (arches, in 2D, and caps, if generalized to 3D spheres). To solve the obtained system of integral equations, a Random Walk procedure is constructed where the random walks are living on the intersecting surfaces. Since the spheres are fixed, it is convenient to construct also discrete random walk methods for solving the system of linear equations approximating the system of integral equations. Here we develop two classes of special Monte Carlo iterative methods for solving these systems of linear algebraic equations that are constructed as a kind of randomized versions of the Chebyshev iteration method and successive over relaxation (SOR) method. It is found that in this class of randomized SOR methods, the Gauss–Seidel method has a minimal variance. In [172] we have concluded that in the case of classical potential theory, the RWFS considerably improves the convergence rate of the standard Ran­ dom Walk on Spheres (RWS) method. More interesting, we succeeded there to extend the algorithm to the system of Lamé equations which cannot be solved by the con­ ventional RWS method. We present here a series of numerical experiments for 2D do­ mains consisting of 5, 10, and 17 disks, and analyze the dependence of the variance on the number of disks and elastic constants. Further generalizations to Neumann and Dirichlet–Neumann boundary conditions are straightforward. There are two main classes of stochastic numerical methods for solving PDEs: (1) methods based on probabilistic representations of solutions in the form of expecta­ tions over diffusion stochastic processes (see, e.g. [48]), (2) methods based on Markov chain simulation technique for solving integral equations; here the crucial point is the reformulation of the original boundary value problem in the form of integral equa­ tion [50; 51; 160]. The probabilistic representations are possible however only for scalar elliptic and parabolic equations, e.g. see [48]. Even in this case, considerable difficulties arise when approximating the random process near the boundary: one should take care that in each step, the simulated diffusion process is inside the domain. This implies a rapid diminishing of the integration step when approaching the boundary, which in turn rapidly increases the computational cost [116]. Exterior boundary value problems

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9.1 Introduction

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| 149

are hard or better to say impossible to solve by a numerical simulation of diffusion pro­ cesses in unbounded regions. The methods based on integral equation reformulation are much more flexible. Generally, any boundary value problem transformed into an integral form can be solved by a Markov chain simulation technique. It should be noted however that the Monte Carlo methods for solving integral equations are traditionally applied to inte­ gral equations v = Kv + f with substochastic kernels when K  < 1. The method was first developed for solving linear radiative transfer equations and the famous Neu­ mann–Ulam scheme works under the condition that the Neumann series converges. The first Monte Carlo method for integral equations with divergent Neumann se­ ries was suggested by Sabelfeld [157] and further developed in [160] and [154]. The method is based on a conformal transformation of the spectral parameter. In this approach the variance analysis is much more difficult than in the Neumann–Ulam method. Nevertheless, it opens new interesting possibilities of applications. For ex­ ample, the Random Walk on Boundary methods were constructed first for Dirichlet problem in [157] and then generalized in [154] to all classical interior and exterior boundary value problems of the electrostatic, heat and elastic potential theory. Note that in this case, there are no difficulties with the boundary conditions and exte­ rior problems. As to the disadvantages of this class of methods, the variance of the method may be large for highly nonconvex domains. However special versions of this method for such domains can be developed, in particular, based on a discretization of the boundary integral equations, or using branching Random Walk on Boundary process [175]. In this chapter, we deal with a new class of Markov chain simulation technique for systems of elliptic equations. The method differs from the conventional RWS method in the following points: (1) the spheres are not randomly chosen, instead, they are deterministically fixed so that the original domain is well approximated by this set of spheres; (2) the randomized evaluation of the solution via the iterations of inte­ grals follows not the Neumann–Ulam scheme, but a different iterative method, e.g. the Chebyshev iterations, or the SOR method; (3) since the phase space is fixed, one may introduce its discretization and construct discrete Random Walks. It should be noted that one can think of different combinations of (1)–(3). For ex­ ample, the choice (1) means that the phase space is fixed and the Random Walk is constructed on the set of fixed spheres; (1) and (2) means that the relevant iteration procedure is constructed directly for integral operators, and finally, with (1)–(3), we turn to the discrete Random Walk method organized according to the relevant itera­ tive procedure. Thus the idea of the method is that the original boundary value problem is refor­ mulated in an integral form derived from the spherical mean value relations for fixed overlapped disks (see also [4; 37]). The basic approach is described in our book [155]. In [171], we have extended this approach by using the Poisson integral formula for overlapping spheres and considered the relevant system of integral equations. The

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150 | 9 Random Walk on Fixed Spheres for Laplace and Lamé equations kernel of the Poisson integral formula was the generating transition probability den­ sity function (pdf) of the Markov chain. The iterative procedure was actually a ran­ domized method of simple iterations. Generally, this iterative procedure diverges in the case of the Lamé equation. Therefore, we have introduced different iteration meth­ ods, in particular, an iterative procedure with random parameters, which coincides in the deterministic limit with Chebyshev’s method and a randomized SOR. A general discrete random walk scheme can be constructed through a discrete approximation of our system of integral equations. Surprisingly, this not only has complicated the method, but also in contrast, we have obtained a convenient fast convergent method with a finite variance. Note that extensions from disks to ellipses can be readily done by using the relevant Poisson formula for ellipses, e.g. see [152]. Generalization to 3D case is also quite simple, since the Poisson kernel is explicitly known. The chapter is organized as follows. In Section 9.2, we present the main idea of the method for a simple case of the 2D Laplace equation and study the main proper­ ties of the generating integral equation. Section 9.3 deals with the system of elliptic equations, the so-called Lamé equation governing the elastic deformations of 2D bod­ ies. In Section 9.4 we present iteration methods we use to construct Random Walk algorithms. For this purpose we use an iteration method with randomly chosen pa­ rameters, introduced by Vorobiev (see, e.g. [107; 211]) and the SOR method that is par­ ticularly efficient for a special class of domains which we call DS2 -domains. Detailed description of Random Walk algorithms are given in Section 9.5 and numerical simu­ lations are presented in Section 9.6.

9.2 Laplace equation To explain the main idea of the method, we present here the case of two-dimensional Dirichlet problem for the Laplace equation. It should be noted that even in this simple case where the conventional RWS works as well, the new method converges much faster and the accuracy achieved is considerably higher.

9.2.1 Integral formulation of the Dirichlet problem Let us consider the boundary value problem Δu (x) = 0 ,

x ∈ D,

u(y) = φ ,

y ∈ Γ = ∂D , (1)

(9.1) (2)

where the domain D consists of two overlapping disks K (x0 , R1 ) and K (x0 , R2 ) cen­ (1) (2) tered at O1 = x0 and O2 = x0 (see Figure 8.2 in Chapter 8): (1)

(2)

D = K ( x0 , R1 ) ∪ K ( x0 , R2 ) ;

(1)

(2)

K (x0 , R1 ) ∩ K (x0 , R2 ) = ∅ .

(9.2)

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9.2 Laplace equation | 151 (1)

(1)

We denote by 𝛾2 the part of the circle S(x0 , R1 ) = ∂K (x0 , R1 ) which belongs to (1) the second disk while Γ1 is the part of the circle S(x0 , R1 ) not belonging to the second disk; analogously 𝛾2 and Γ2 are defined. So the boundary of the domain D consists of Γ1 and Γ2 and 𝛾1 ∪ 𝛾2 is the phase space of the integral equation to be constructed. The regular solution to the harmonic equation satisfies the spherical mean value relation in each of the two disks:  u (y)dS y R2 − r2 . (9.3) u(x) = 2πR |x − y|2 S ( O,R )

Here R = R1 in the first and R = R2 in the second disk, and the same for r = (i) |x − x0 |, the distance from x ∈ 𝛾i to the circle’s center O = O i , i = 1, 2. It is not difficult to find out that the function (1)

p(y; x) =

R21 − |x − x0 |2 1 · 2πR1 |x − y|2

(1)

(9.4)

(1)

is a pdf of the variable y ∈ S(x0 , R1 ), for all x ∈ K (x0 , R1 ). This immediately follows from the representation of the solution u = 1 to the Dirichlet problem for the Laplace equation Δu (x) = 0, u (y) = 1 through the Poisson integral. From the probabilis­ tic representation of the Dirichlet boundary value problem considered, the density (1) p(y; x) coincides with the pdf of the first passage on S(x0 , R1 ) of a Wiener process (1) starting at x ∈ K (x0 , R1 ). Simple simulation of the transition according to (9.4) is given in [80]. It is possible to find explicitly the distribution function P(x → y ∈ 𝛾) – the prob­ ability for a particle starting at x ∈ S(x0 , R), with r = |x − x0 |, to reach an arc 𝛾 ∈ S(x0 , R) defined by the limit angles α 1 and α 2 , say, α 1 < α 2 , since (see, e.g. [63]): 

 α2 dS y α  1 R+r  = arctg , tg |x − y|2 π R−r 2 α1 𝛾     1 R + r α2 1 R + r α1 = arctg . − arctg tg tg π R−r 2 π R−r 2

R2 − r2 P(x → y ∈ 𝛾) = 2πR

(9.5)

Let us now write down the Poisson formulae for both disks in the form  (1) R2 − |x − x0 |2 u(y) u(x) = 1 dS y , x ∈ 𝛾1 , 2πR1 |x − y|2 (1)

S ( x 0 ,R 1 )

u(y) =

R22 − |y − 2πR2

(2) x0 |2



(2 ) S ( x 0 ,R 2 )

u(x ) dS x , |y − x |2

y ∈ 𝛾2 .

(9.6)

We can give different equivalent formulations of the boundary value problem starting from these Poisson formulae. First, let us derive a scalar Fredholm linear

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152 | 9 Random Walk on Fixed Spheres for Laplace and Lamé equations integral equation of the second kind for the solution u (x). To this end, we define the kernel K (x, y), x, y ∈ 𝛾1 ∪ 𝛾2 as follows. For x ∈ 𝛾1 : ⎧ ⎨0,

K (x, y) = ⎩

y ∈ 𝛾1

k 11 (x, y) =

(1)

R 21 −| x − x 0 |2 1 2πR 1 | x − y |2 ,

y ∈ 𝛾2 ,

and for x ∈ 𝛾2 : ⎧ ⎨0 , K (x, y) = ⎩ k (x, y) = 22

y ∈ 𝛾2 (2 )

R 22 −| x − x 0 |2 1 2πR 2 | x − y |2

y ∈ 𝛾1 .

,

Using this notation we can rewrite formulae (9.6) as follows:  u(x) = K (x, y)u (y)dS(y) + f (x) ,

(9.7)

𝛾1 ∪𝛾2

where (1)

f (x) =

R21 − |x − x0 |2 2πR1 (2)

f (x) =

R22 − |x − x0 |2 2πR2

 Γ1



Γ2

φ(y) dS y , |x − y|2

for

x ∈ 𝛾1 ,

φ(y) dS y , |x − y|2

for

x ∈ 𝛾2 .

(9.8)

Note that this equation is not symmetric, but we can show that it can be symmetrized. Indeed, let us introduce a new function by : ; ; Ri w ( x ) = u ( x )< 2 , x ∈ 𝛾i , i = 1, 2 . (i) R i − |x − x0 |2 Then we get the following equation:  ˜ (x, y)w(y)dS(y) + f (x) , w( x ) = K

(9.9)

𝛾1 ∪𝛾2

where for x ∈ 𝛾1 ⎧ ⎪ ⎨0,

˜ (x, y) = K ⎪ ⎩

A

1 1 2π | x − y |2

(1)

R 21 −| x − x 0 |2 R1

A

y ∈ 𝛾1 (2 )

R 22 −| y − x 0 |2 R2

,

y ∈ 𝛾2 ,

and for x ∈ 𝛾2 : ⎧ ⎪ ⎨0,

˜ (x, y) = K ⎪ ⎩

1 1 2π | x − y |2

A

(1) R 21 −| y − x 0 |2

R1

A

y ∈ 𝛾2 (2 ) R 22 −| x − x 0 |2

R2

,

y ∈ 𝛾1 .

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˜ (x, y) Thus we come to the integral equation with the symmetric kernel K  ˜ (x, y)w(y)dS(y) + f (x) . K w( x ) =

(9.10)

𝛾1 ∪𝛾2

This implies that the eigenvalues of the integral operator defined by the kernel K (x, y) in (9.7) are all real, moreover, we will give below the explicit expression for the princi­ pal eigenvalue of this integral operator. As mentioned above, it is possible to give a different equivalent integral equation formulation of the problem under study. Let us introduce the notation: v1 (x) = u (x) for x ∈ 𝛾1 and v2 (x) = u (x) for x ∈ 𝛾2 . Then, (9.7) reads  v1 (x) = p(y; x)v2 (y)dS y + f1 (x) , 

v2 (x) =

𝛾2

p(x ; x)v1 (x )dS x + f2 (x) ,

(9.11)

𝛾1

where



f1 (x) =



p(y; x)φ(y)dS y ,

f2 (x) =

Γ1

p(x ; x)φ(x )dS x .

(9.12)

Γ2

It is convenient to rewrite the system (9.11) in the matrix form v = Gv + F ,

(9.13)

where v = (v1 , v2 )T , F = (f1 , f2 )T , and G is the matrix-integral operator which acts on v as follows:

G11 G12 v1 Gv = G21 G22 v2 

0 v1 𝛾2 p ( y; x ) ∗ ( y ) dS y =  v p ( x ; x ) ∗ ( x ) dS 0 (9.14) 2 . x 𝛾 ⎞ ⎛ 1 p(y; x)v2 (y)dS y ⎟ ⎜ 𝛾2 ⎟ ⎜ =⎝ ⎠  𝛾1 p ( x ; x ) v 1 ( x ) dS x The integral equation (9.7) and its equivalent vector counterpart (9.13) with the integral operator G have nice properties. First of all, the L1 -norm of G is less than 1,  for any configuration of the two overlapping disks, since S(x,R) p(y; x)dS y = 1. Hence (E − G)−1 exists and is represented as a convergent Neumann series. This also follows from the next assertion that presents a nice property of the Poisson kernel and gives simultaneously an interesting characterization of the Wiener process.

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154 | 9 Random Walk on Fixed Spheres for Laplace and Lamé equations Theorem 9.1. For any x ∈ 𝛾1 and any y ∈ 𝛾2   θ∗ θ∗ θ∗ p(y; x)dS y = p(y ; y)dS y = 1 − 1 − 2 = 12 , π π π 𝛾2

(9.15)

𝛾1

∗ ∗ where the angles θ∗ 1 and θ 2 are defined as follows (see Figure 9.1): 2θ 1 is the angle of ∗ view of the arc 𝛾2 from the center of the first circle, and 2θ2 is the angle of view of the arc ∗ ∗ 𝛾1 from the center of the second circle. The angle θ∗ 12 = π − θ 1 − θ 2 is the angle of view of the segment (O1 , O2 ) from the intersection point P1 or P2 . This property characterizes the Wiener process as follows. For a Wiener process starting from x ∈ 𝛾1 , the probability to reach the arc 𝛾2 does not depend on the starting point x and is explicitly given by (9.15). Moreover, the same is true for the Wiener process starting from y ∈ 𝛾2 : the probability to reach the arc 𝛾1 is not depending on the starting point y and is equal to the same constant θ∗ 12 / π given in (9.15).

Proof. Partly, we presented this result in [172]. Here we give a different proof of this elegant result but first let us recall the main idea of the proof in [172]. For any x ∈ 𝛾1 we have obviously: cos(ψ) 1 − , (9.16) p(y; x) = π|x − y| 2πR1 which follows from the relation R21 − r21 + |x − y|2 = 2R1 |x − y| cos(ψ) where ψ is the angle between the vectors x − y and ny , the inner normal vector at the point y which (1) is collinear to x0 − y. Using relation (9.16) we can write for any x ∈ 𝛾1 : ⎧ ⎫ ⎪ ⎪   ⎬ 1 ⎨ cos(ψ) 1 p(y; x)dS y = ⎪ dS y ⎪ − dS y . π ⎩ |x − y| ⎭ 2πR1 𝛾2

𝛾2

𝛾2

On the right-hand side, the first integral in the braces is the double layer potential integral which is equal (see, e.g. [160]) to the angle of view of the arc 𝛾2 from the point x, ∗ i.e. to (2π −2θ∗ 2 )/2. The second integral is simply θ 1 / π. This completes the proof, since by symmetry, exactly the same result is obviously obtained for the second disk, when y ∈ 𝛾2 . Let us give another proof of this assertion, based on the series representation of the kernel p(y; x). It is not so elegant as the above proof, but the technique can be used in solving the full eigenvalue problem. The expansion of the kernel we need is [81]: p(y; x) =

∞ 1 − ρ2 1 1  k 1 = + ρ cos[k (θ − α )] . 2π 1 − 2ρ cos(θ − α ) + ρ 2 2π π k=1

(9.17)

Here we use the polar coordinates: the point x is specified by (r, θ), ρ = r/R, and the point y by (R, α ).

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P1

y Θ*2

Θ*1

P3

O2

θ x

O1

P2 Figure 9.1. Illustration to the proof of Theorem 9.1.

Let us introduce some notations. The first disk is centered at the origin O1 = 0 and O2 is the center of the second disk. We denote the point of intersection of the coordi­ nate axes O1 X with the second circle by P3 . Let P1 and P2 be the points of intersections of our circles. We introduce the following angles (see the picture of Figure 9.1): ψ1 is the angle between the vector O1 P1 and x − P1 and ψ2 is the angle between the vector O1 P2 and x − P2 . Further, φ1 is the angle between x − P1 and P3 − P1 ; φ2 is the angle between x − P2 and P3 − P2 . Routine calculations yield ∗

θ1 

I= −θ∗ 1

1 − ρ2 dα 2π 1 − 2ρ cos(θ − α ) + ρ 2 ∗

θ1  ∞ 2θ∗ 1  k 1 = + ρ cos[k (θ − α )]dα 2π π k =1 −θ∗ 1

θ∗ 1 = 1 + π π θ∗ 1 = 1 + π π

∞ 



ρ

k

k =1



sin[k (θ∗ sin[k (θ∗ 1 − θ )] 1 + θ )] + k k



ρ sin(θ∗ ρ sin(θ∗ 1 − θ) 1 + θ) + arctg arctg ∗ 1 − ρ cos(θ1 − θ) 1 − ρ cos(θ∗ 1 + θ)



.

(9.18)

ρ sin(θ∗ 1 + θ) . 1 − ρ cos(θ∗ 1 + θ)

(9.19)

It is readily seen that ψ2 = arctg

ρ sin(θ∗ 1 − θ) , 1 − ρ cos(θ∗ 1 − θ)

ψ1 = arctg

∗ ∗ ∗ Simple geometric analysis shows that θ∗ 1 + θ 2 /2 + ψ 1 + φ 1 = π, θ 1 + θ 2 /2 + ψ 2 + φ 2 = π ∗ ∗ and φ1 + φ2 = π. This yields ψ1 + ψ2 = π − 2θ1 − θ2 , which, in view of (9.18), (9.19) completes the proof.

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156 | 9 Random Walk on Fixed Spheres for Laplace and Lamé equations In constructions of iterative numerical procedures, we will need the information about the principal eigenvalue of the integral operator. In the next theorem we find this eigenvalue explicitly. This result was formulated in [172]; however the proof there was incomplete. We give here the revised version of the proof. Theorem 9.2. The integral operator G is a Fredholm operator with the kernels p(y; x), p(x ; y), continuous on x ∈ 𝛾1 , y ∈ 𝛾2 , with integrable singularities at the points of ∗ intersection of 𝛾1 and 𝛾2 of the type p(y; x)  (sin (θ∗ 1 + θ 2 ))/( π |x − y |) as x → y. The eigenvalues of G, λ i , are all real, and moreover, λ i = ±σ i ρ (G) where σ i ≤ 1 are some positive constants and ρ (G) is the spectral radius of G given explicitly by ρ (G) = 1 −

θ∗ θ∗ θ∗ 1 − 2 = 12 . π π π

The integral equation (9.13) has a unique solution which solves the Dirichlet prob­ lem (9.1). Proof. First let us show that the singularities have the form p(y; x)  (sin(θ∗ 1 + 2 2 θ∗ ))/( π | x − y |) as x → y. Simple geometrical considerations show that R − r = 1 2 ∗ ∗ |x − y|· b, where b = |x − y | and y is the second point of intersection of the line x − y with the circle S(x0 , R1 ). Thus p(y; x) = (b )/(2πR1 |x − y|). Now, as x ∈ 𝛾1 → y ∈ 𝛾2 , ∗ we have asymptotically b  2R1 sin (θ∗ 1 + θ 2 ). Let us now consider the eigenvalue problem. Note that the integral operator G is not symmetric, but we can symmetrize it if we follow the symmetrization we used above. Indeed, introducing the new functions : : ; ; ; ; R1 R2 < < w1 ( x ) = v1 ( x ) × , w ( x ) = v ( x ) × , 2 2 (1) 2 (2) 2 2 R1 − |x − x0 | R2 − |x − x0 |2 we come to the eigenvalue problem for the symmetric integral equation ¯ , λw = Gw ¯ is defined by where the matrix-integral operator G ⎛

¯ = ⎝ Gw

0

g 21 ( x,y ) 𝛾2 | y − x |2

Here

and

∗(x)dS x



g 12 ( x,y ) 𝛾1 | x − y |2

∗(y)dS y

0

⎞ ⎛w ⎞ 1 ⎟ ⎠⎜ ⎝ ⎠. w2

: : ; ; (1) ; 2 (2) 2 1 ; < R1 − |x − x0 |2 < R2 − |y − x0 |2 g12 (x, y) = , 2π R1 R2 : : ; ; (1) 2 ; 2 (2) 2 1 ; R − | y − x | < 1 < R2 − |x − x0 |2 0 . g21 (x, y) = 2π R1 R2

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So our system of integral equations is symmetric since g12 (x, y) = g21 (y, x), and hence, the eigenvalues λ are real, moreover, they are concentrated in the interval (−ρ, ρ ) symmetrically relative to the origin, where ρ = ρ (G) < 1 is the spectral ra­ dius. Indeed, if λ is an eigenvalue with the corresponding eigenfunction (w1 , w2 ), then −λ is also an eigenvalue with the corresponding eigenfunction (w1 , −w2 ). Let us now evaluate the spectral radius of our system of integral equations. Taking the eigenfunction as a constant (1, 1)T , we see that the corresponding eigenvalue is given by  θ∗ θ∗ θ∗ λ0 = p(y; x)dS y = 1 − 1 − 2 = 12 , (9.20) π π π 𝛾i

which does not obviously depend on x. It is not difficult to show that ρ (G) = λ0 . Indeed, let λ be an arbitrary eigen­ value and (w1 , w2 )T – the corresponding eigenfunction. For any x ∈ 𝛾1 we can write |λ||w i (x)| ≤ |w2 (y∗ )|λ0 , where y∗ is a point where |w2 | reaches its maximum. For any y ∈ 𝛾2 we analogously have |λ||w2 (y)| ≤ |w1 (x∗ )|λ0 , where x∗ is the point of maximum of |w1 |. From these two inequalities we get the desired result: |λ|2 ≤ λ20 . Thus ρ (G) = λ0 . Finally, the equivalence of the integral equation and the Dirichlet problem is ob­ vious; the solution of the Dirichlet problem satisfies the integral equation whose so­ lution is unique. This completes the proof.

9.2.2 Approximation by linear algebraic equations Having the integral equation (9.7) or its symmetric version (9.10), we can construct a standard Random Walk based on the Neumann–Ulam scheme, with the phase space 𝛾1 ∪ 𝛾2 . Moreover, we can also construct different iterative methods, e.g. the SOR method, directly for these equations. However, for more general domains, it is convenient to deal with discrete Random Walks. To this end, we need to approximate the integral equations by the relevant system of linear algebraic equations . So let us approximate the system of integral equations (9.13) by a system of linear algebraic equations. We choose a set of nodes x1 , . . . , x m1 +1 uniformly on the arc 𝛾1 and y1 , . . . , y m2 +1 on 𝛾2 generating by the uniform polar angles distributions (the end points are included). These meshes subdivide 𝛾1 and 𝛾2 in the set of arches 𝛾1(i) , i = 1, . . . , m1 and 𝛾2(i) , i = 1, . . . , m2 , respectively. Of course, the nodes can be chosen not uniformly, say, according to some distribution which generates the nodes more densely around the singular points where the arches do intersect.

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158 | 9 Random Walk on Fixed Spheres for Laplace and Lamé equations Since the Poisson kernel p(y; x) has a singularity, it is convenient to take the ap­ proximation in the form 

p(y; y k )v2 (y)dS y =

m 1 +1

(1)

p i (x i , y k )v2 (x i ) ,

k = 2, . . . , m2 ,

i =1

𝛾1

and analogously, 

p(x ; x k )v1 (x )dS x =

m 2 +1

(2)

p i (y i , x k )v1 (y i ) ,

k = 2, . . . , m1 ,

i =1

𝛾2

where



(1)

p i (x i , y k ) =

p(y; y k )dS y ,

(2)



p i (y i , x k ) =

(i) 𝛾1

p(x ; x k )dS x .

(9.21)

(i) 𝛾2

These coefficients can be evaluated explicitly, using formula (9.5). The same ap­ proximation is used to calculate the right-hand sides f1 and f2 in all grid points. Thus we come to a discrete approximation of (9.13) in the form of the following system of linear algebraic equations: w(k) =

m 1 +m2

a ik w(k) + F (k) ,

k = 1, . . . , m1 + m2 ,

(9.22)

i =2

or in a matrix form w = Aw + F. Here the column vector w = (w1 , w2 )T consists of two column vectors: w1 , whose components approximate the function v1 and w2 approximating the function v2 . The same for the vector F. Note that the matrix A is a square matrix, it has a 2 × 2-block form, with zero diagonal blocks, and rectangular blocks A12 and A21 relating the vectors w1 and w2 . In our calculations, we also use a slightly different approximation, by applying an appropriate interpolation of the integrated functions v and F and applying a 12-point refinement Gauss approximation formula at the end points of the arc.

9.2.3 Set of overlapping disks Generalization to connected domains consisting of n arbitrarily overlapping disks is not difficult; the main problem is to choose a convenient numeration of the arches inside the disks. Generally this is a tricky problem but not too difficult for computer implementation. We have written a code that automatically generates a numeration and the relevant matrix A of the generated system of linear equations. There are domains for which the structure of the matrix kernel of the system of in­ tegral equations is very simple and convenient both for theoretical analysis and com­ puter implementation. For example, assume that each disk is overlapping only with

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Figure 9.2. A chain of five disks, generating a system of eight integral equations.

two immediate neighbor disks which are nonoverlapping – see Figure 9.2. The relevant system of integral equations is written in a block matrix form, whose general structure is shown in (9.25) for the case when the domain is a chain of five overlapping disks. This is a particular example belonging to a more general class of domains which we will call DS2 -domains (DS stands for disks, index 2 means the generated matrix is cyclic of index 2). Definition 9.1. A DS2 -domain is defined as follows: (1) the domain is a connected union of overlapping disks, (2) each disk may overlap with an arbitrary number of disks but each intersection is a result of overlapping of only two disks, (3) any subset of disks that is a closed family of disks (a chain of successively overlapping disks where the first disk overlaps with the last disk (see, e.g. Figure 9.3) consists of an even number of disks.

2 1

γ11 γ6 6

γ10

γ1

γ12

3

γ2 4

5

γ7 γ5

γ9 γ4

γ3

γ8

Figure 9.3. “Red–Black” indexation. A DS 2 -domain as a closed chain of six disks.

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2

γ8

4

3

1

γ6

γ7

5

γ5 γ4

γ3 γ1

γ2

Figure 9.4. “Red–Black” indexation. A DS 2 -domain with five disks.

Property A1. In a DS2 -domain, the numbering of the arches can be chosen so that the generating matrix kernel G of the integral operator G is a cyclic matrix of order 2 and hence it has the following block form:

0 G12 G= . (9.23) G21 0 This numbering of arches we call a consistent numbering. Indeed, the consistent numbering is constructed as follows. Let us index the disks successively following say a clockwise direction, as 1, 2, . . . , n. We divide all the disks in two different classes of disks: the first class (say, “red” disks) includes the disks with odd indices 1, 3, 5, . . . and the second class includes the disks (say, “black” disks) with even indices. Now, the numbering of arches is as follows: first we number successively along the chosen direction the arches of the disks belonging to the first class, and then turn, in the last disk, to numbering successively the arches of the disks in the second class. For illustration we show in Figures 9.4–9.5 examples of DS2 -domains, with the red–black indexation: a simple chain of five disks in Figure 9.4, with the matrix A given schematically in Figure 9.6 , left picture; a closed set of six disks in Figure 9.3 and a more complicated set of 17 disks in Figure 9.5. By the construction of the indexation it is clear that the matrix G has the desired block structure (9.23) where G12 and G21 relate the arches of the first and second groups of disks. By Property A1 and from the block structure (9.23) the following property readily follows. Property A2. Let D be an arbitrary DS2 -domain, with a consistent numbering of the arches and let us take the decomposition G = L + U where L is the left and U the right triangular operators whose matrix kernel is given by (9.23). Then the following equalities are true: (E − ωL)−1 = E + ωL

and L2 = U2 = 0 ,

(9.24)

where ω is an arbitrary parameter.

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9.2 Laplace equation |

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2 1 γ11 γ10

γ6

3

γ1

6 γ12

γ2 4

5

γ7 γ5

γ3 γ9

γ4

7

γ8 10 8 9

17 13

11

16

12 14 15

Figure 9.5. “Red–Black” indexation. A DS 2 -domain with 17 disks, which includes four closed subsets of even number of disks.

Figure 9.6. General structure of block matrices for the consistent numbering through “Red–Black” indexation. Left picture: five disks shown in Figure 9.2. Right picture: the same geometry but for six disks.

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162 | 9 Random Walk on Fixed Spheres for Laplace and Lamé equations This property will be used in Section 9.4 where we construct the SOR method for solv­ ing this kind of linear equations. To illustrate how different can be the generating matrix for different numbering, we consider the domain consisting of n = 5 disks presented in Figure 9.2. First, let us choose the simplest successive numbering of all arches of our phase space, say, from left to right. Introducing k = 2(n − 1) functions v i and writing the Poisson formulae in each disk we come to a system of k integral equations v = Gv + f where the kernel of the matrix integral operator G is a k × k-matrix G which has the following structure: in the first row, only the kernel G12 is not zero, the second and the third rows have the following nonzero kernels: G21 , G24 and G31 , G34 . The same for the rows 4 and 5: the nonzero entries are G43 , G46 and G53 , G56 , etc., so that the jth row nonzero entries (j is even) are G j,j−1 and G j,j+2, while the j + 1th row nonzero entries are G j+1,j−1 and G j+1,j+2. The last row has only one nonzero entry: G k,k−1. So the kernel matrix G has the following structure (n = 5 disks and number of arches k = 8): ⎞ ⎛ 0 G12 0 0 0 0 0 0 ⎟ ⎜ ⎜G21 0 0 G24 0 0 0 0 ⎟ ⎟ ⎜ ⎜G 0 0 G34 0 0 0 0 ⎟ ⎟ ⎜ 31 ⎟ ⎜ ⎟ ⎜ 0 0 0 G 0 0 0 G 43 46 ⎟. ⎜ G=⎜ (9.25) ⎟ 0 0 G 0 0 G 0 0 53 56 ⎟ ⎜ ⎟ ⎜ ⎜ 0 0 0 0 G65 0 0 G68 ⎟ ⎟ ⎜ ⎜ 0 0 0 0 G75 0 0 G78 ⎟ ⎠ ⎝ 0 0 0 0 0 0 G87 0 We will now show that to ensure that the system of linear algebraic equations u = Au + f obtained as described above is a good approximation to the exact system of integral equations it is enough to prove that (E − A)−1 exists. This in turn is ensured by the fact that our matrices are all substochastic and their spectral radii are all less than 1. Let D be a DS2 -domain with the boundary Γ = ∂D. Let us consider an arbitrary disk of this domain, say, K (x k , R k ) with S(x k , R k ) = ∂K (x k , R k ) which is overlapped with say m k other disks. Then, S(x k , R k ) consists of two sets of arches: one set (we denote it by 𝛾˜k ) consists of arches lying in the overlappings and the second one (we ˜ k ) is a part of the boundary Γ. denote it by Γ Assume that we have fixed some numbering of the arches in our domain D and the relevant numbering of the functions so that the function v j (x) is defined on the arc 𝛾j . This numbering generates a block matrix A whose entries are constructed as follows. Let us fix an arc 𝛾j ∈ K (x k , R k ), then the jth block row consists of m k nonzero # blocks; we denote the integer set of numbering these blocks by J j , so that l∈J j means that the sum is taken over all blocks in jth block row. Now we estimate the difference between v j (x i ), the exact solution of the system (j) of integral equations taken on jth arc at a point x i ∈ 𝛾j and the approximation u i

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taken as the ith component of the solution of our linear equation (ith row in the jth (j) (j) block of the matrix A): ϵ i = u i − v j (x i ) . Hence the error vector ϵ has in jth block (j) the components ϵ i . (𝛾˜j )

˜j) (Γ

Let us also define the error vectors δ and δ f with entries δ i and δ i , the errors of approximation of the Poisson integrals (of the function v and of the boundary func­ ˜ l . Thus we can write tion φ, respectively) for jth arc taken over the set of arches 𝛾˜l and Γ for the ith row in the jth block: nl   ˜j) (Γ (j) (l) (l) a ik u k − p(y; x i )v l (y)dS(y) + δ i u i − v j (x i ) = l ∈ J j k =1

=

nl 

l ∈ J j 𝛾l (l)

(𝛾˜j )

(l)

a ik (u k − v l (x k )) + δ i

˜j) (Γ

+ δi

.

l ∈ J j k =1

Thus written in the matrix form these relations are ϵ = Aϵ + δ + δ f .

(9.26)

Let Δφ = maxi (φ i+1 − φ i ) be the maximum difference taken over the all angular meshes. For simplicity we take simple estimations δ < C1 Δφ and δ f  < C2 Δφ. Therefore, we have ϵ ≤ (E − A)−1 (C1 + C2 )Δφ . (9.27)

9.2.4 Estimation of the spectral radius The explicit expression for the spectral radius given in Theorem 9.2 could be obtained only for two overlapped disks. For domains with three and more disks, we will get an estimation for the spectral radius of the integral operator of our problem. Let us first analyze the eigenvalue problem for the system of integral equations (9.13) for a DS2 -domain consisting of three disks, with a consistent numbering of the arches. In this case we have four integral equations for the function u 1 , u 2 , u 3 , u 4 de­ fined on the arches 𝛾1 , 𝛾2 , 𝛾3 and 𝛾4 , respectively. Let P be a 4 × 4 matrix whose nonzero entries are defined by  P ij = p(y; x)dS y , x ∈ 𝛾i , (i, j) = (1, 2), (2, 1), (2, 4), (3, 1), (3, 4), (4, 3) . 𝛾j

So by definition, P ij is the probability of the transition x ∈ 𝛾i → 𝛾j , for the pairs of arches (i, j) = (1, 2), (2, 1), (2, 4), (3, 1), (3, 4), (4, 3). The integrals   P12 = p(y; x)dS y , P43 = p(y; x)dS y , 𝛾1

𝛾3

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164 | 9 Random Walk on Fixed Spheres for Laplace and Lamé equations do not depend on the point x and P12 = P21 , P34 = P43 , see Theorem 9.1. But P24 and P31 do depend on the point x. It is not difficult to find a point x∗ where P24 reaches its ¯ 24 ; the same for P31 . maximum P Indeed, let us denote by x1 and x2 the points of intersection of the second and third disks, S(O2 , R2 ) and S(O3 , R3 ). We now construct a disk S(O23 , R23 ) that goes through the points x1 and x2 and touches the first disk at a point x∗ ∈ 𝛾2 . Then the point x∗ is ¯ 24 . Indeed, of all the points of the the desired point where P24 reaches its maximum P ∗ arc 𝛾2 , the point x is the point with a maximum angle of view of the arc 𝛾4 . As follows from the proof of Theorem 9.1, πP24(x) equals the difference between this angle of view ∗ and θ∗ 23 , the angle of view of the arc 𝛾4 from the center O 2 . This implies that x is indeed ¯ 24 is equal the point where P24 (x) reaches its maximum. Moreover, by Theorem 9.1, P ¯ ¯ to θ23 /π where θ23 is the angle of view of the segment (O2 , O23 ) from the intersection ¯ 31 = θ¯ 31 /π. point x1 or x2 . The same is true for P31 : P ¯ which is obtained from P by changing Let us find the eigenvalues of the matrix P ¯ 24 and P ¯ 31 , respectively. Simple calculations give P24 and P31 with P 9 ¯ 24 P43 P ¯ 31 P212 + P234 ± (P212 − P234 )2 + 4P12 P 2 λ1,2 = . (9.28) 2 The estimation for any eigenvalue of the eigenvalue problem λu = Gu is now carried out directly, as we have done in the case of two disks: from the first equation we have |λ||u 1 (x)| ≤ P12 |u 2 (y∗ )| for any x, y∗ being the point of maximum of u 2 . The value |u 2 (y∗ )| is estimated from the second equation, etc. Simple calculations finally yield ¯ 24 P34 P¯ 31 . (λ2 − P212 )(λ2 − P234 ) ≤ P12 P (9.29) The form of this parabola shows that λ21,2 given by (9.28) are exactly the values where inequality (9.29) becomes an equality. Thus for any eigenvalue λ we have the estima­ tion: |λ| ≤ maxi=1,2 |λ i |. This shows that the spectral radius is mainly defined by the ¯ 24 and P ¯ 31 are small, compared to P12 and value max {P12 , P34 }. Indeed, the values P P34 , as explained above, so we conclude from (9.28) that ρ (G) ≈ max {P12 , P34 } + ε, where ε is a small value. Analogous estimations can be obtained for the general case of DS2 -domains. For arbitrary connected domains consisting of a finite number of overlapped disks one can use simple estimation m  P kj , (9.30) ρ (G) ≤ max k =1,...,n

j =1

where n is the total number of rows and m is the total number of columns in the system of matrix-integral equations; here it is assumed that for all arches that are not the near ¯ kj . neighbors, P kj are changed with a majorant of type P

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9.3 Isotropic elastostatics In this section, we will extend the approach presented in Section 9.2 to a system of elliptic equations (Lamé equation) governing a static 2D elasticity problem. As men­ tioned in Section 9.1, for elliptic systems, probabilistic representations in the form of an expectation over diffusion processes are not known. We mention some efforts to treat this problem. In [148], a direct generalization of the Walk on Spheres process to the Lamé equation was attempted; however, the main problem of divergence there was unperceived. In [19], the authors applied the Malliavian calculus to construct an iterative procedure, but the authors have not estimated the variance that actually is increasing to infinity with the number of iterations. Note also that a model to treat a crack problem described in [131] appeals to an analogy with the diffusion limited aggregation but without any convergence analysis. Suppose a homogeneous isotropic elastic body D ⊂ Rn with a boundary Γ is given, whose state in the absence of body forces is governed by the classical static elasticity equation, the Lamé equation, see, e.g. [96; 155]: Δu(x) + α grad div u(x) = 0 ,

x ∈ D,

(9.31)

where u(x) = (u 1 (x1 , . . . , x n ), . . . , u n (x1 , . . . , x n )) is a vector of displacements, whose components are real-valued regular functions. The elastic constant α α=

λ+μ μ

is expressed through the Lamé constants of elasticity λ and μ. It can be expressed through the Poisson ratio ν = λ/2(λ + μ ) as follows: α = 1/(1 − 2ν). The Poisson ratio characterizes the relative amount of the change of the transverse to longitudinal dis­ placements. It is known that due to thermodynamical reasons ν is bounded between −1 ≤ ν < 0.5. This implies for α: 1/3 ≤ α < ∞. So there are materials with negative values of ν (α varies in 1/3 ≤ α ≤ 1) and materials with ν ≈ 0.5. The last case is very difficult for computational treating. The first boundary value problem for the Lamé equation consists in finding a vec­ ¯ ) satisfying the boundary condition tor function u ∈ C2 (D) ∩ C(D u(y) = g(y),

y ∈ Γ,

(9.32)

where g ∈ C(Γ) is the given vector function. In a full analogy with the Laplace equation, we will use the integral formulation of the given boundary value problem which is based on the spherical mean value re­ lation which is a generalized Poisson formula. In what follows we deal with the twodimensional case. Let us consider an arbitrary point x = (x1 , x2 ) with polar coordinates (r, φ ) inside a disk K (x0 , R) centered at x0 = (x01 , x02 ). The point y = (y1 , y2 ) situated on the circle

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166 | 9 Random Walk on Fixed Spheres for Laplace and Lamé equations S(x0 , R) has the coordinates (R, θ), where θ = φ + β and z is defined by z = y − x, β is the angle between the vectors x and y; ψ is the angle between x and z. Also define the angle φ by φ = φ + ψ. Let us rewrite the Poisson formula for the Lamé equation in a form we use in the construction of the algorithm. Theorem 9.3. The solution to equation (9.31) satisfies the following mean value relation, x being an arbitrary point in K (x0 , R):  2 b ij (x, y)u j (y) R2 − |x − x0 |2  u i (x) = dS y , i = 1, 2 , (9.33) 2πR |x − y|2 j =1 S ( x 0 ,R )

where b ij are functions of x, y, explicitly represented as the entries of the following ma­ trix: α B= α+2

2 |x−y| |x−y| 2 2 cos φ sin φ + R sin (θ + φ) α + 2 cos φ + R cos ( θ + φ ) × . |x−y| |x−y| 2 cos φ sin φ + R sin (θ + φ) α2 + 2 sin2 φ − R cos (θ + φ) Since by definition we have y1 − x01 y2 − x02 , sin θ = , cos θ = R R

cos φ =

y1 − x1 , |x − y|

sin φ =

we get b 11 = 1 + 𝛾1 , b 22 = 1 − 𝛾1 , b 12 = b 21 = 𝛾2 where $ α ( y 1 − x 1 )2 − ( y 2 − x 2 )2 𝛾1 = α+2 |x − y|2 (y1 − x1 )(y1 − x01 ) − (y2 − x2 )(y2 − x02 ) R2 $ α (y1 − x1 )(y2 − x2 ) 𝛾2 = 2 α+2 |x − y|2

.

+

+

(y2 − x2 )(y1 − x01 ) + (y1 − x1 )(y2 − x02 ) R2

y2 − x2 , |x − y|

,

.

.

In the notation of p(y; x) introduced in (9.4), relation (9.33) reads in the matrix form  p(y; x)Bu(y)dS(y) . (9.34) u(x) = S ( x 0 ,R )

Taking this representation for two overlapping disks (see Figure 8.2), we can derive a system of four integral equations defined on the arches 𝛾1 and 𝛾2 . Indeed, let us intro­ (1) (2) (1) duce the notations: v1 (x) = u 1 (x) and v1 (x) = u 2 (x) for x ∈ 𝛾1 and v2 (x) = u 1 (x) (2) and v2 (x) = u 2 (x) for x ∈ 𝛾2 . Then the analog of the system (9.13) can be written as v = Gv + F, or in more details, ⎛ (1) ⎞ ⎛ ⎞ ⎛ (1) ⎞ ⎛ (1) ⎞ v1 v1 f1 0 0 B11 B12 ⎜ (2) ⎟ ⎜ ⎟ ⎜ (2) ⎟ ⎜ (2) ⎟ ⎜v1 ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ 0 B B 21 22 ⎟ ⎜ v 1 ⎟ ⎜ ⎟ ⎜ ⎜f1 ⎟ (9.35) (1) ⎟ + ⎜ (1) ⎟ , ⎜ (1) ⎟ = ⎜ B ⎟ ⎜ ˆ ˆ 0 0 ⎠ ⎝ v2 ⎠ ⎝ f2 ⎠ ⎝ v2 ⎠ ⎝ 11 B12 (2) (2) (2) ˆ 22 0 ˆ 21 B 0 B v2 v2 f2

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where the integral operators B ij , i, j = 1, 2 are defined, according to (9.33), for the (1) points of the first disk x ∈ K (x0 , R1 ):  (j) (j) B ij v2 (x) = p(y; x)b ij (x, y)v2 (y)dS(y), i, j = 1, 2 , 𝛾2

ˆ ij , i, j = 1, 2 are defined for the points of the second disk while the integral operators B (2) x ∈ K ( x 0 , R 2 ):  ˆ ij v(1 j) (x) = p(y; x)b ij (x, y)v(1 j) (y)dS(y), i, j = 1, 2 . B 𝛾1 j

The functions f i are defined analogously: (j) f i (x)

=

2   k =1 Γ

p(y; x)b jk (x, y)g k (y)dS(y),

i, j = 1, 2 .

i

It should be noted that the equivalence of the system (9.35) and the boundary value problem (9.31), (9.32) is not evident, in contrast to the case of the Laplace equa­ tion. Indeed, the L1 -norm of the integral operator is generally larger than 1, so we have to use finer properties. Indeed, let us estimate the L1 -norm. Simple evaluations yield

θ∗ θ∗ 1 2 GL 1 ≤ Q α 1 − − (9.36) π π where Qα =

√ 2 + 4 2α . α+2

(9.37)

In the general case of a DS2 -domain we obtain by (9.30) GL 1 ≤ Q α max

k =1,...,n

m 

P kj .

(9.38)

j =1

This estimation shows that GL 1 can be made less than 1 for a fixed value of α by ∗ ∗ a proper choice of θ∗ 1 , θ 2 , . . . , θ n which would imply a restriction of the overlapping configuration. To be free of such a restriction, we turn to the spectral radius estimation. Theorem 9.4. The integral operator G of the system (9.35) is a Fredholm operator with kernels continuous on x ∈ 𝛾1 and y ∈ 𝛾2 , with the same type of singularities at the points of intersections of the arches 𝛾1 and 𝛾2 as the singularities in the case of Laplace equation. The spectral radius of G is less than 1 for any nonempty overlapping, which ensures the equivalence of the system (9.35) and the boundary value problem (9.31), (9.32). The first part of the statement immediately follows from the fact that the kernel func­ tions of the integral operator G are represented as products of bounded functions b ij

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168 | 9 Random Walk on Fixed Spheres for Laplace and Lamé equations and the function p(y; x). The property ρ (G) < 1 can be derived as a consequence of the result obtained by Sobolev in [200]: the Schwarz alternation procedure for two overlapped domains, for the boundary value problem (9.31), (9.32), constructed by the simple iteration of the Green formula representations for these domains, is conver­ gent. For details of this derivation see Chapter 8. Note that it seems quite plausible that Sobolev’s arguments hold true for arbitrary connected domains consisting of a finite number of overlapping disks, see [87]. The discrete approximation of the system of integral equations (9.34) in the form of a linear system of algebraic equations is straightforward: using exactly the same nodes we obtain an analog of (9.22). The error vector will also have the same form (9.26), with the estimation of type (9.27). This follows from the structure of the kernel (9.34) represented as a product of the Laplace kernel p(y; x) and smooth functions b ij (x, y). The difference with the Laplace equation is in the entries: instead of a scalar element a ij we have a 2 × 2 matrix {b ij }. Hence, all the stochastic iteration procedures we present in the next sections are equally applicable to systems of linear algebraic equations generated by both the Laplace and Lamé integral kernels. The principal difference of the Laplace and Lamé integral kernels is that the sys­ tem of algebraic equations generated by the Lamé integral kernel is not substochastic, in contrast to the case of the Laplace integral kernel.

9.4 Iteration methods In this section we present different Monte Carlo iterative procedures for solving linear systems of equations, generally being integral equations, with specific details for sys­ tem of linear algebraic equations. First we present a general iterative procedure with random parameters which in the deterministic limit tends to the iterative procedure with Chebyshev parameters. In the next subsection we describe a randomized version of the SOR-type method. Both classes of methods will then be used to solve our sys­ tems of linear equations.

9.4.1 Stochastic iterative procedure with optimal random parameters Assume we have to solve a linear, generally, integral equation of the second kind:  u (x) = k (x, y)u (y)dy + f (x) (9.39) X

or in the operator form u = Ku + f . Standard Monte Carlo algorithms (known also as the Neumann–Ulam scheme) for solving this kind of equations usually require that ρ (|K |) < 1, where the integral

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operator |K | is defined by its kernel |k (x, y)|, ρ (|K |) is the spectral radius. Sabelfeld (see [157] and [160]) has extended the Neumann–Ulam methods by applying confor­ mal transformation of the spectral parameter. This generates different iteration pro­ cedures which are convergent even if ρ (|K |) ≥ 1. However the main problem – the variance finiteness of the relevant Monte Carlo estimator – was resolved under certain restrictive assumptions. Further developments of this approach can be found in [112]. Here we suggest to use a nonstationary iterative procedure, starting with u 0 = 0, u 1 = β 0 f : u j+1 = α j u j + β j (f + Ku j ) , j = 1, . . . , (9.40) where α j , β j are some constants which we choose so that α j + β j = 1. It should be stressed that we will deal here with two stochastic elements in the Monte Carlo evaluation of the iterative procedure (9.40): the first one introduced by Vorobiev [211] suggests the sampling of parameters β j at random, according to a cer­ tain optimal probability distribution. The second one is introduced by a Markov chain for a Monte Carlo calculation of the iterations K j . Of course, these two elements can be used independently. For instance, in [211] the random parameters were used to solve linear algebraic iterations where the matrix iteration A j were calculated directly. On the other side, in our first algorithms, we have constructed a numerical procedure where in (9.40) the iterations K j were calculated by a Markov chain simulation, while the parameters were deterministic, satisfying a convergence condition (9.42). Simple analysis shows that if we assume that the eigenfunctions ϕ l (defined through ϕ k = λ k Kϕ k ) form a complete system in the space L2 (X ) of square-integrable functions on space X, then the following estimation of the error can be made. # Let the initial error be ϵ0 = ∞ i =1 c i ϕ i , then ⎤ ⎡ ⎡ ⎤  ∞ n ∞ n   /  / λ − 1 i 1 ⎦ ⎦ ϕi . ϵn = c i ⎣ ( α j + β j λ− ci ⎣ 1 − βj (9.41) i ) ϕi = λi i =1 j =1 i =1 j =1 Hence if for all λ i there exists a set of numbers β such that     1 − β λ k − 1  = q k < 1 − δ , δ > 0  λk 

(9.42)

then for all β j belonging to this set the method converges.

Algorithm NIRP: nonstationary iterations with random parameters It is possible to construct different Monte Carlo estimators following this iterative pro­ cedure. We prefer to construct-biased estimators: first, we fix n, the number of iter­ ations we will perform and choose the numbers β 0 , β 1 , . . . , β n−1 at random. How to make such a choice optimal, we will discuss later.

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170 | 9 Random Walk on Fixed Spheres for Laplace and Lamé equations Then we proceed as follows. We use for convenience the reversed indexation, so let β k = β n−k , k = 1, . . . , n. First we choose p(x, y), an arbitrary transition density function for our Markov process such that p(x, y) = 0 for (x, y) where k (x, y) = 0. Start our Markov chain from the point where the solution u (x0 ) should be found, say, x0 and take the current state as X = x0 . The current value of the iteration index is j = 1. Take the initial value of the weight as Q = 1. The initial value of the random estimator is ξ = f (x0 )β 1 . Then, make the following steps: 1. Sample uniformly in (0, 1) a random number rand and check if rand > β j . If so, then calculate the random estimator ξ := ξ + Qf (X )β j+1 ,

2.

and go to the next iteration which means that we put j := j +1 and go to 1 (provided j < n). Otherwise if rand ≤ β j , we simulate the transition from the current state X to the next state Y according to the transition density p(x, y). Then recalculate the weight Q := Qk (X, Y )/p(X, Y ) and the random estimator is scored as ξ := ξ + Qβ j+1 f (Y ). The current state is now renewed as i = k, X = Y; we turn to the next iteration again by putting j := j + 1 and go to 1 if j < n.

After n steps we finish the evaluation of our random estimator ξ (x0 ). It is not difficult to show that the constructed random estimator ξ is unbiased: u n (x0 ) = Eξ . Indeed, for i = 1, 2 this is obvious since u 0 = 0, u 1 = β 0 f . The next step is also the next step in the Markov chain method since u j+2 = (α j+1 E + β j+1 K )(α j E + β j K )u j + (α j+1 E + β j+1 K )β j f + β j+1 f , and so on. After n steps we will have ⎧ ⎫ n ⎨/ ⎬ u j + n = ⎩ ( α j + n − i E + β j + n − i K )⎭ u j i =1 ⎫ ⎡⎧ ⎤ n −1 ⎨ n/ −k ⎬ ⎣ + ( α j + n − i E + β j + n − i K ) β j + k −1 f ⎦ + β j + n −1 f . ⎭ ⎩ k =1

(9.43)

i =1

Written in this form, the iteration procedure can be clearly evaluated as described in the above algorithm.

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Optimal random parameters β k As mentioned above, the parameters β i can be chosen deterministically, say, accord­ ing to the Chebyshev iteration method which is based on polynomials uniformly close to zero, e.g. see [107]. However in our method, it is quite natural to choose these pa­ rameters randomly, according to a minimization of the probabilistic error (see [211]). Remarkably, the Chebyshev choice of parameters will follow from this probabilistic approach. To analyze the error, it is convenient to work with the operator H = E − K. We introduce the corresponding polynomial by P n (t) =

n / (1 − β i t ) .

(9.44)

i =1

By the definition of iterations (9.40) and taking into account that α j = 1 − β j , Hu = f , and ε j = u − u j , we readily find that ϵ j+n = P n (H )ϵ j .

(9.45)

It is the general idea, in the iterative methods, to make the polynomial P n (t) as close to zero as possible and in the deterministic approach the problem was solved by Markov and Chebyshev (see, e.g. [107]). There is another approach suggested in [211] where the parameters β k are sampled at random and it is then natural to measure the error in the probabilistic sense. So let us assume that we have chosen n random numbers β 1 , . . . , β n which are equally and independently distributed on some interval. Then the polynomial P n is a random variable and we can write as ln |P n | =

n 

ln p k ,

(9.46)

k =1

where p k = |1 − β k t|. Note that the random numbers ln p k are equally and indepen­ dently distributed, so we can apply the central limit theorem. This implies that as n increases, the distribution density function of ln |P n | tends to a Gaussian distribution density   (x − na)2 1 exp − K (x) = √ , 2nD 2πnD where a = ln |1 − β k t| is the expectation and D – the variance of ln p k . Standard considerations yield   ln(ϵ) − na √ P (|P n | > ϵ) ≈ 1 − Φ 2nD  2 x where Φ is the function Φ = √2π 0 e−t dt. From this follows that to ensure that the probability of deviation of P n (t) tends to zero we have to require that the expectation a is negative.

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172 | 9 Random Walk on Fixed Spheres for Laplace and Lamé equations Let φ(x) be the distribution density of β k which is defined on the interval [M −1 , m−1 ] where m and M are the lower and upper boundaries of the spectrum of the operator E − K. Thus we have 1 /m

a=

1 /m

ln |tx − 1|φ(x)dx, 1/ M

(ln |tx − 1| − a)2 φ(x)dx .

D=

(9.47)

1/ M

In [211] it is suggested that the expectation a should not depend on t, which implies that 1 /m xφ(x) da dx = 0 . = (9.48) dt tx − 1 1/ M

−1

−1

A density function on [M , m ] which solves (9.48) has the form φ(x) = This gives

and

1  . πx (1 − mx)(Mx − 1)

(9.49)

√ √ M+ m a = − ln √ √ , M− m

(9.50)

A m m + O(( )3/2 ) . D < π + 8 ln 2 M M

(9.51)

2

From this, an estimation of the number of iterations n required to reach the error ϵ can be derived ln(ϵ) , n> a where the expectation a is given by (9.50). One might argue that Vorobiev’s suggestion to choose the density φ under the condition that the expectation a is independent of t looks unjustified. However in his second paper [212] it was shown that in some sense, this choice cannot be improved. Sampling from the density φ is simple: by the inversion method we find first the 1 simulation formula for the random number β − k , which finally yields βk =

2 , (M − m) cos(π randk ) + M + m

(9.52)

where randk are random numbers uniformly distributed on (0, 1). A variance reduc­ tion can be achieved by the following modification: the interval is uniformly divided into n equal subintervals, and then, change in the simulation formula (9.52) randk with (j − randj )/n where j are integer numbers which cyclically vary with period n as j = 1, . . . , n and randj are random numbers uniformly distributed on (0, 1). Remark­ ably, if randj are changed with their expectations 0.5, we come to the method with optimal Chebyshev parameters, see [211].

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9.4.2 SOR method Let us start with the simple case of two overlapping disks and the governing system of integral equations (9.13). The matrix kernel G can be represented as G = L + U where L and U are the lower and upper triangular operators, respectively: ⎞ ⎛ ⎞ ⎛

v1 0 ⎟ 0 0 ⎜ ⎟ ⎜ ⎟ Lv =  ⎝ ⎠=⎜ ⎝ ⎠ p ( x ; y ) ∗( x ) dS 0 x 𝛾1 v2 𝛾1 p ( x ; y ) v 1 ( x ) dS x and

Uv =

0 0

 𝛾1

⎞ ⎛ ⎞ ⎛

v1 p(x ; y)v2 (x )dS x 𝛾 2 ⎟ p(y; x) ∗ (y)dS y ⎜ ⎟ ⎜ ⎟. ⎝ ⎠=⎜ ⎝ ⎠ 0 v2 0

Introducing a scalar parameter ω we rewrite our equation v = Gv + F in the form v = (E − ωL)−1 [(1 − ω)E + ωU]v + ω(E − ωL)−1 F .

(9.53)

This is a general form of transformation which is used to construct SOR method as a simple iteration method for (9.53) (see, e.g. [223] and [107]). Note that in the case we consider here, i.e. for DS2 -domains, with a consistent numbering of arches, (E − ωL)−1 = E + ωL, therefore, our equation has the following simple form: v = Tω v + d ,

(9.54)

where Tω = (E + ωL)[(1 − ω)E + ωU],

d = ω(E + ωL)F .

Now we notice that all this is true for any DS2 -domain, since here we have used only (9.24), the Property A2, which holds for such domains. Now, if ρ (|Tω |) < 1, we can apply the standard Neumann–Ulam scheme. It can be applied directly to the integral form, or to the approximating system of linear algebraic equations. Here it is convenient again to use a Markov chain of length n, to evaluate the nth approximation. Note that in the case of matrix operators, there are well-known interrelations be­ tween the spectra of G and Tω , (see, e.g. [223; 126; 64]) which can be used to analyze the convergence and variance of stochastic methods. Here we show an analogous re­ sult for integral operators. Theorem 9.5. Assume that D is an arbitrary DS2 -domain. The eigenvalues λ T of the in­ tegral operator Tω and the eigenvalues μ of the original integral operator G = L + U are related by (λ T + ω − 1)2 = λ T ω2 μ 2 . (9.55)

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174 | 9 Random Walk on Fixed Spheres for Laplace and Lamé equations Hence, for each i, the two numbers ⎧ ⎫2 9 ⎪ 9 ⎨ ωμ i ± ω2 μ 2i − 4(ω − 1) ⎪ ⎬ μ2 ω2 μi ω 2 2 =1−ω+ i ± μ i ω + 4 − 4ω (9.56) λ± T,i = ⎪ ⎪ 2 2 2 ⎩ ⎭ are eigenvalues of the operator Tω . Proof. In the transformations below we use property (9.24) which is true for an arbi­ trary DS2 -domain. Simple transformation of the eigenvalue problem Tω v = λ T v yields (1 − ω − λ T )Ev + ωUv + λ T ωLv = 0 ,

hence, (U + λ T L)v =

λT + ω − 1 v. ω

(9.57)

Second iteration of (9.57) results in  (U + λ T L)2 v = λ T (UL + LU)v =

λT + ω − 1 ω

2

v.

(9.58)

Note that the second iteration of the original eigenvalue problem (L + U)w = μw reads (U + L)2 w = (LU + UL)w = μ 2 w. (9.59) Comparison of (9.58) and (9.59) yields 1 λT



λT + ω − 1 ω

2 = μ2

(9.60)

which proves (9.55). Relation (9.56) follows immediately from (9.55). It is well known (see, e.g. [223]) that the necessary condition for the convergence of the SOR method is |ω − 1| < 1 and the minimum of λ T is attained at ωopt =

2 9 . 1 + 1 − ρ 2 (G)

Moreover, for any ω in the range 0 < ω < 2, ⎧7 9 8 ⎨ ωρ (G) + ω2 ρ 2 (G) − 4(ω − 1) 2 /4 ρ (T ω ) = ⎩ ω−1

(9.61)

if 0 < ω ≤ ωopt , if ωopt ≤ ω < 2 .

(9.62)

We will use relation (9.56) to estimate the variance of our stochastic algorithm for solv­ ing the Lamé equation in the second subsection of Section 9.2.

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Remark 9.1. Estimation (9.68) obtained in the next section and calculations of Sec­ tion 9.6 will show that the best results are obtained with a Random Walk method based on the Gauss–Seidel method, i.e. the SOR method with ω = 1, when ρ (Tω ) = ρ 2 (G). For illustration, we show in Tables 9.1 and 9.2 the general structure of the relevant ker­ nel matrices T ω for the case of a DS2 -domain consisting of five disks (see Figure 9.4), where A ij stand for the relevant kernels of the original system of integral equations.

Table 9.1. Matrix R ω in the kernel matrix of SOR: T ω = (1 − ω) E + R ω , for five disks shown in Figure 9.4 (here ω2 = ω(1 − ω)).

Rω = 0

0

0

0

0

0

0

ωA 18

0

0

0

0

0

ωA 26

ωA 27

0

0

0

0

0

0

ωA 26

ωA 27

0

0

0

0

0

ωA 45

0

0

0

0

0

ω2 A 53

ω2 A 54

ω2 A 54 A 45

ω2 A 53 A 36

ω2 A 53 A 37

0

0

0

ω2 A 63

ω2 A 64

ω2 A 64 A 45

ω2 A 63 A 36

ω2 A 63 A 37 ω2 A

ω2 A 71

ω2 A 72

0

0

0

ω2 A

ω2 A 81

ω2 A 82

0

0

0

ω2 A 82 A 26

72 A 26

72 A 27

ω2 A 82 A 27

0 ω2 A

71 A 18

ω2 A 81 A 18

Table 9.2. The kernel matrix of the Gauss–Seidel method (ω = 1), for five disks shown in Figure 9.4.

Tω = 0

0

0

0

0

0

0

A 18

0

0

0

0

0

A 26

A 27

0

0

0

0

0

0

A 26

A 27

0

0

0

0

0

A 45

0

0

0

0

0

0

0

A 54 A 45

A 53 A 36

A 53 A 37

0

0

0

0

0

A 64 A 45

A 63 A 36

A 63 A 37

0

0

0

0

0

0

A 72 A 26

A 72 A 27

A 71 A 18

0

0

0

0

0

A 82 A 26

A 82 A 27

A 81 A 18

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176 | 9 Random Walk on Fixed Spheres for Laplace and Lamé equations

9.5 Discrete Random Walk algorithms In this section, we present stochastic algorithms for solving systems of linear algebraic equations constructed as discrete approximations to the relevant integral equations, as described in Sections 9.2 and 9.3. The stochastic algorithms are based on discrete versions of the iteration methods described in the first subsection of Section 9.2 (iter­ ation method (9.40) with its stochastic implementation in Algorithm NIRP) and in the second subsection of Section 9.3 (SOR, based on the transformation (9.54)). These algorithms can be considered as a Random Walk approach for solving the relevant system of linear algebraic equations on the basis of relevant iteration method which is different from the conventional Monte Carlo method based on the convergent Neumann series.

9.5.1 Discrete Random Walk based on the iteration method Let us consider a system of linear algebraic equations (LAE) that approximates the relevant system of integral equations for our domain: this can be, for example, the system of type (9.7) in the case of Laplace equation, or (9.35) – for the Lamé equation. The LAE can be written in its direct form, or in the form related to the appropriate indexation generated by the consistent numbering for the DS2 -domains. We stress that the form of LAE is not important for the stochastic methods we suggest below. So assume that we have to construct a Monte Carlo algorithm for a system of m linear algebraic equations xi =

m 

a ij x j + b i ,

i = 1, . . . , m

j =1

or in the matrix form x = Ax + b .

(9.63)

We assume that max, min, the maximal, and minimal eigenvalues of the matrix A are known or at least estimated. Here we adopt the algorithm described in Section 9.4 for integral equations, to LAE (9.63). We will construct unbiased random estimators ξ n for u n , the nth iteration of the process (9.40) to the solution x and more precisely, to its lth component x l . First of all, we have to choose a nonnegative transition density matrix p(i → j); #m i, j = 1, . . . , m, j=1 p(i → j) = 1 for all i, which is consistent with the matrix A, i.e. p(i → j) = 0 if a ij = 0. It is convenient to take |a ij | p(i → j) = p ij = #m . j =1 |a ij |

This ensures that the random walk will be concentrated only on nonzero elements which is important since we deal with sparse block matrices. We will not have absorp­ tions in our random walk. The Random Walk algorithm can be presented as follows:

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9.5 Discrete Random Walk algorithms

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0. The initial score is set to zero: S = 0. 1. Choose n random parameters according to the formula: β (i) =

2 , max + min +(max − min) cos(π rand(i))

i = 1, . . . , n

where rand(i), i = 1, . . . , n are independent samples generated by a rand-gener­ ator. Calculate the initial value of the estimator as ξ n = b l β (1). 2. Set the initial weight Q = 1 and the initial number of iteration j = 1; fix the initial state as i = l. 3. Take a sample α j = rand(j); if α j > β (j), then calculate ξ n := ξ n + Qb i β (j + 1) and make the next iteration, i.e. j := j + 1 and go to step 3 if j, the number of iterations is less than n; otherwise, if j = n, make a score S := S + ξ n and start the new statistics from step 1. 4. Otherwise, if α j ≤ β (j), we simulate the transition from the old state i to the new state k according to the density p(i → k ). If the dimension of the problem is very large, we can use the economic algorithm described below in Section 9.2. Recal­ culate the weight and the random estimator: Q := Qa ik /p ik ,

ξ n := ξ n + Qβ (j + 1)b k ,

then, renew the state as i = k and go to the next iteration, i.e. j := j + 1 and go to step 3, if j, the number of iterations is less than n; otherwise make a score S := S + ξ n and start the new statistics from step 1. Averaging the estimator over statistics of size N gives the result: x l ≈ S/N.

9.5.2 Discrete Random Walk method based on SOR Here we present two variants of the Random Walk algorithm. Let D be a DS2 -domain, so that (E − ωL)−1 = E + ωL and hence our system (9.63) can be rewritten in the form x = Tx + f ,

(9.64)

where T = (E + ωL)((1 − ω)E + ωU ) and f = (E + ωL)b. The first algorithm for calculation of nth approximation is based on a direct ran­ domized calculation of the finite number of iterations of the operator T, i.e. by evalua­ tion of the Neumann series f + Tf + T 2 f + · · · + T n f + · · · . As in the previous section, we do not introduce absorption in our Markov chain. So to calculate the component x l of the solution to (9.64), we suggest the following algorithm. 0. The initial score is set to zero: S = 0. 1. Fix n, the number of iterations to be made and choose the parameter ω, say, equal to ωopt given by (9.61), or to 1, as in the Gauss–Seidel method. Calculate the ma­ trix T and the vector f .

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178 | 9 Random Walk on Fixed Spheres for Laplace and Lamé equations 2. 3.

Set the initial weight Q = 1, the number of iteration j = 1 and the current state of the Markov chain i = l. The initial value of the estimator is set as ξ n = f l . Simulate the transition from the state i to the new state k according to the density p(i → k ) which is chosen, for example, as in the method of the previous section: |t ik | p(i → k ) = p ik = #m . j =1 |t ij |

Recalculate the weight and the random estimator: Q := Qt ik /p ik ,

ξ n := ξ n + Qf k ,

then, renew the state as i = k and go to the next iteration, i.e. j := j + 1 and go to step 3, if j, the number of iterations is less than n; otherwise make a score: S := S + ξ n and start the new statistics from step 1. Averaging the estimator over statistics of size N gives the result x l ≈ S/N. Another version of this is algorithm follows from the factorization of the SOR oper­ ator. Let l(i, j) and u (i, j) be the entries of the triangular matrices L and U, respectively and let al(i, j) = δ ij + ωl(i, j), au (i, j) = (1 − ω)δ ij + ωu (i, j) where δ ij is the Kronecker symbol. According to the representation T = (E + ωL)[(1 − ω)E + ωU ], we make the transition from the state i to state k in two steps: first, sample the transition from i to a state i according to the matrix E + ωL (i.e. the transition i → i is sampled from the pdf pl(i, i ) defined below in (9.65)) and then make the transition i → k according to the matrix ((1 − ω)E + ωU ) (i.e. the transition i → k is sampled from the pdf pu (i , k ) also defined in (9.65)). In each step, the weight is recalculated, so that in the first step Q := Qal(i, i )/pl(i, i ) and then, Q := Qau (i , k )/pu (i , k ), with the final random estimator ξ n := ξ n + Qf (k ). The transition densities are defined by |al(i, i )| , pl(i, i ) = #m j =1 |al ( i, j )|

|au (i , k )| pu (i , k ) = #m . j =1 |au ( i , j )|

(9.65)

9.5.3 Sampling from discrete distribution In the discrete random walks, we use the discrete distributions p ij are fixed, and there­ fore, to sample from the discrete distributions, it is very convenient to use the algo­ rithm suggested by Walker (see [213]). We suggest below one of the possible imple­ mentations of this method. So let p1 , . . . , p n be a discrete distribution. First, we arrange two arrows: a real array q1 , . . . , q n and an integer array a1 , . . . , a n . This is done in the following procedure:

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Let q i = p i n, i = 1, . . . , n. ¯ = Construct two initial sets of the set of indices 1, . . . , n by Q = {i : q i < 1}, Q {1, . . . , n} \ Q. ¯ and recalculate q j := q j − 3. For each element i of Q, do the following: take j ∈ Q (1 − q i ); ¯ \ {j}. ¯ =Q put a i = j; if q j < 1 then the set Q is extended by Q = Q ∪ {j}, while Q Note that here under “For each element i of Q” we understand that we have to go through all the elements of the continuously renewed set of indices Q. 4. Recalculate q i := q i + i − 1, i = 2, . . . , n.

1. 2.

Having constructed these two arrays, q1 , . . . , q n and a1 , . . . , a n , the procedure of mod­ eling the required random integer number l from the distribution p1 , . . . , p n looks sim­ ple and uses only one sample of the random generator rand, namely: 5. Put v = n · rand, nn = Int{v} + 1 if v < q nn , then l = nn, else l = a nn . Note that we arrange the arrays “out of the loop.” This makes the algorithm extremely efficient because the time to sample one transition i → k does not depend on the dimension of the distribution p1 , . . . , p n . In other words, the computer time is essen­ tially not depending on the number of nodes on the arches we use to approximate the integral equations by a system of linear algebraic equations.

9.5.4 Variance of stochastic methods The described algorithms are based on the convergent iteration processes (the nonsta­ tionary process (9.40) and the stationary process SOR (9.54)), and on unbiased random estimators of the iterations. However, all this does not guarantee that the stochastic method is numerically stable since the variance can be increasing with the number of iterations. In the nonstationary process, the variance has an estimation (9.51), provided we can control the variance coming from the randomized estimation of each iteration. Let us consider the variance of the SOR method described in Section 9.5. It is well known (see, e.g. [112]) that the Neumann–Ulam scheme for a linear equation x = Tx + f has a finite variance if ρ (T 2 /p) < 1 where under T 2 /p we understand here a matrix whose entries are defined through the entries of the matrix T and the transition prob­ abilities p ik by t2ik /p ik . We can derive an estimation of the spectral radius ρ (T 2 /p) using the form of the # transition probabilities we have chosen in our scheme: p ik = |t ik |/ m j =1 |t ij |. Indeed, for such a choice, we have obviously ρ (T 2 /p) ≤ T 1 ρ (|T |) ,

(9.66)

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180 | 9 Random Walk on Fixed Spheres for Laplace and Lamé equations #m where T 1 = maxi=1,...,m j=1 |t ij | and |T | is a matrix with the entries |t ij |. For the norm T 1 , we obtain the estimation T 1 ≤ |1 − ω| + ωC1 + ω|1 − ω|C2 + ω2 C3

(9.67)

where C1 , C2 , and C3 are some constants expressed through the probabilities P ij and P ik P kj . These constants can be expressed through L1 and U 1 . Indeed, from the definition of the matrix T we conclude that C2 = L1 , C1 = U 1 , and C3 = L1 U 1 . For the spectral radius ρ (|T |) we use the expression given by (9.56). Putting these estimations in (9.66) we obtain the desired estimation ( ) ρ (T 2 /p) ≤ |1 − ω| + ωC1 + ω|1 − ω|C2 + ω2 C3   (9.68) μ2 ω2 μω 9 2 2 × 1−ω+ + μ ω + 4 − 4ω , 2 2 where μ = ρ (A), A being the original matrix of our linear system. Note that for two disks, in the case of the Laplace equation, an explicit expression can be obtained: since ρ (G) = λ0 (see (9.20)), we get for the Gauss–Seidel scheme (ω = 1) ρ (T 2 /p) = λ40 . The behavior of ρ (T 2 /p) on the parameter ω in the general case of an arbitrary DS2 -domain is well described by the estimation (9.68) both for the Laplace and Lamé equations. In the interval 0 < ω < 1 it decreases polynomially, with the principal linear term and for ω > 1 it increases polynomially as well, with the principal term C3 ρ (G)ω4 . This is confirmed in our calculations, see Figure 9.7 (Laplace equation, five disks – left panel, and Lamé equation with α = 5, for five disks – right panel). ρ

ρ 0.9

1.4 0.8 1.2

0.7

ρ(A)=0.599

0.6 0.5

ρ(T2/p)

1 0.8

ρ(T)

ρ(A)=0.79

2

0.4 ρ(A /p)=0.379

ρ(T)

0.6

0.3

0.4

ρ(T2/p)

0.2

ρ(A2/p)=1.11

0.2 0.1 0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3 ω

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2 ω

Figure 9.7. The dependence of the spectral radii ρ( T ) and ρ( T 2 / p) on the parameter ω. Geometry: five disks shown in Figure 9.4. Left panel: Laplace equation, right panel: Lamé equation, α = 5. For comparison, the spectral radii ρ( A ) and ρ( A 2 / p) are shown where A is the matrix of the untrans­ formed system.

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9.6 Numerical simulations We first present the results of numerical experiments for the Laplace equation that illustrate the convergence acceleration when using the SOR-based RWFS in compar­ ison to the conventional RWS method. Note that under RWS we understand here a discrete random walk on the fixed disks, constructed according to the Neumann–U­ lam scheme. The main calculation results however concern the Lamé equation: here we give a detailed numerical analysis of the new methods suggested. In particular, we analyze the behavior of the spectral radii of the SOR-based integral operators, in particular, how they depend on the parameters ω and α, the rate of overlapping, and the number of disks. For both Laplace and Lamé equations we analyze the error of the method as a function of the number of iterations.

9.6.1 Laplace equation The domain consists of four disks of radii 1, all equally pairwise overlapped with θ∗ = ∗ θ∗ 1 + θ 2 = 0.5. The Laplace equation is solved by the standard Random Walk on Spheres (RWS) method and by the SOR-based RWFS. In Figure 9.8 we show the relative error ε as a function of the number of iterations. It is clearly seen that the SOR-based RWFS method reaches its steady-state error ε0 more than two times faster than that of the standard RWS. In addition, the steady-state error ε0 of the SOR-based RWFS is considerably smaller than that of the standard RWS. ε(n) 10

0

10

–1

10

–2

10

–3

10

–4

10

−5

Standard RWS SOR−based RWFS

0

10

20

30

40

50

n 60

Figure 9.8. The relative error ϵ ( n ) as a function of n, the number of iterations, for the standard RWS method (RWS) – upper curve and for the SOR-based RWFS. Geometry: four equal disks of radii 1, all ∗ equally pairwise overlapped with θ∗ 1 + θ 2 = 0.5. Laplace equation.

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182 | 9 Random Walk on Fixed Spheres for Laplace and Lamé equations

9.6.2 Lamé equation The following model boundary value problem is solved: Δu(x) + α grad div u(x) = 0 ,

x ∈ D,

(9.69)

with the Dirichlet boundary conditions u(y) = g(y), for y ∈ ∂D, the domain D con­ sists of five overlapping disks shown in Figure 9.4. We have chosen the case with the exact solution u i (x1 , x2 ) = 1 + 21 1+α α x2i − x1 x2 + x j , i = 1, 2, with j = i. First we study how the rate of convergence of the SOR-based RWFS depends on the parameter ω. In Figure 9.7, we show the spectral radii of the following operators: A – the original (untransformed) matrix which generates the standard RWS, with the relevant operator A2 /p, – a matrix with the entries {a2ij /p ij }, where p ij is the relevant transition probability. Analogously is defined the matrix T 2 /p where T is the matrix of the SOR method. For comparison, in Figure 9.7 we show the results both for the Laplace (left panel) and Lamé (right panel) equations. As seen from the results of Figure 9.7, right panel, the standard RWS diverges in the case of Lamé equation, because ρ (A2 /p) ≈ 1.11. For the SOR-based RWFS, the spectral radius ρ (T ) monotonically decreases with ω; however, ρ (T 2 /p) reaches its minimum ρ (T 2 /p) ≈ 0.6 at ω = 1. Thus the SOR method with ω = 1, i.e. the Gauss–Seidel method, is optimal here. The almost linear decrease in the interval 0 ≤ ω ≤ 1 and the polynomial increase with the principal term of order Cω4 for ω > 1 are theoretically explained at the end of Section 9.5. The Gauss–Seidel method has shown the best results in all the calculations, in particular, let us discuss the results presented in Figure 9.9. Here we show the relative error as a function of the number of iterations. Two cases are considered, left panel: the domain consists of five disks shown in Figure 9.2 and the right panel: six disks shown in Figure 9.3. The Lamé equation was solved by the RWS and SOR-based RWFS methods. It is seen that for both α = 3.5 and α = 6, the SOR-based RWFS converges, and reaches its steady-state error in about 20 iterations. The standard RWS shows a divergence for α = 6, while for α = 3.5, it shows a kind of stable results, but the error is considerably larger. In Figure 9.10 it is shown how the spectral radii do depend on the amount of overlapping. Here we have solved the Lamé equation with α = 2.5, for ∗ 2 two disks of unit radii and denote θ∗ = θ∗ 1 + θ 2 . It is clearly seen that ρ ( T / p ) < 1 for 2 ∗ any overlapping while ρ (A /p) < 1 only if θ > 1.1. Detailed information about the spectral radii is presented in Table 9.3: here α varies from 1/32 to 15, the number of overlapping disks is 2, 5, and 10. It is seen that the spectral radii depend very slowly on the number of disks. The dependence on α is more pronounced. It is interesting to note that the standard RWS method diverges after α reaches the value of about 2, while the SOR-based RWFS starts to diverge only after α approaches to 10.

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0

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10 0

10

ε(n)

ε(n) −1

10

10−1

−2

10

10−2

−3

10

0

10

20

30

40

50

n 10−3

60

0

10

20

30

40

50

60

n

Figure 9.9. The relative error, as a function of n, the number of iterations. Left panel: the Lamé equa­ tion, for five disks presented in Figure 9.2; solid line: SOR, for α = 3.5; here the spectral radii are: ρ( T ) = 0.575 and ρ( T 2 / p) = 0.503. Dashed line: SOR, for α = 6; the spectral radii are: ρ( T ) = 0.65 and ρ( T 2 / p) = 0.68. Circles: RWS, for α = 3.5; the spectral radii are: ρ( A ) = 0.76 and ρ( A 2 / p) = 0.99. Stars: RWS, for α = 6; the spectral radii are: ρ( A ) = 0.81 and ρ( A 2 / p) = 1.17. Right panel: the same as in left panel, but for six disks presented in Figure 9.6. The spectral radii are: solid line, ρ( T ) = 0.72 and ρ( T 2 / p) = 0.64; dashed line, ρ( T ) = 0.79 and ρ( T 2 / p) = 0.80; circles: ρ( A ) = 0.85 and ρ( A 2 / p) = 1.04; stars: ρ( A ) = 0.89 and ρ( A 2 / p) = 1.2.

1.6 ρ ρ(A 2/p)

1.4 untransformed operator:

ρ(A)

1.2 1 0.8 0.6 ρ(T)

0.4

SOR:

ρ(T2/p)

0.2 0 0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2 θ*

Figure 9.10. The spectral radii of the SOR operator T and T 2 / p and for the simple iteration opera­ tor A and A 2 / p, as functions of θ∗ , for two overlapping disks of unit radii. The Lamé equation with α = 2.5.

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184 | 9 Random Walk on Fixed Spheres for Laplace and Lamé equations Table 9.3. The spectral radii of the simple iteration operators A and A 2 / p and the SOR operators T and T 2 / p, for different values of the elasticity parameter α, for DS 2 -domains consisting of 2, 5, and 10 disks (it is seen that the standard RWS diverges already for α = 2.5 even for two disks. The SOR-based RWFS converges in this case for all α ≤ 15 and for 10 disks, for α < 9. α 1 32

1.

2.5

5.

9.

10.

15.

N disks

ρ(T )

ρ(T 2 /p)

ρ(A)

ρ(A2 /p)

2

0.547

0.322

0.740

0.718

5

0.549

0.333

0.741

0.731

10

0.536

0.319

0.732

0.713

2

0.569

0.355

0.755

0.783

5

0.576

0.375

0.759

0.801

10

0.565

0.361

0.751

0.782

2

0.662

0.527

0.813

1.06

5

0.689

0.586

0.830

1.10

10

0.683

0.574

0.826

1.08

2

0.737

0.705

0.858

1.28

5

0.781

0.805

0.884

1.34

10

0.779

0.797

0.883

1.32

2

0.791

0.863

0.889

1.43

5

0.847

0.997

0.920

1.51

10

0.853

1.00

0.924

1.51

2

0.799

0.890

0.894

1.46

5

0.854

1.02

0.924

1.54

10

0.859

1.02

0.927

1.51

2

0.826

0.983

0.909

1.54

5

0.890

1.14

0.944

1.62

10

0.898

1.15

0.947

1.63

Of course, a question arises if the method can be improved, to guarantee the con­ vergence for larger values of α. For example, it would be quite suggestive to apply the symmetrized version of the SOR method [223] because Q α in (9.36) varies between 1 √ and 4 2, as α increases.

9.7 Conclusion and discussion Random Walk on Fixed Spheres (RWFS) method is developed for Laplace and systems of Lamé equations. The method is especially efficient for 2D domains represented as a family of overlapping disks, but it works well also for practically arbitrary 2D do­ mains. The method is based on the reformulation of the original differential boundary

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9.7 Conclusion and discussion

|

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value problem into a system of integral equations, starting with the Poisson-type inte­ gral formula for a disk. The derived system of integral equations can be solved by the standard Neumann–Ulam scheme, but it works only under some restrictions which are satisfied in the case of Laplace equation and are not satisfied in the case of Lamé equation. To overcome this difficulty, we have constructed two different stochastic it­ erative procedures: (1) a Chebyshev-type iterations with random parameters and (2) a SOR-based iteration procedure. The calculations have shown that the new SOR-based RWFS method considerably accelerates the convergence of the standard random Walk on Spheres method and provides much higher accuracy, when applying to the Laplace equation. More interesting, the new SOR-based RWFS is the first convergent method with finite variance for solving the system of Lamé equations. Generally, there are no probabilistic representations for system of elliptic equations and we believe that the idea behind our approach will give a rise to new attempts in constructing such prob­ abilistic solutions. As mentioned above, the RWFS method is especially efficient for domains repre­ sented as a family of overlapping disks (or spheres, in 3D), but it works well also for domains which can be approximated by this kind of families. It should be noted that the standard RWS method where the last random sphere stops in an ε-boundary deals also with these types of domains. The difference in this sense is that in RWS, a huge number of random spheres are involved, while in RWFS, we approximate the domain by a family of deterministic spheres the number of which cannot be taken large.

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10 A stochastic spectral projection method for solving PDEs for some classes of domains 10.1 Introduction There are many physical processes in which the solution of Laplace’s equations in­ side or outside of a body of general shape is central to the theoretical description [42]. The isotropic elastostatics governed be the Lamé equation is another very broad application field where numerous computational methods are developed. Threedimensional problems of equilibrium of elastic bodies bounded by spherical sur­ faces are classical subjects of the theory of elasticity. Many interesting problems in physics, chemistry and biology deal with domains which can be well approximated by a set of overlapping disks, spheres, half-spaces and combination of such bodies (e.g. see [42; 111; 118; 186; 41; 225; 226]). Often, not the whole solution field, but some simple numerical characteristics are of the main interest. For example, in electrostat­ ics, evaluation of the capacitance, which is the integral of the normal derivative of the solution over the boundary, is an important practical problem. Similar to that is the problem of calculation of the hydrodynamic friction of a variety of objects [42; 110; 71]. The boundary element methods are the most popular in these fields, but they need to recover the whole solution, so they need a lot of memory. Spectral methods are broadly used, in particular, in the elasticity problems (e.g. see [72]), and stochastic projection methods are well applied in high dimensions [209] and in stiff problems [12]. Concerning the mesh-free methods, Random Walk on Spheres (RWS) methods and other Monte Carlo Markov chain based methods (e.g. see [51; 50; 160; 54]) are well de­ veloped for the Laplace equation with the Dirichlet boundary conditions. This method is well suited for high-dimensional problems with rather complicated boundaries, es­ pecially when the local solution in one or several points only is desired (e.g. see [27]). The commonly mentioned advantage of the RWS methods is that they have minimum storage requirements. Based on the central limit theorem, these methods have gen­  erally weak convergence, since the error behaves like ε ∼ 1/ computational budget, which implies in practice that the typical accuracy cannot be much higher than a few percents. To construct a modified version of RWS method with higher accuracy we in­ troduced in our paper [170] a Random Walk on Fixed Spheres (RWFS) method which was further studied and generalized in [172] for for Laplace and Lamé equations gov­ erning static elasticity problems, and presented in more details in the previous chap­ ter. It should be noted that the kernels of the integral equations have singularities on the boundaries of the caps, the overlapping of spheres, so an additional refinement of the mesh near the singularities should be made. Then, we loose the advantage of the method mentioned above since for complicated geometries, we are back to the memory problem.

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10.2 Laplace equation | 187

To overcome this difficulty, we describe here a different approach introduced first in [164]: we invert the system of integral equations by a spectral expansion of the ker­ nels. The relevant system of linear algebraic equations for the coefficients of this ex­ pansion can then be solved by any standard linear solver, if the dimension of the sys­ tem is not too large (say, below 1000). For matrices of higher dimension we used the randomized singular value decomposition (SVD) technique we developed in [178], and the random projection method we presented in [176] in the case of very high dimen­ sions. Thus the new method proposed is mesh free, in contrast to the method we sug­ gested earlier in our papers [170; 172; 174]. Note that in [182] we have suggested another mesh-free stochastic spectral method for solving PDEs based on the method of funda­ mental solutions (MFS), see for details the next chapter. This method is applicable for arbitrary domains and any boundary conditions, but the linear systems for the coeffi­ cients are ill-conditioned, in contrast to the case considered in this chapter where the linear systems are well conditioned and can be stably resolved. This chapter is organized as follows. Section 10.2 is dealing with the Laplace equa­ tion. In Section 10.2.1 we explain the main idea of the spectral expansion method on the very simple case of two overlapping disks, Section 10.2.2 presents the extension to many overlapping disks. Generalization to Neumann and Robin boundary conditions is given in Section 10.2.3 In Section 10.2.4 we consider further extensions, in partic­ ular, to an overlapping of a half-plane with many disks. In Section 10.3 we consider the system of Lamé equations: Section 10.3.1 treats in details the case of elastic disk, and Section 10.3.2 presents the elastic half-plane. Three-dimensional problems are solved in Section 10.4: in Section 10.4.1 we consider the Laplace equation, and Sec­ tion 10.4.2 presents the case of elastic half-space. In the last section we discuss briefly the stochastic projection method we used for solving the linear algebraic equations for the coefficients in the spectral expansions.

10.2 Laplace equation 10.2.1 Two overlapping disks Let us start with the following boundary value problem for the Laplace equation: Δu (x) = 0 ,

x ∈ D,

u (y) = g(y) ,

y ∈ Γ = ∂D , (1)

(10.1) (2)

where the domain G consists of two overlapping disks D(x0 , R1 ) and D(x0 , R2 ) cen­ (1) (2) tered at O1 = x0 and O2 = x0 (see Figure 10.1): (1)

(2)

G = D ( x 0 , R 1 ) ∪ D ( x 0 , R 2 );

(1)

(2)

D(x0 , R1 ) ∩ D(x0 , R2 ) = ∅ .

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188 | 10 Stochastic spectral projection method for solving PDEs

Γ2 Γ1 R2

R1

θ*2

θ*1

O2

O1 γ1

γ2

Figure 10.1. Two overlapping disks illustrating the main notation.

(1)

(1)

We denote by 𝛾2 the part of the circle S(x0 , R1 ) = ∂D(x0 , R1 ) which belongs to (1) the second disk while Γ1 is the part of the circle S(x0 , R1 ) not belonging to the second disk; analogously 𝛾1 and Γ2 are defined. So the boundary of the domain D consists of Γ1 and Γ2 , and 𝛾1 ∪ 𝛾2 is the phase space of the integral equation to be constructed. The regular solution to the harmonic equation satisfies the integral relation (known as the Poisson formula) in each of the two disks:  u (y)dS y R2 − r2 . (10.2) u(x) = 2πR |x − y|2 S ( O,R )

Here R = R1 in the first, and R = R2 in the second disk, and the same for r = (i) |x − x0 |, the distance from x ∈ 𝛾i to the circle’s center O = O i , i = 1, 2. Based on the integral representations (10.2), it is possible to derive a system of in­ tegral equations in the phase space 𝛾1 ∪ 𝛾2 which can be then solved numerically, see the previous chapter and [170]. However, the relevant integral equation has singular­ ities in the two points of intersection of the disks. This issue was addressed in [172] where special technique to overcome this difficulty was developed. In particular, a nonuniform mesh which concentrates near the singular points can be used. This tech­ nique is applicable in the case of two disks, but not really possible in the case of many overlapping disks since it rapidly increases the need in the storage capacity. Therefore, we present here another approach based on the singular decomposi­ tion of the Poisson kernel in (10.2) (1)

K (y, x) =

R21 − |x − x0 |2 1 · . 2πR1 |x − y|2

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10.2 Laplace equation |

The expansion of the kernel we need is K (y, x) =

∞ 1 1 − ρ2 1 1  k = + ρ cos[k (θ − φ)] . 2π 1 − 2ρ cos(θ − φ) + ρ 2 2π π k =1

Here we use the polar coordinates: the point x is specified by (r, θ), ρ = r/R, and the point y by (R, φ). This series converges exponentially for r < R, so we take the approximation K (y, x) ≈ K m (r, θ − φ) =

m   1 1  r k + cos[k (θ − φ)] . 2π π k=1 R

This function can be written in the form K m (r, θ − φ) = (V (θ) · W (φ)) = V k (θ)W k (φ), where V (θ) and W (φ) are 2m + 1-vectors: ⎛

1

⎞

2π ⎟ ⎜ ⎜ ϱ 1 cos θ ⎟ ⎟ ⎜ ⎜ ϱ 1 sin θ ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ϱ 2 cos 2θ ⎟ ⎟ ⎜ ⎟ V (θ) = ⎜ ⎜ ϱ 2 sin 2θ ⎟ , ⎟ ⎜ ⎜ ··· ⎟ ⎟ ⎜ ⎜ ··· ⎟ ⎟ ⎜ ⎟ ⎜ ⎝ϱ m cos mθ ⎠ ϱ m sin mθ

#2m+1 k =1

⎞ 1 ⎟ ⎜ ⎜ cos φ ⎟ ⎟ ⎜ ⎜ sin φ ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ cos 2φ ⎟ ⎟ ⎜ ⎟ W (φ) = ⎜ ⎜ sin 2φ ⎟ , ⎟ ⎜ ⎜ ··· ⎟ ⎟ ⎜ ⎜ ··· ⎟ ⎟ ⎜ ⎟ ⎜ ⎝cos mφ⎠ sin mφ ⎛

where we denote ϱ k = π1 ( Rr )k , k = 1, . . . , m. Note that when varying the point inside the disk (say, along an arc), the distance r is a function of the angle, so generally, we  k have ρ k (r) = π1 r(Rθ ) . Our goal is now to derive a series representation of the solution. Let us write u for the solution on 𝛾1 , and use v for the solution on 𝛾2 ; we keep this notation for the approximations, when the exact kernel K (y, x) is changed with its approximation K m . For arbitrary points (r1 , θ1 ) ∈ 𝛾1 and (r2 , θ2 ) ∈ 𝛾2 we get from the Poisson formula 

2m +1 

⎤

V k (θ1 )W k (φ)⎦ v(φ) dφ + g1 (r1 , θ1 )

(10.3)

⎤  ⎡2n +1 v( r2 , θ2 ) = ⎣ V k (θ2 )W k (ψ)⎦ u (ψ) dψ + g2 (r2 , θ2 )

(10.4)

u ( r1 , θ1 ) = 𝛾2

and

⎡

𝛾1

⎣

k =1

k =1

where the functions g1 and g2 are explicitly given by   g1 (r1 , θ1 ) = K (r, θ − φ)g(φ) dφ , g2 (r2 , θ2 ) = K (r, θ − ψ)g(ψ) dψ . Γ1

Γ2

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190 | 10 Stochastic spectral projection method for solving PDEs From this we obtain 2m +1 

u ( r1 , θ1 ) =

⎡ ⎤  ⎢ ⎥ V k (θ1 ) ⎣ v(φ) W k (φ) dφ⎦ + g1 (r1 , θ1 )

k =1

and v( r2 , θ2 ) =

2n +1 

(10.5)

𝛾2

⎤ ⎡  ⎥ ⎢ V k (θ2 ) ⎣ u (ψ) W k (ψ) dψ⎦ + g2 (r2 , θ2 ) .

k =1

(10.6)

𝛾1

Now we are going to multiply the equation (10.5) by W j (ψ), j = 1, . . . , 2n + 1 and integrate it over 𝛾1 . Note that in fact it means, we integrate with respect to dψ, where ψ is the central angle of the disc S(O2 , R2 ). The same is done with the equation (10.6): it is multiplied by W i (φ), i = 1, . . . , 2m + 1 and integrated over 𝛾2 with respect to dφ, where φ is the central angle of the disc S(O1 , R1 ). Thus we have to express the angle θ1 through ψ, and θ2 through φ. Let us consider the triangle O1 O2 S where O1 , O2 are the centers of the discs, and S is the point (r1 , θ1 ). Let us denote by d the distance between the centers. Simple geometric considerations give the relation between the angles θ1 and ψ, as well as the analogous relation between the angles θ2 and φ: θ1 (ψ) = arsin[(R2 /t1 ) sin ψ],

(10.7)

θ2 (φ) = arsin[R1 /t2 ) sin φ] ,

(10.8)

where t1 = (d2 + R22 − 2dR2 cos ψ)1/2 , and t2 = (d2 + R21 − 2dR1 cos φ)1/2 . So we are now ready to carry out the integrations mentioned above: ⎡ ⎤   2m +1  ⎢ ⎥ u (·, ψ))W j (ψ) dψ = c k ⎣ V k (θ1 (ψ)) W j (ψ) dψ⎦ + G1j , k =1

𝛾1

and



v(·, φ)W i (φ)dφ =

2n +1  k =1

𝛾2

(10.9)

𝛾1

⎡ ⎤  ⎢ ⎥ ¯2 c¯ k ⎣ V k (θ2 (φ))W i (φ)dφ⎦ + G i ,

(10.10)

𝛾2

where we use the notation:  c k = v(·, φ)W k (φ)dφ,

k = 1, . . . , 2m + 1 ,

𝛾2



c¯ k =

u (·, ψ)W k (ψ)dψ,

k = 1, . . . , 2n + 1 ,

𝛾1



and G1j =

g1 (·, θ1 (ψ)) W j (ψ) dψ ,

j = 1, . . . , 2n + 1 ,

g2 (·, θ2 (φ)) W i (φ) dφ ,

i = 1, . . . , 2m + 1 .

𝛾1



¯ 2i = G 𝛾2

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10.2 Laplace equation | 191

Table 10.1. The matrix A for the two overlapping disks.

A= 0

0

...

0

a11

a12

...

a12m+1 a22m+1

0

0

...

0

a21

a22

...

...

...

...

...

...

...

...

...

0

0

...

0

a2n+11

a2n+12

...

a2n+12m+1 0

¯ 11 a

a¯ 12

...

a¯ 12n+1

0

0

...

a¯ 21

¯ 22 a

...

a¯ 22n+1

0

0

...

0

...

...

...

...

...

...

...

...

¯ 2m+11 a

¯ 2m+12 a

...

¯ 2m+12n+1 a

0

0

...

0

The systems (10.9), (10.10) form a 2(m + n + 1) × 2(m + n + 1) system of algebraic equations for unknowns c i , i = 1, . . . , 2m + 1, and c¯ j , j = 1, . . . , 2n + 1. It has the form: c¯ j =

2m +1 

a jk c k + G1j ,

j = 1, . . . , 2n + 1 ,

¯ 2i , a¯ ik ¯c k + G

i = 1, . . . , 2m + 1

k =1

ci =

2n +1  k =1

where the non-zero entries a jk and a¯ ik of the matrix A (see Table 10.1) are defined by  j = 1, . . . , 2n + 1, k = 1, . . . , 2m + 1 , a jk = V k (θ1 (ψ)) W j (ψ) dψ , 𝛾1



a¯ ik =

V k (θ2 (φ)) W i (φ) dφ ,

i = 1, . . . , 2m + 1, k = 1, . . . , 2n + 1 .

𝛾2

This system can be written in the matrix form C = AC + G , where the vector of the unknown solution is C = (c¯1 , . . . , c¯ 2n+1 , c1 , . . . , c2m+1 )T , the ¯ 21 , . . . , G ¯ 22m+1 )T . After this is done, the solu­ right-hand vector is G = (G11 , . . . , G12n+1 , G tion in any point is calculated from the Poisson integral in the disk where this point is lying. Having resolved this system by one of the methods we get the solution according to (10.3), (10.4) in the form u ( r1 , θ1 ) =

2m +1 

c k V k ( θ1 ) + g1 ( r1 , θ1 )

k =1

and v( r2 , θ2 ) =

2n +1 

¯c k V k (θ2 ) + g2 (r2 , θ2 ) .

k =1

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192 | 10 Stochastic spectral projection method for solving PDEs In conclusion we note that the extension to domains consisting of many overlapping disks can be made following the scheme presented in the previous chapter.

10.2.2 Neumann boundary conditions The Neumann boundary conditions are treated similarly, but in this case we have to relate through an integral relation not the solution, but its derivatives. Let us consider the inner problem for the disk D = D(x0 , R): Δu (x) = 0 ,

x ∈ D,

∂u (y) = g(y) ∂n

y ∈ Γ = ∂D ,

where n is the internal normal vector. The Poisson-type formula in polar coordinates centered at x0 has the form [179] 2π

u (r, θ) =

K (ρ, θ − φ)g(φ)dφ + const

(10.11)

0

where ρ = r/R, const is an arbitrary constant which we further take equal to zero, and

K (ρ, θ − φ) = −

∞ 1 1  ρk ln(1 − 2ρ cos(θ − φ) + ρ 2 ) = cos[k (θ − φ)] . 2π π k =1 k

Using this expansion we now derive an analogous Poisson integral relation for the derivatives. Thus let us again consider two overlapping disks shown in Figure 10.1. We are go­ ing to relate the normal derivative of u for points on the arc 𝛾1 with the normal deriva­ tive of v on the arc 𝛾2 where the normal derivative of u means the derivative with respect to R2 , and the normal derivative of v with respect to R1 . More precisely, the derivative of u is taken along the unit vector n2 directed from the point on 𝛾1 to O2 , the center of the second disk. The same for v: the derivative is taken with respect to the unit vector n1 directed to O1 , the center of the first disk. So let us consider the function u (r1 , θ1 ) at a point (r1 , θ1 ) ∈ 𝛾1 . We have to calcu­ late the derivative ∂u /∂n2 for n2 = (n2x , n2y ) which is, by definition, ∂u ∂u ∂u = n2x + n2y . ∂n2 ∂x ∂y To obtain the derivative we need, according to (10.11), just to find the derivative of the Poisson kernel. Since the Poisson kernel is defined in polar coordinates, we use the relation ∂ 1 ∂ ∂ = cos θ − sin θ , ∂x ∂r r ∂θ

∂ 1 ∂ ∂ = sin θ + cos θ . ∂y ∂r r ∂θ

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10.2 Laplace equation | 193

Using the series representation of the Poisson kernel K 1 in the first disk we find ∞ ∞   1 1 ∂K 1 = cos θ1 ρ 1k−1 cos[k (θ1 − φ)] + sin θ1 ρ 1k−1 sin[k (θ1 − φ)] ∂x πR1 πR 1 k =1 k =1

=

∞ 1  k −1 ρ cos[(k − 1)θ1 − φ)] , πR1 k=1 1

∞ ∞   ∂K 1 1 1 = sin θ1 ρ 1k−1 cos[k (θ1 − φ)] − cos θ1 ρ 1k−1 sin[k (θ1 − φ)] ∂y πR1 πR1 k =1 k =1

=−

∞ 1  k −1 ρ sin[(k − 1)θ1 − φ)] . πR1 k=1 1

By n2x = (L − r cos θ1 )/R2 , n2y = cos θ1 /R2 we conclude ∞ ∞ L  k −1 1  k ∂K 1 = ρ 1 cos[(k − 1)θ1 − φ] − ρ cos[(k − 2)θ1 − φ] . ∂n2 πR1 R2 k=1 πR2 k=1 1

The same arguments lead us to the derivative in the second disk ∞ ∞ L  k −1 1  k ∂K 2 = ρ 2 cos[(k − 1)θ2 − φ] − ρ cos[(k − 2)θ2 − φ] . ∂n1 πR1 R2 k=1 πR1 k=1 2

Substituting these series in ∂u ( r1 , θ1 ) = ∂n2 ∂v ( r2 , θ2 ) = ∂n1

 𝛾2



𝛾1

∂K 1 (r1 , θ1 − φ)v(φ)dφ , ∂n2 ∂K 2 (r2 , θ2 − φ)u (φ)dφ , ∂n1

we arrive at the desired spectral expansion. The rest of the results presented above for the Dirichlet problem remains unchanged. Remark 10.1. Note that the method can be easily extended to the case when on the boundaries of some disks the Dirichlet condition is imposed, and for other disks the Neumann boundary condition is prescribed. Also, the third boundary value problem can be treated analogously. Indeed, for the Robin problem Δu (x) = 0

x ∈ D,

α

∂u + βu = g(φ) ∂n

on

∂D ,

where α 2 + β 2 = 0, we need a series expansion of the Poisson kernel. This can be obtained by standard variable separation. Let us take the Fourier series for the given function ∞ c0  + [c k cos kφ + d n sin kφ] , g(φ) = 2 k =1

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194 | 10 Stochastic spectral projection method for solving PDEs where 1 ck = π

2π

g(φ) cos kφdφ

k = 0, 1, . . . ,

0

1 dk = π

2π

g(φ) sin kφdφ . 0

It can be found that the solution u (r, φ) has the following representation: u (r, φ) =

∞ 

r k [A k cos kφ + B k sin kφ] ,

(10.12)

k =1

where A0 =

c0 , 2β

Ak =

ck , αkR k−1 + βR k

Bk =

dk . αkR k−1 + βR k

Now by taking the derivative in (10.12) in arbitrary direction l we can obtain analogous representation for the function α∂u /∂l + βu, and the rest is similar to the case of the Neuman problem considered above.

10.2.3 Overlapping of a half-plane with a set of disks Let us consider the case when one of the disks has an infinite radius, i.e. the domain D is obtained as a union of a half-plane and a set of disks which have nonzero intersec­ tions with this half-plane, see Figure 10.2. To obtain the series representation we need the Poisson formula for the solution of the Dirichlet problem for the upper half-plane. This formula (e.g. see [210]) gives

Figure 10.2. Intersection of three disks with one disk (left panel) and with a half-plane (right panel).

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10.2 Laplace equation | 195

the solution at a point (x, y) ∈ R2+ : y u (x, y) = π

∞ −∞

g(z)dz . ( x − z )2 + y 2

(10.13)

Using the Fourier transform y π

∞ −∞

cos(kx)dx = e−| k| y , y2 + x2

(10.14)

we can expand the solution in a series. Indeed, since by assumption the function is rapidly decreasing as |x| → ∞ we cut-off the integration so that the integration is performed on an interval (−a, a) for sufficiently large a. Approximating the integral by a finite sum we obtain the approximate expansion by taking the inverse Fourier transform in (10.14): . m  1 − πk y kπ(x − ξ ) 1 y a ≈ cos , e π [(x − ξ )2 + y2 ] k=1 a a

where the number of retained terms should be sufficiently large as well depending on the desired accuracy ε (e.g. see [179] where estimation ε ∼ 1/m is given). So we have # the desired representation of the kernel in the form K = k V k W k where ⎞ ⎞ ⎛ ⎛ ϱ 1 cos θ cos φ ⎟ ⎟ ⎜ ⎜ ⎜ ϱ 1 sin θ ⎟ ⎜ sin φ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ϱ cos 2θ ⎟ ⎜ cos 2φ ⎟ ⎟ ⎟ ⎜ 2 ⎜ ⎟ ⎟ ⎜ ⎜ ⎜ ϱ 2 sin 2θ ⎟ ⎜ sin 2φ ⎟ ⎟ ⎟ ⎜ V (x, y) = ⎜ (10.15) ⎜ · · · ⎟ , W (ξ ) = ⎜ · · · ⎟ , ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎜ ··· ⎟ ⎜ ··· ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ϱ cos mθ ⎟ ⎜cos mφ⎟ ⎠ ⎠ ⎝ m ⎝ ϱ m sin mθ sin mφ where φ = πξ /a, θ = πx/a, and ϱ k = a1 exp(−πky/a). Here the phase space consists of 6 arches (see the dashed arches in Figure 10.2), 𝛾1 , . . . , 𝛾6 . For instance, it is conve­ nient to choose the “red–black” indexation: in the first disk, the dashed part of the half-plane boundary (lying in this disk) is 𝛾1 , and the part of the boundary of this disk which does not belong to the domain D is 𝛾6 . Analogously, the relevant two arches in the second disk are 𝛾2 and 𝛾5 , and in the third disk they are 𝛾3 and 𝛾4 . The transition ma­ trix T has the form presented in Figure 10.3. Note that exactly the same structure has the transition matrix for four disks shown in Figure 10.2. The advantage of the method is that having written down the column-vectors V, W, and the transition matrix, the rest of the algorithm remains unchanged.

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196 | 10 Stochastic spectral projection method for solving PDEs

Figure 10.3. General structure of the block matrix for the domain taken as an intersection of a halfplane (Figure 10.2) and three disks in “Red-Black” indexation (left). Right: for the exterior of a tetragon shown in Figure 10.4.

1

2

5

8

6

Γ1 1 Γ2

4 3

7

2

Figure 10.4. Left panel: Exterior to a tetragon obtained as an intersection of four half-planes. Right panel: A wedge as an intersection of two half-planes.

Note that with the Poisson formula (10.13) we can extend the method for solving exterior problems for polygons. Especially simple structure appears in the case of con­ vex polygons. Let us consider a simple example of a convex polygon with four sides, see Figure 10.4, left picture. The boundaries 𝛾1 , . . . 𝛾8 where the unknown functions are defined, are shown as dashed half-lines. The transition matrix corresponding to the chosen indexation is presented in Figure 10.3, right panel. Note that the domains can be unbounded, e.g. see the wedge shown in Figure 10.4. Here the phase space con­ sists of the dashed half-lines 𝛾1 and 𝛾2 , the boundary conditions are imposed on the boundary of the wedge, Γ1 and Γ2 . Topologically, this case is equivalent to the case of two overlapping disks, see Figure 9.6, so the solution can be obtained exactly from the formulae presented in formulae (10.3)–(10.6) where the column-vectors V and W are defined by (10.15).

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10.3 Extension to the isotropic elasticity: Lamè equation | 197

10.3 Extension to the isotropic elasticity: Lamè equation 10.3.1 Elastic disk Let us consider the plane elasticity problem in the disk D(x0 , R): μΔu(x) + (λ + μ ) grad div u(x) = 0 ,

x ∈ D(0, R) ,

u(y) = g(y), y ∈ S(0, R) ,

(10.16)

where u = (u 1 , u 2 )T is the displacement column-vector which is prescribed on the boundary as a column-vector g = (g1 , g2 )T , λ and μ are the elasticity constants. Let x = reiθ be a point in the disk Disc(0, R), and let ρ = Rr . Here it is convenient to work in the polar coordinates, with the displacement func­ tions (u θ , u r ). The Poisson integral formula (6.26) presented in Section 6.1.2 reads

2π u r (r, θ) L11 (ρ; θ − φ) = u θ (r, θ) L21 (ρ; θ − φ) 0

L12 (ρ; θ − φ) L22 (ρ; θ − φ)



g r (Reiφ ) dφ . g θ (Reiφ )

(10.17)

This system is an equation with a separable matrix kernel, hence, we can apply the same approach we used in the case of Laplace equation. We take the first m terms in the series, and write the relevant approximations as follows: L11 = L21 =

2m +1 

V k11 (θ)W k (φ) ,

L12 =

2m +1 

k =1

k =1

2m +1 

2m +1 

V k21 (θ)W k (φ) ,

L22 =

k =1

V k12 (θ)W k (φ) , V k22 (θ)W k (φ) ,

(10.18)

k =1

where we use the notation: ⎛

ρ 2π

⎞

⎟ ⎜ ii ⎜ B1 cos θ ⎟ ⎟ ⎜ ii ⎜ B1 sin θ ⎟ ⎟ ⎜ ⎟ ⎜ ii ⎜ B2 cos 2θ ⎟ ⎟ ⎜ ii ⎟ V ii (θ) = ⎜ ⎜ B2 sin 2θ ⎟ , ⎟ ⎜ ⎜ ··· ⎟ ⎟ ⎜ ⎜ ··· ⎟ ⎟ ⎜ ⎟ ⎜ ii ⎝B m cos mθ ⎠ B iim sin mθ

⎛

⎞ 0 ⎜ ⎟ ⎜ B1ij sin θ ⎟ ⎜ ⎟ ij ⎜ −B cos θ ⎟ ⎜ ⎟ 1 ⎜ ij ⎟ ⎜ B sin 2θ ⎟ ⎜ 2 ⎟ ⎜ ⎟ V ij (θ) = ⎜ −B2ij cos 2θ ⎟ , ⎜ ⎟ ⎜ ⎟ ··· ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ · · · ⎜ ij ⎟ ⎜ B sin mθ ⎟ ⎝ m ⎠ ij −B m cos mθ

where we denote in the first column B iik = ij

1 k ρ λ ii (ρ, π

⎞ 1 ⎟ ⎜ ⎜ cos φ ⎟ ⎟ ⎜ ⎜ sin φ ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ cos 2φ ⎟ ⎟ ⎜ ⎟ W (φ) = ⎜ ⎜ sin 2φ ⎟ , ⎟ ⎜ ⎜ ··· ⎟ ⎟ ⎜ ⎜ ··· ⎟ ⎟ ⎜ ⎟ ⎜ ⎝cos mφ ⎠ sin mφ ⎛

k ), i = 1, 2, k = 1, . . . , m, and

B k = π1 ρ k λ ij (ρ, k ), k = 1, . . . , m, for i = j, i, j = 1, 2 in the second column.

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198 | 10 Stochastic spectral projection method for solving PDEs Thus we have derived explicitly the column-vectors V and W, and the method now is the same as in the case of Laplace equation. For instance, in the case of two overlapping disks the solution is represented as u r ( r1 , θ1 ) =

2m +1 

V k11 (θ1 )c1k +

k =1

u θ ( r1 , θ1 ) = v r ( r2 , θ2 ) = v θ ( r2 , θ2 ) =

2m +1 

V k12 (θ1 )c2k + g1r (r1 , θ1 )

k =1

2m +1 

V k21 (θ1 )c1k +

2m +1 

k =1

k =1

2n +1 

2n +1 

V k11 (θ2 )c¯1k +

k =1

k =1

2n +1 

2n +1 

V k21 (θ2 )c¯1k +

k =1

V k22 (θ1 )c2k + g1θ (r1 , θ1 )

V k12 (θ2 )c¯2k + g2r (r2 , θ2 ) V k22 (θ2 )c¯2k + g2θ (r2 , θ2 ) .

(10.19)

k =1

Here, for i = 1, 2, 

g ir (r i , θ i ) =



L11 (r i , θ i − φ)g r (φ)dφ + Γi



g iθ (r i , θ i ) =

L12 (r i , θ i − φ)g θ (φ)dφ Γi



L21 (r i , θ i − φ)g r (θ)dφ + Γi

L22 (r i , θ i − φ)g θ (φ)dφ .

(10.20)

Γi

The vectors of coefficients c1k and c2k are obtained from the system of linear equa­ tions C = AC + G where the 2 × 2-block matrix A consists of four blocks A kl , k, l = 1, 2, with the entries 2π

A kl ij

V ikl (θ(ψ))W j (ψ)dψ ,

= (1 − δ kl )

k, l = 1, 2 ,

(10.21)

0

where i = 1, . . . , 2n + 1, j = 1, . . . , 2m + 1 for A12 , and i = 1, . . . , 2m + 1, j = 1, . . . , 2n + 1 for A21 , and δ kl is the Kronecker symbol which stands to ensure that A11 = 0, A22 = 0. ¯ 1, G ¯ 2 , G1 , G2 )T have 2m + 2n + 2 The column-vectors C = (c¯1 , ¯c2 , c1 , c2 )T and G = (G components, C = (c¯11 , . . . , c¯12n+1 , c¯21 , . . . , c¯22n+1 , . . . , c11 , . . . , c12m+1 , . . . , c21 , . . . , c22m+1 )T ¯ 11 , . . . , G ¯ 12m+1 , . . . , G ¯ 21 , . . . , G ¯ 22m+1 )T , G = (G11 , . . . , G12n+1 , G21 , . . . , G22n+1 , . . . , G where G1j





g1r (r1 ,

=

θ1 )W j (θ1 )dθ1 ,

G2j

=

𝛾1

g2θ (r2 , θ2 )W j (θ2 )dθ2 ,

j = 1, . . . , 2m + 1 .



g2r (r2 , 𝛾2

j = 1, . . . , 2n + 1 ,

𝛾1



¯ 1j = G

g1θ (r1 , θ1 )W j (θ1 )dθ1 ,

θ2 )W j (θ2 )dθ2 ,

¯ 2j = G 𝛾2

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10.3.2 Elastic half-plane To finish the generalization of the method to the Lamè equation it remains to consider the case of the elastic half-plane. In [173] we have obtained the Poisson formula for the Lamè equation governing the displacements of the elastic half-plane: ∞

u(x, y) =

L(x − x , y)g(x )dx .

(10.22)

−∞

Here the 2 × 2-matrix kernel L is defined by L = K (x − x , y)Q(x − x , y) where K (x − x , y) =

y π((x − x )2 + y2 )

is the kernel of the Poisson formula for the Laplace equation and

( x − x )2 − y 2 2(x − x )y β Q(x − x , y) = I + , 2(x − x )y −((x − x )2 − y2 ) ( x − x )2 + y 2 λ+ μ

where I is the identity matrix, and β = λ+3μ . The Fourier transform of the Poisson kernel L with respect to x − x has the fol­ lowing explicit form [173]:

|ξ |y ıξy 1 −| ξ | y F [L(τ, y)](ξ ) = I−β , e ıξy1 −|ξ |y 2π where τ = x − x . As in the case of Laplace equation, we can obtain from this represen­ tation the approximating series expansion by applying the discrete inverse transform. This yields 7 8 7 8⎞ ⎛ m  λ11 (y, k ) cos kπ(xa−x ) λ12 (y, k ) sin kπ(xa−x ) πk 1 7 8 7 8⎠ , L(x − x , y)) ≈ e− a y ⎝ a λ21 (y, k ) sin kπ(x−x ) λ22 (y, k ) cos kπ(x−x ) k =1 a

a

where πk y, a πk λ22 (y, k ) = 1 + β y , a λ11 (y, k ) = 1 − β

λ12 (y, k ) = β

πk y, a

λ21 (y, k ) = λ12 (y, k ) .

Thus we have obtained the desired spectral representation with +1   2m  ij L ij (x − x , y), y) = V ij (x, y) · W (x ) = V k (x, y)W k (x ) ,

i, j = 1, 2 ,

k =1

where the uneven components of the column-vectors V ii , have the form . (2k + 1)π (x − x ) ii − π(2ka+1) y V2k λ ii (y, 2k + 1) cos , k = 0, 1, . . . , m , +1 = e a

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200 | 10 Stochastic spectral projection method for solving PDEs i = 1, 2 and the even components are . π(2k) (2k )π (x − x ) ii = e− a y λ ii (y, 2k ) sin , V2k a

k = 1, . . . , m ,

The components of the column-vectors V ij , for i ≠ j: . π(2k+1) (2k + 1)π (x − x ) ij , V2k+1 = e− a y λ ij (y, 2k + 1) sin a

i = 1, 2 .

k = 0, 1, . . . , m ,

for uneven components, and ij

V2k = e−

π(2k) a

y

λ ij (y, 2k ) cos

(2k )π (x − x ) a

.

,

k = 1, . . . , m ,

for the even components. The vector W: it has uneven components . (2k + 1)π (x − x ) , k = 0, . . . , m W2k+1 (x ) = cos a and W2k (x ) = sin

-

(2k )π (x − x ) a

.

,

k = 1, . . . , m .

10.4 Extension to 3D problems The three-dimensional case is treated similarly, starting with the relevant expansion of the Poisson kernel for a sphere and half-space.

10.4.1 A sphere Let us consider the Laplace equation. The Poisson kernel for a sphere is [210]  g(y)dS y R2 − r2 u(x) = . 4πR |x − y|3 S ( x 0 ,R )

In spherical coordinates centered at x0 the Poisson formula reads 1 − ρ2 u (r, θ, φ) = 4π

2π π 0 0

sin(θ )g(θ , φ )dθ dφ [1 − 2ρ cos(ψ) + ρ 2 ]3/2

(10.23)

where ρ = r/R, and ψ is the angle defined by cos(ψ) = cos θ cos θ + sin θ sin θ cos(φ − φ ) .

(10.24)

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10.4 Extension to 3D problems |

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The following expansion is well known (e.g. see [210]): K (ρ, ψ) ≡

∞  1 − ρ2 = 1 + ρ k (2k + 1)Pk (cos(ψ)) . [1 − 2ρ cos(ψ) + ρ 2 ]3/2 k =1

(10.25)

For brevity, let us introduce the notation for the unit vectors, s , and s defined by its direction angles (θ , φ ), (θ, φ), respectively, and let (s , s) = cos(ψ1 ) = cos θ cos θ + sin θ sin θ cos(φ − φ ) .

The Legendre polynomials we denote by Pl (cos θ), – recall that these functions are defined on (−1, 1) as follows: Pl (μ) =

1 dl (μ 2 − 1)l , 2l l! dμ l

l = 0, 1, . . . .

The desired variable separation follows from the fact that Pl ((s, s )) =

l 

κ lm Y lm (s)Y lm (s ) .

(10.26)

m =− l

Here the coefficients are given by κ lm =

(l − |m|)! 2 , (1 + δ0m ) (l + |m|)!

(10.27)

and the system of spherical harmonics functions is defined by {Y lm (θ, φ)}, l = 0, 1, . . . ; m = 0, ±1, ±2, . . . , ±l is defined as follows: Y lm (θ, φ) = Plm (cos θ) cos(mφ) , Y lm (θ,

φ) =

m = 0, 1, 2, . . . , l;

Plm (cos θ) sin(|mφ|) ,

m = −1, −2, . . . .

(10.28)

Thus we conclude that the Poisson kernel is represented again as a sum of sepa­ rable functions and the rest is similar to the two-dimensional case. The expansion of the Poisson kernel for an elastic sphere follows from the Somigliana’s formulae and is given explicitly in [96] and [118]. Let us consider now the elas­ tic half-space.

10.4.2 Elastic half-space Let us consider the Dirichlet problem for the system of Lamé equations in the domain D+ ⊂ R3 , the upper half-space with the boundary Γ = {z : z = 0}: Δu(x) + α grad div u(x) = 0 , u(x ) = g(x ) ,

x ∈ D+ ,

x ∈ Γ = ∂D+ ,

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202 | 10 Stochastic spectral projection method for solving PDEs where u(x) = (u 1 (x, y, z), . . . , u 3 (x, y, z))T is a column vector of displacements, and g(x ) = (g1 (x , y , 0), . . . , g3 (x , y , 0))T is the vector of displacements prescribed on the boundary. The Poisson formula for the problem (13.75) has the form (e.g. see [96]): ∞ ∞

u(x, y, z) =

L(x − x , y − y , z)g(x , y )dx dy ,

−∞ −∞

where L( x − x , y − y , z) = ⎛ ⎛ ( x − x )2 z ⎜ 3β ⎜ ⎝(1 − β )I + 2 ⎝(x − x )(y − y ) 2πr3 r (x − x )z

⎞⎞ (x − x )(y − y ) (x − x )z ⎟⎟ ( y − y )2 ( y − y ) z ⎠⎠ , 2 (y − y )z z 9 λ+ μ where I is the identity matrix, β = λ+3μ and r = (x − x )2 + (y − y )2 + z2 . To extend the method to this 3D problem we have to obtain the spectral series expansion of the matrix kernel L(x − x , y − y , z). As in the 2D case, we use the Fourier transform with respect to τ x = x −√x τ y = y − y . In [193] we have it derived in the 2 2 explicit form F [K ](τ x , τ y , z) = e−z ξ x +ξ y G(ξ x , ξ y , z), where ⎛ ⎞ 2 √ ξ2x 2 √ξ x2ξ y 2 ıξ x ξ x +ξ y ⎜ ξ x +ξ y ⎟ ⎜ ξx ξy ⎟ ξ y2 ⎟. √ √ G(ξ x , ξ y , z) = I − βz ⎜ ıξ y ⎜ ξ x2 +ξ y2 ⎟ 2 +ξ 2 ξ x y ⎝ ⎠ 9 2 2 ıξ x ıξ y − ξx + ξy

Now to get the desired spectral series expansion we take the discrete inverse trans­ form where we take the integration on a rectangular (−a1 , a1 ) × (−a2 × a2 ), where a1 and a2 are sufficiently large. This yields for the all diagonal entries L ii , i = 1, 2, 3, and for L12 = L21 the following series representation: L ij (τ x , τ y , z) ≈

𝛾=

1 4a1 a2

( m 1 ,m 2 ) 

k1 , k2 = −(m1 , m2 ) (k1 , k2 ) ≠ (0, 0)

k2 τ y k1 τ x + , a1 a2



e−πzR k1 k2 G ij (k 1 , k 2 , z) cos(π𝛾)

9 R k 1 k 2 = ( k 1 / a 1 )2 + ( k 2 / a 2 )2 ,

where G ij , i, j = 1, 2, 3 are the entries of the matrix ⎛ 2 k1

⎜ a21 R k1 k2 ⎜ G(k 1 , k 2 , z) = I − βzπ ⎜ a ak1Rk2 ⎝ 1 2 k1 k2 k1 a1

k1 k2 a 1 a 2 R k1 k2 k 22 a22 R k1 k2 k2 a2

k1 a1 k2 a2

−R k1 k2

⎞ ⎟ ⎟ ⎟. ⎠

The same representation is true for L13 = L31 and L23 = L32 , with the following difference: the cosine function should be changed with the sine function.

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10.5 Stochastic projection method for large linear systems

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10.5 Stochastic projection method for large linear systems of algebraic equations In this section we discuss shortly the methods we applied to solve the system of alge­ braic equations C = AC + b. If the size of the linear system is not large, we can of course use one of the direct solvers like the Gauss method. For very large dimensions, which we have for instance when solving PDEs in domains with many overlapping disks, spheres, and/or half-domains, we use the stochastic projection method we suggested in [176]. Let us shortly describe this method. The row action iteration process suggested first by Kaczmarz [76] converges for any system of linear equations with nonzero rows, even when it is singular and incon­ sistent. The arithmetic operations required in an iteration of the method are compar­ atively few (e.g. see [205]). So let us consider a system of linear algebraic equations Ac = b where A is a rectangular m × n matrix with m ≥ n, and b ∈ IRm , c ∈ IRn . We denote by a i the i-th row of A, with the Euclidean norm a i , and aTi is the relevant column-vector, the transpose of a i , and assume that a i  > 0 for all i. The stochastic projection process we used is written as follows: c k +1 = c k + ω k

b ν(i ) − ( a ν(i ) · c k ) T a ν(i ) ,  a ν ( i ) 2

k = 1, 2, . . .

where ω k are some parameters, the indices ν(i) are sampled at random among ran­ dom subsets of indices lying in (1, 2, . . . , m). The distribution of indices can be chosen so that the method converges with expected exponential rate, not depending on the number of equations in the system. The solver does not even need to know the whole system, but only some random rows of the matrix, therefore, it is well suited for solving very large systems of linear algebraic equations. Assume that we have chosen a partition of our matrix A into s row blocks, and the relevant partition of the vector b. So let us denote by A i , i = 1, . . . , s the row blocks of the matrix A, A = (A1 , . . . , A s )T , and b i are here the relevant blocks of the vector b. In our case these block are naturally generated by the relevant blocks of the transition matrix which relates the neighbor disks. In the row projection method discussed above, the previous iteration was orthog­ onally projected successively on the hyperplanes. The projection of this vector on the intersection of hyper-planes defined by equations of the block A i is defined by the projection operator P i = ATi (A i ATi )−1 A i . It is assumed here that (A i ATi )−1 exist for all i. So the solution process will be simply the method of successive projections ap­ plied to calculating the intersection of the sets H i = {x : A i x = b i } .

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204 | 10 Stochastic spectral projection method for solving PDEs The randomized iteration method is based on a random sampling of the block, say, having a random index ν(i). Then the iteration step is obtained by the following pro­ jection: x k+1 = x k + ATν(i) (A ν(i) ATν(i) )−1 (b ν(i) − A ν(i) x k ) . Of course, the method assumes that the matrix A ν(i) ATν(i) can be efficiently in­ verted. For instance, in the case of two row block sampling when the blocks consist of two rows, this inversion is explicit, and the method is easily implemented. Our calcu­ lations show that even in this simplest case the method converges much faster than the single row sampling. Note that the strategy of random sampling of pairs of equations may be different: one may sample them uniformly, each pair being two successive rows, or the pairs may be formed by random rows sampled according to the weights proportional to norms of the blocks, or they may be randomly chosen without repeti­ tion, etc. This sampling strategy depends of course on the structure of the matrix. For instance, in the case of band matrices it is often useful to sample larger blocks which are inverted by efficient direct solvers. For more details see our paper [176].

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11 Stochastic boundary collocation and spectral methods 11.1 Introduction Traditional Markov-chain-based Monte Carlo methods such as the Random Walk methods for integral and differential equations (see, e.g. [199; 52; 160]) are applied when dealing with high dimensions. However, these methods have strong limitations and cannot be applied to many interesting practical problems. It should be noted that during the last two decades, randomized methods based on general concentration of measure phenomenon (e.g. see [100]) are intensively de­ veloped. In linear algebra, the Random Projection has emerged as a powerful method for dimensionality reduction, see, for instance, the review [65; 209; 26] and [38]. A fundamental result of Johnson and Lindenstrauss [74] says that any n point subset of Euclidean space can be embedded in k = O(log{n}/ε2 ) dimensions without distort­ ing the distances between any pair of points by more than a factor of (1 ± ε), for any 0 < ε < 1. This transformation was essentially simplified in [1] by showing that this matrix can be changed with a matrix whose entries r ij are independent discrete ran­ dom variables with the distribution P(±1) = 1/6, P(0) = 1/3 which greatly sparsifies the matrix. Some further improvements are reported in [43]. Different matrix operations such as matrix multiplication and singular value de­ composition (SVD) for large matrices based on randomization idea has been reported in different papers, e.g. see [57; 43; 8; 44; 45; 177; 178; 103; 88; 17; 218; 103; 122]. The randomized SVD methods are used in principal component analysis: by forming an empirical covariance matrix from a collection of statistical data, one computes then the SVD and finds the directions of maximal variance. The application of SVD to the method of fundamental solution is described in [143; 197; 28], see also [20]. Other ap­ plications of randomization in different fields can be found in [211; 133; 47; 1; 109; 146; 36; 28; 134]. In this chapter, we deal with the randomization technique for the method of fun­ damental solutions (MFS), first described in [182]. The constructed stochastic versions appear to be highly efficient, stable, and applicable to high-dimensional boundary value problems with complicated boundaries. In the MFS, the solution is approxi­ mated by linear combinations of fundamental solutions of the governing equations that are expressed in terms of sources located outside the domain of the problem, to avoid singularities. The unknown coefficients in the linear combination of the fun­ damental solutions and the final locations of the sources are determined so that the boundary conditions are satisfied in a least squares sense. Note that in [32], the sin­ gularity positions were randomly distributed and the simulated annealing algorithm was applied to least squares minimization, see also [147].

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206 | 11 Stochastic boundary collocation and spectral methods The MFS was first proposed by Kupradze and Aleksidze [94; 3]. They called this technique as “generalized Fourier method”. The convergence was addressed in [10]. A similar and in a sense more general approach based on the boundary condition fitting of linear combinations of a complete set of functions is used in the Trefftz method [206], see also [16] and [86]. An analysis for a general boundary element method framework can be found in [14]. We also mention applications of MFS to the eigenvalue problem for PDEs given in [144] and [153]. In this chapter, we present different versions of MFS based on the idea of ran­ domization when dealing with very high dimensions. The chapter is organized as fol­ lows. Section 11.2 introduces potential integral representations and explains how the Monte Carlo integration can be used. As an example of completely meshless method we present in Section 11.3 the Random Walk on Boundary (RWB) method and explain how the RWB can be combined with MFS. The general scheme of MFS is described in Section 11.4. A MFS with separable kernel generated by the Poisson integral formula is given in Section 11.5. Here we also explain how the Monte Carlo integration can be used to extend the MFS to inhomogeneous PDEs. This method is extended in Section 11.6 to the system of elasticity equations. Section 11.7 presents the SVD method and its randomized version which is the main method we suggest to apply in MFS when solv­ ing large-scale problems. Some results of numerical experiments and a discussion of the performance of the suggested methods are given in Section 11.8.

11.2 Surface and volume potentials Let us recall the main idea of the general potential theory for solving a PDE Lu = 0 with boundary conditions Bu = g. Given a domain D ⊂ Rn with a boundary Γ = ∂D, the simple layer potential V (x), for x ∈ D is defined by the surface integral  V (x) = E(x, y)μ (y)dσ (y) , Γ

where E(x, y) is the fundamental solution of the equation LE(x, y) = δ(x − y), μ (y) is the surface density defined on Γ and integration is carried out over the boundary Γ with respect to the surface element dσ (y). The double layer potential is another sur­ face integral  ∂E(x, y) W (x) = μ (y)dσ (y) , ∂n Γ

where n is the unit inward normal vector at y ∈ Γ.

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11.2 Surface and volume potentials | 207

The volume potential is defined by the volume integral  N (x) = E(x, y)ρ (y)dy , D

where ρ (y) is the volume density function defined in D. By the construction, the volume potential, as well as the surface potentials, solve the equation Lu = 0, under some general conditions on the boundary, namely, Γ should be a piecewise Lyapunov surface (e.g. see [154]). The unknown density functions are uniquely defined from the first and second kind boundary integral equations with singular kernels, since the singularities y of the fundamental solution belong to the boundary. The classical boundary element methods are based on this type of integral equations, e.g. see [21] and [154]. Both the Trefftz and MFS are based on the desingularization idea. One takes a ˆ such that D ¯ ⊂D ˆ and in the potential integrals, the singularities y are placed domain D ¯ outside the domain D. This can be written as follows. Let us denote the kernel of the surface potential (simple or double layer) by Φ (x, y), i.e. Φ(x, y) = E(x, y) in the case of simple layer and Φ (x, y) = ∂E(x, y)/∂n for the double layer. We define the potential M (x) by  M (x) =

Φ(x, y)μ (y)dσ (y) , ˆΓ

ˆ where Γˆ is the boundary of the domain D. Represent the unknown density μ (y) on Γˆ as a linear combination of some inter­ polation functions φ j (y), j = 1, . . . , n as μ(y) =

n 

c j φ j (y) ,

ˆ, y∈Γ

j =1

then the potential is approximated by M (x) =

n  j =1



Φ(x, y)φ j (y)dσ (y) .

cj

(11.1)

ˆ Γ

The surface integrals here need to be evaluated numerically. They can be calculated very conveniently by the Monte Carlo method, as explained in the next section. By the construction, the approximations M (x) are automatically satisfying the equation LM (x) = 0 and it remains to fit the boundary conditions. This can be done by fitting in the mean square sense, or simply by imposing the boundary conditions at m points on the boundary Γ. Note that the functions φ j can be taken as delta func­ ˆ The same considerations tions, i.e. the singularities are placed on the boundary Γ. can be done with the Newton potential, then the singularities can be distributed in ¯ the domain outside D.

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11.3 Random Walk on Boundary Algorithm In this section, we give an example of a fully meshless method, the RWB we first sug­ gested in [157] and presented in details in [160] and [154]. In this stochastic method, there are no collocation points on the boundary, and moreover, there is even no con­ ventional integration over the surface elements. The method is based on a random­ ized solution of the classical singular boundary integral equation. The implementa­ tion is extremely simple, and remarkably, it is almost the same for 2D and 3D problems. The memory requirements are minimal and the method is obviously parallelizable. We present the method shortly, only for the case of the interior Dirichlet problem for the Laplace equation. Extensions to other boundary conditions and other equations can be found in [160] and [154]. This method shows us that the surface integrals can be conveniently calculated by a stochastic integration, without introducing any mesh. Let us consider the Dirichlet problem in a domain D ⊂ R3 : Δu (x) = 0 ,

x ∈ D,

u (y) = g(y) ,

y ∈ Γ = ∂D.

The solution is constructed in the form of a double-layer potential  ∂E(x, y) u(x) = 2 μ (y)dσ (y) , ∂n

(11.2)

Γ

where

cos ψ yx ∂E(x, y) = , ∂n 4π|x − y|2

n is the unit inward normal vector at y ∈ Γ and E(x, y) = 4π| x1−y| is the fundamental solution of the Laplace equation and ψ yx is the angle between the vector x − y and n. The singularities y are placed on the boundary and the unknown density μ (y) is uniquely determined from the second-kind boundary integral equation  cos ψ y y μ (y )dσ (y ) + g(y) . μ(y) = − 2π|y − y |2 Γ

cos ψ



If the domain D is convex, the kernel 2π| y−yy y|2 has a simple probabilistic interpre­ tation: it can be considered as a transition probability density function of distribution of the point y ∈ Γ from the point y ∈ Γ and the random view angle from y to y is isotropic, i.e. y = y + |y − y|Ω, where Ω is the unit isotropic vector in R3 . The same interpretation has the kernel in (11.2) with the difference that y is changed with x ∈ D. Using a randomization of a special iterative procedure for solving the integral equation, we suggested in [157] the Random Walk on Boundary Algorithm which can be formulated as follows.

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11.3 Random Walk on Boundary Algorithm

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Assume that we are interested in calculation of the solution in a point x ∈ D. First, we simulate m random points {y0 , y1 , . . . , y m } on the boundary as follows: The point y0 ∈ Γ is determined as y0 = x + |x − y0 |Ω, where Ω is a sample of a unit isotropic vector. Then, the Markov chain of points {y1 , . . . , y m } is defined by y k +1 = y k + | y k +1 − y k | Ω k , −1 where {Ω k }m k =1 is a set of independent unit isotropic vectors. Thus, the next point y k +1 in the Markov chain is obtained just as the intersection of the ray with the origin in the previous point y k and having random isotropic direction, with the boundary. On the simulated random points {y0 , y1 , . . . , y m } we define the following random estimator: & ' (11.3) ξ m (x) = 2 g(y0 ) − g(y1 ) + g(y2 ) − · · · + . . . + (−1)m g(y m ) .

This is an ε-based random estimator and the solution is approximated by u(x) ≈

N 1  (k) ξ (x) , N k =1 m

(11.4)

(k)

where ξ m (x) are random samples constructed on N independent Markov chains {y0 , y1 , . . . , y m }, N being large enough, N ∼ 1/ε2 where ε is the error of the method. Note that the number of transitions m in the Markov chain can be taken quite small since the error of approximation (11.4) behaves like O(q m ) where q < 1 is the constant. In practice, even with m = 3, the algorithm provides an accuracy of about 1–2% which is typical for Monte Carlo methods, so we take usually m ∼ 5  7 to guarantee the error of 0.1%. Note that the method described works well for convex domains. For nonconvex domains the method is also applicable, see the details in [154], but the variance in this case may grow rapidly. Note that the statistical error of approximation (11.4) is determined by the variance of estimator (11.3), which can be estimated by m2 Var(g). This implies, the accuracy of the method can be increased by decreasing of the vari­ ance Var(g). To decrease the variance Var(g), we expand the solution of the Dirichlet problem as u (x) = u 1 + u 2 where u 1 is a harmonic function obtained by MFS whose boundary values g¯ are close to g: u 1 |Γ = g¯ ≈ g. Then, the function u 2 is calculated by the RWB method, with the variance controlled by Var(g − g¯ ), so MFS and RWB can be combined to get an optimal result. The same variance reduction technique can be applied in the Random Walk on Spheres methods [160]. It should be noted that the surface integrals in (11.1)  I j (x) ≡ Φ(x, y)φ j (y)dσ (y) ˆΓ

can be calculated very efficiently in the case when Φ(x, y) is the double-layer kernel and the domain is convex. Indeed, for any x ∈ Γ, find y on the boundary Γˆ as the

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210 | 11 Stochastic boundary collocation and spectral methods ˆ i.e. y = x + |x − y|Ω intersection of the random isotropic vector with the boundary Γ, where Ω is a random unit isotropic vector. Then, the approximation of the integral takes the form N 1  I j (x) ≈ φ j (y k ) , N k =1 where y k are independent samples of the random boundary point y simulated as de­ scribed above. Clearly, the same points y k can be used to calculate the integrals for all j and for all collocation points x ∈ Γ.

11.4 General scheme of the method of fundamental solutions (MFS) Let us first present the basic idea of the method of functional equations as it was de­ scribed by Kupradze and Aleksidze in [95]. Assume that we are solving the Laplace equation in a bounded domain B i ⊂ R3 ¯ i = B i + S and B e is the domain exterior to B i having thus with the boundary S, i.e. B the same boundary S. Then we can write for the harmonic function u (x), x ∈ B i the integral relation   1 ∂ 1 1 1 )φ(y)dS − ( ψ(y)dS , x ∈ B i , (11.5) u(x) = 4π ∂n y r(x, y) 4π r(x, y) S

S

where φ(y) = u |S , ψ(y) = ∂u ∂n , n( y ) is the inner normal vector at the point y ∈ S and r(x, y) is the distance between the points x and y, with 1/r(x, y) being the fundamen­ tal solution of the Laplace equation. For the points outside of B i , i.e. for x ∈ B e the following integral equation is true:   1 ∂ 1 1 1 0= ( φ(y)dS , x ∈ B e . (11.6) )ψ(y)dS − 4π ∂n y r(x, y) 4π r(x, y) S

S

The authors prove in [95] that from this equation, it is possible to find the solutions of the Dirichlet and Neumann problems, both interior and exterior. Moreover, for the Dirichlet problem this also gives the normal derivative of the solution on the boundary and for the Neumann problem, the solution on the boundary can be found from (11.6). For our purpose, we use the following version of this method. Let us consider a boundary value problem (interior or exterior): Lu (x) = 0,

x ∈ D ⊂ Rd ,

(11.7)

z ∈ Γ = ∂D ,

(11.8)

with boundary conditions Bu (z) = g(z) ,

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11.4 General scheme of the method of fundamental solutions (MFS)

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where L is an elliptic operator and B is the boundary operator. Domain D and its boundary Γ are assumed to satisfy the existence and uniqueness conditions. It is as­ sumed that the fundamental solution E(x, y) of the operator L is known explicitly. Recall that the fundamental solution is defined as a singular function satisfying in the whole space the equation LE(x, y) = δ(x − y) ,

x, y ∈ Rd ,

where δ(x − y) is the Dirac delta function. In MFS, the solution to (11.7) and (11.8) is constructed as a linear combination of fundamental solutions m  c k E(x, y k ) (11.9) u m (x) = k =1

¯ = D ∪ Γ. In the placed outside the domain of the problem D most popular versions of the MFS, the singularities of the fundamental solutions are placed on a prescribed pseudoboundary, i.e. the boundary Γ a of a domain D a which embraces D. The locations of the singularities are preassigned or determined along with the coefficients y k of the fundamental solutions so that the approximate solution satisfies the boundary conditions as well as possible. This is achieved by a least squares fit of the boundary conditions (11.8), so the system of linear equations with singularities {y k }m k =1

g(z k ) = Bu m (z k ),

zk ∈ Γ ,

k = 1, 2, . . . n

(11.10)

is solved for c k by introducing a set of collocation points {z j }nj=1 on the boundary and applying one of the method for solving the system (11.10). This system is not neces­ sarily bad conditioned, but often it is, so an appropriate regularization method and least squares solver are used in such cases. In this chapter, we apply a randomized SVD method we developed in [178]. Other possibility is the use of the stochastic pro­ jection method we suggested in [176], see also [209; 26] and [38]. The uniqueness of the solution in the MFS is addressed in [95], see also [31]. As mentioned above, the locations of the singularities are usually placed on an auxiliary boundary ∂D a of a domain D a containing D, often taken as a sphere in 3D or a circle in 2D. It is known from the estimations that with the increase of the diameter of D a the convergence rate increases, but the condition number of the matrix of the system for the coefficients c k is increasing, too. Hence, it is a reasonable assumption that an optimal choice of D a exists. There is a second approach suggested by the same authors which is especially efficient for calculation of the derivatives of the solution, for instance, the charge dis­ tribution and the capacitance in electrostatic problems. We describe this approach shortly in the next section.

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11.4.1 Kupradze–Aleksidze’s method based on first-kind integral equation Let us describe the second MFS-type method based on the first-kind integral equation suggested by Kupradze and Aleksidze in [95], see also [3; 94]. Assume that we are solving the Laplace equation in a domain D i ⊂ R3 with the ¯ i = D i + Γ and D e is the domain exterior to D i having thus boundary Γ = ∂D i , i.e. D the same boundary Γ. Then we can write for the harmonic function u (x), x ∈ D i the integral relation   1 ∂ 1 1 1 u(x) = ( ψ(y)dS(y) , x ∈ D i , (11.11) )φ(y)dS(y) − 4π ∂n y r(x, y) 4π r(x, y) Γ

Γ

   , n(y) is the exterior normal vector at the point y ∈ Γ where φ(y) = u |Γ , ψ(y) = ∂u ∂n Γ and r(x, y) is the distance between the points x and y, 1/r(x, y) being the fundamen­ tal solution of the Laplace equation. For the points outside of D i , i.e. for x ∈ D e the following integral equation is true:   1 ∂ 1 1 1 0= )φ(y)dS(y) − ( ψ(y)dS(y) , x ∈ D e . (11.12) 4π ∂n y r(x, y) 4π r(x, y) Γ

Γ

Thus for the Dirichlet problem, φ is given, and we can find from (11.12) ψ(y), the normal derivative on the boundary. For the Neumann problem, ψ(y) is given, so we can find from (11.12) φ(y), the solution on the boundary. The solution at any point inside the domain is then obtained just by taking the integrals in (11.11) and (11.5). The authors have shown in [95] that this method gives a stable algorithm for calculating the solutions of the Dirichlet and Nuemann problems, both interior and exterior. In the last section, we apply this method for solving an exterior Dirichlet problem for the Laplace equation and calculation of the capacitance of polymers modeled as chains composed by a set of overlapped spheres. In conclusion, we note that the calculation of the derivative on the boundary is known as a very difficult problem for all known methods, including the standard MFS. Thus the main idea of MFS is similar to that of the Trefftz method: find a set of particular solutions to the equation and fit the boundary conditions by a proper linear combination of these particular solutions. In Section 11.5, we suggest a different approach based on the Poisson integral for­ mula for a sphere (or a disk, in 2D). In this approach, we find the unknown function on the auxiliary circle by fitting the boundary conditions. It is an interesting feature of this approach that here we deal with integral equations with separable kernels; hence they can be efficiently inverted. We apply this approach for the Laplace equation and the system of Lamè equations of isotropic elasticity.

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11.4 General scheme of the method of fundamental solutions (MFS) |

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11.4.2 MFS for Laplace and Helmholz equations Let us consider the boundary value problem Δu (x) + λ2 u (x) = 0 ,

z ∈ D,

Bu (z) = 0 ,

z ∈ Γ,

where the following three main boundary conditions are defined by ⎧ ⎪ ⎪ g( z) + u ( z) z ∈ Γ1 (Dirichlet boundary cond.) ⎪ ⎨ Bu (z) = ⎪ g(z) + ∂u∂n(z) z ∈ Γ2 (Neumann boundary cond.) ⎪ ⎪ ⎩ g(z) + β (z)u (z) + 𝛾(z) ∂u(z) z ∈ Γ (Robin boundary cond.) , 3

∂n

where Γ = Γ1 ∪ Γ2 ∪ Γ3 . In the case of Laplace equation, when λ = 0, the fundamental solution is E(x, y) = −

For λ = 0,

1 log |x − y| 2π

in R2 ,

and E(x, y) =

⎧ ⎨− i H (2) (λ|x − y|) , 4 0 E(x, y) = ' & ⎩− 1 exp −iλ|x − y| , 4π| x − y |

1 4π|x − y|2

in R3 .

in R2 , in R3 ,

(2)

where H0 (λ|x − y|) is the zero-order Hankel function of the second kind. For the modified Helmholz equation (Δ − λ2 )u (x) = 0, ⎧ ⎨− 1 K0 (λ|x − y|) , in R2 , 4 E(x, y) = ' & ⎩− 1 exp −λ|x − y| , in R3 , 4π| x − y |

where K0 is the modified Bessel function of the second kind of order zero. The solution is approximated by u m (x) =

m 

c k E(x, y k ) ,

k =1

where the coefficients c k and the locations of the singularities {y k }m k =1 are determined by minimizing the functional F ({c k }, {y k }) =

M 

|Bu m ({c k }, {y k }; x i )|2 ,

(11.13)

i =1

where the collocation points {x i }M i =1 are preselected on the boundary. The minimiza­ tion of this functional can be carried out using a nonlinear least squares algorithm. If however we also preselect the singularities {y k }m k =1 , say, on the auxiliary boundary, or randomly distributed in the domain, exterior to D, then we have a standard linear least squares problem. In our calculations we solved it by the SVD algorithm, when the values M and m were not too large. For very large matrices, we have applied the randomized SVD method we developed in [178] and the method reported in [146].

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11.4.3 Biharmonic equation The same approach is applicable to equations of higher order and systems of equa­ tions. Let us consider the biharmonic equation Δ2 u ( x) = 0 , u(y) = f (y) ,

x ∈ D, ∂u (x, y) = g(y), ∂n

y ∈ Γ.

The solution is constructed in the form u m (x, y) =

m 

c i E1 (x, y i ) +

i =1

m 

d i E2 (x, y i ) ,

i =1 1

E1 (x, y) is the fundamental solution of the Laplace equation, E2 (x, y) = 8π |x − y|2 log |x − y| is the fundamental solution of the biharmonic equation. The coeffi­ cients c i and d i are determined from the boundary conditions

u m (x k , y) = f (x k ) ,

∂u m (x k , y) = g(x k ) , ∂n

k = 1, . . . M ,

which results in the system of linear equations for c = (c1 , . . . , c m ) and d = (d1 , . . . , d m ): $ % G11 G12 c f = . G21 G22 d g Here (G11 )s,p = E1 (x s , y p ) , (G21 )s,p =

∂E1 (x s , y p ) , ∂n

(G12 )s,p = E2 (x s , y p ) , (G22 )s,p =

∂E2 (x s , y p ) . ∂n

11.5 MFS with separable Poisson kernel As described above, in the traditional MFS one approximates the solution as a linear combination of fundamental solutions and the unknown coefficients are determined by fitting the boundary conditions and solving the least squares problem, say, through the SVD, or applying other stable solvers to the relevant system of linear equations which is generally generated by a rectangular matrix with the number of rows equal to the number of collocation points on the boundary and the number of columns equal to the number of singularities sampled outside the domain. The main problem is then to improve the poor condition property of the system of linear equations for the unknown coefficients. Here we describe a different approach where the system of linear equations is de­ rived from the Poisson integral representation of the solution in a sphere (disk, in R2 ).

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The kernel of the Poisson integral is approximated by a separable function which sim­ plifies the structure of the linear system. More important, the derived system is better conditioned.

11.5.1 Dirichlet problem for the Laplace equation We start with the presentation of the method for the interior Dirichlet problem for the Laplace equation with the prescribed function on the boundary u |Γ = g. Let us choose a disk Disk (0, R) of radius R, centered at 0, containing the domain D, D ⊂ Disk (0, R). ¯ In polar coordinates, For the exterior problem, we had to choose Disk (0, R) ⊂ R2 \ G. the solution at a point x = (r, θ) ∈ D is represented by the Poisson integral formula: R2 − r2 u (r, θ) = 2π

2π 0

f (φ)dφ . R2 − 2Rr cos(θ − φ) + r2

(11.14)

The idea is simple: fitting the function u (r, θ), calculated in a set of collocation points on the boundary Γ by (11.14), with the given values g(z) on the boundary Γ, we obtain an approximation to the unknown function f (θ) on the circle S(0, R). In the spirit of MFS, we could do the fitting numerically, by inverting the linear system for unknown values f (x j ) on the circle S(0, R). However, it can be done much more efficiently using the fact that in (11.14) we deal with the separable Poisson kernel ∞   1 1 1  r k R2 − r2 K (r, θ − φ) = = + cos[k (θ − φ)] . (11.15) 2π R2 − 2Rr cos(θ − φ) + r2 2π π k=1 R This series converges exponentially for r < R, so we take the approximation m   1 1  r k + cos[k (θ − φ)] . K m (r, θ − φ) = 2π π k=1 R This function can be written in the form K m (r, θ − φ) = (V (θ) · W (φ)) = V k (θ)W k (φ) where V (θ) and W (φ) are 2m + 1-vectors: ⎞ ⎞ ⎛ ⎛ 1 1 2π ⎟ ⎟ ⎜ ⎜ ⎜ ϱ 1 cos θ ⎟ ⎜ cos φ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ϱ 1 sin θ ⎟ ⎜ sin φ ⎟ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎜ ϱ 2 cos 2θ ⎟ ⎜ cos 2φ ⎟ ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ V (θ) = ⎜ ϱ 2 sin 2θ ⎟ , W (φ) = ⎜ sin 2φ ⎟ ⎟, ⎟ ⎟ ⎜ ⎜ ⎜ ··· ⎟ ⎜ ··· ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ··· ⎟ ⎜ ··· ⎟ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎝ϱ m cos mθ ⎠ ⎝cos mφ ⎠ ϱ m sin mθ sin mφ where we denote ϱ k =

1 π

 k r R

(11.16) #2m+1 k =1

, k = 1, . . . , m.

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216 | 11 Stochastic boundary collocation and spectral methods Now the boundary condition implies that for z = (r, θ) ∈ Γ: 2π

g(z) = u (z)|z∈Γ =

K (r, θ − φ)f (φ)dφ .

(11.17)

0

Our goal is now to find the unknown function f (φ) from this equation without introducing any collocation points. Using expansion (11.16) we obtain from (11.17): 2π

g( z) =

K m (r, θ − φ)f (φ)dφ + ε , 0

where the error ε can be easily estimated as a quantity of O

   m +1 r R

. Omitting the

error ε we find the function f (φ) from the following second-kind integral equation: 2π

g(r, θ) =

K m (r, θ − φ)f (φ)dφ .

(11.18)

0

Here, to be rigorous, more correct there would be to write f m,𝛾,ε instead of f , to indicate that the exact solution f differs from the solution to (11.18). However to simplify the notation, we keep writing f for the solution to (11.18). The relevant approximation to the solution u (r, θ) is denoted further by u m (r, θ). Let us show that the solution u has the following approximation: u m (r, θ) =

2m +1 

c i V i (θ) ,

i =1

where the coefficients c1 , . . . , c2m+1 are solutions to the system of linear equations Ac = b where the matrix A and the right-hand side vector b are defined by the en­ tries 2π 2π b i = g(φ)W i (φ)dφ , A ij = V i (φ)W j (φ)dφ, 0

0

i, j = 1, . . . 2m + 1 . Indeed, the solution u m (r, θ) can be now written in the form u m (r, θ) =

2m +1  i =1

2π

V i (θ)

W i (φ)f (φ)dφ

(11.19)

0

 2π and we have to find the weights c i = 0 W i (φ)f (φ)dφ. To do that, we take representation (11.19) on the boundary of our domain, multiply this equation by W j and integrate over (0, 2π), this yields 2π

g(r, θ)W j dθ = 0

2π 2m +1   i =1 0

2π

V i (θ)W j (θ)dθ

W i (φ)f (φ)dφ ,

(11.20)

0

j = 1, . . . , 2m + 1. Relations (11.19) and (11.20) prove our assertion.

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11.5 MFS with separable Poisson kernel | 217

Thus, having solved the system Ac = b, with precalculated matrix A, we can cal­ culate the solution of the Dirichlet problem in any set of points. Moreover, if we need to find the solution of the Dirichlet problem for many boundary functions g, we need to prepare different vectors b and then the calculations are carried out with the same matrix A. Note that the method described is highly efficient yet quite technical. For 3D, it contains even more technical efforts dealing with the evaluation of spherical harmon­ ics. Therefore, a direct numerical scheme can be used, based on the randomized cal­ culation of the surface integrals with the Poisson kernel involved. Indeed, let us seek the unknown function f (φ) in the form f (φ) =

n 

c j h j (φ) ,

j =1

where h j (φ) is a set of complete functions on [0, 2π]. Substituting it in the Poisson formula (9.4) yields n  c j I j (r, θ) , u (r, θ) = j =1

where

2π

I j (r, θ) =

K (r, θ − φ)h j (φ)dφ . 0

These integrals can be calculated very fast, using its probabilistic interpretation, for all collocation points x i = (r i , θ i ) ∈ Γ, i = 1, . . . m. This leads to a system of linear equations n  I j (r i , θ i )c j , i = 1, . . . m , (11.21) ui = j =1

where u i are the values of the boundary function g in the collocation points x i ∈ Γ. The linear equation (11.21) which is generally a rectangular system, can be solved by one of the mean square minimization methods, e.g. by a truncated SVD as will be described in Section 11.8. For large-scale problems governed by very large matrices the deterministic SVD is costly, so we apply the randomized version of SVD.

11.5.2 Evaluation of the Green function and solving inhomogeneous problems In many practical problems it is desired to evaluate Green functions. In particular, inhomogeneous problems can be efficiently solved by the integration  u (x) = G(x, y)f (y)dy , D

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218 | 11 Stochastic boundary collocation and spectral methods where G(x, y) is the Green function and f is the right-hand side of the PDE in question. Note that in the literature, MFS is applied to homogeneous problems. Important ap­ plication of the Green functions is the derivation of the integral representation in the form of Poisson formula which involves the normal derivative of the Green function on the boundary. So let us consider an inhomogeneous problem for the Laplace equation Δu (x) = −f (x) ,

x ∈ D,

u ( z) = g( z) ,

x ∈ Γ.

Clearly, we can expand the solution as u (x) = v(x) + w(x) where v is a harmonic function with the boundary conditions v(z) = g(z) and w solves the Poisson equa­ tion Δw(x) = −f (x) with zero boundary conditions w(z) = 0. The solution v can be calculated by MFS as described above, so let us focus on the evaluation of w. By the Green formula,  w(x) = G(x, y)f (y)dy , (11.22) D

where G(x, y) is the Green function defined by ΔG(x, y) = −δ(x − y) , G(z, y) = 0 ,

x, y ∈ D , z ∈ Γ, y ∈ D .

(11.23)

Let us define a function U (x, y) by U (x, y) = G(x, y) + E(x, y) where E(x, y) is the fundamental solution. Then, U (x, y) is a harmonic function with the boundary conditions U (z, y) = E(z, y), z ∈ Γ, y ∈ D; hence, it can be solved by MFS, for any y ∈ D. Assume that we are interested in calculation of w(x) at a point x ∈ D. To apply (11.22), we need to solve (11.23) for a set of points y ∈ D which can be done by applying MFS to find the function U (x, y) and then substituting G(x, y) = U (x, y) − E(x, y) in (11.22). This straightforward scheme would need to solve, by MFS, the Dirichlet prob­ lem for U (x, y) for many points y ∈ D. There is however another, more elegant method which uses the symmetry of the Green function, suggested in [160]. Indeed, let us change the integration order in (11.22). Then, to calculate the integral, we need to find the solution in many points y for only one source placed in the point x, so the Dirichlet problem for U (x, y) is solved only once. This significantly improves the efficiency of the method. To calculate the integral in (11.22), it is convenient to apply the Monte Carlo method. Indeed, we write the integral (11.22) in the form of a mathematical expecta­ tion % % $ $  G(x, ξ )f (ξ ) (U (x, ξ ) − E(x, ξ ))f (ξ ) =E w(x) = G(x, y)f (y)dy = E p(ξ ) p(ξ ) D

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11.5 MFS with separable Poisson kernel |

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219

where the expectation E is taken over random points ξ distributed in D with a den­ sity p. This density is quite arbitrary, chosen so that p(y) = 0 if G(x, y)f (y) = 0. Variance reduction is usually applied to find an optimal choice of p, but a simply uni­ form distribution is also possible. Using the symmetry of the Green function we have $ % (U (ξ, x) − E(ξ, x))f (ξ ) . (11.24) w( x ) = E p(ξ ) 1. 2.

3.

Implementation of the method following (11.24) can be carried out as follows: Sample a set of N random points ξ1 , . . . , ξ N in the domain D according to a prob­ ability density p. By the MFS, find the harmonic function U (ξ, x) with the prescribed boundary function E(z, x), z ∈ Γ. As follows from the MFS, the values U (ξ i , x), i = 1, . . . N can be calculated simultaneously, having once calculated the matrix A and find­ ing the coefficients c j . Calculate the approximation of the solution as $ % N 1  (U (ξ k , x) − E(ξ k , x))f (ξ k ) . w( x ) = N k =1 p(ξ k )

We have described the algorithm for calculating the solution in an arbitrary point x ∈ D. Note that if we need to calculate the solution in a set of points, say, x1 , . . . , x M , we have to find, by MFS, harmonic functions with M different boundary functions. Since however the matrix A of the method does not depend on the boundary conditions, we need only to precalculate M vectors b. Thus the described method calculates the solution of the inhomogeneous problem for any set of points in D, using the matrix A precalculated only once, at the beginning of the calculations.

11.5.3 Evaluation of derivatives on the boundary and construction of the Poisson integral formulae The integral Poisson formulae for PDEs are well known, important, and broadly used in many applied fields. For instance, the Poisson formula is very convenient if it is necessary to solve a boundary value problem for a set of functions prescribed on the boundary, e.g. when solving boundary value problems with random boundary con­ ditions [156]. In the applied probability, the distribution of the Wiener process first hitting the boundary can be obtained from the Poisson formula. We also mention the Random Walk on Spheres and other canonical domains – they are based on the inte­ gral Poisson formulae [160]. The Poisson formulae can be explicitly constructed only for a few canonical domains like a sphere, half-space, and some others. For some domains it can be

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220 | 11 Stochastic boundary collocation and spectral methods represented in the form of a series which however often converges very slowly. The mesh-based construction of the Poisson formulae is possible, but when approaching the boundary, the mesh size should be drastically decreased which much complicates the application of the method. Here we suggest to construct the Poisson integral formulae by the MFS for any domain D with a finite boundary Γ. Generally, the Poisson formula reads  ∂G(x, y) u(x) = g(y)dy , ∂n Γ

where n(y) is the exterior normal at a point y ∈ Γ. It solves the Dirichlet problem for a homogeneous PDE Lu (x) = 0, u |y∈Γ = g(y). ∂G ( x,y ) The function ∂n can be constructed by the algorithm we described above for the Green function. Here we again use the symmetry of the Green function. From the expansion we used above we have ∂U (x, y) ∂E(x, y) ∂G(x, y) = − . ∂n ∂n ∂n ∂G ( x,y )

So to calculate the function ∂n we have to calculate the normal derivative at the boundary of the harmonic function U (x, y) with the prescribed boundary condition U (z, y) = E(z, y), z ∈ Γ. This can be done by the MFS directly. However using the symmetry of U (x, y), this can be done more efficiently, as described above for the Green function calculation. A more direct method for calculation of derivatives can be constructed from the second Kupradze–Aleksidze approach. Let us illustrate it by calculation of capaci­ tance for polymers composed by a set of overlapping spheres.

11.6 Hydrodynamics friction and the capacitance of a chain composed by a set of overlapped spheres In this section, we apply the second Kupradze–Aleksidze’s method presented in Sec­ tion 2.2 to solve a 3D exterior Dirichlet problem for a bounded domain D and calculate the hydrodynamics friction coefficient and the capacitance of a chain of overlapped spheres. This problem is discussed in details in [42]. Throughout the section, we use the boldface characters for the points in R3 , e.g. x = (x1 , x2 , x3 ), y = (y1 , y2 , y3 ), and z = (z1 , z2 , z3 ), and for the unit normal vector exterior to the domain D at a point y ∈ Γ = ∂D we will use the notation n y .

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11.6 Hydrodynamics friction and the capacitance of a chain of spheres | 221

Let us consider the exterior Dirichlet problem for the Laplace equation ⎧ ⎪ B, ⎪  u (z ) = 0 , z∈D ⎪ ⎨  u (z)z→y = 1 , y ∈ Γ , ⎪ ⎪ ⎪ ⎩ u (z )  z→∞ = 0 , B is the exterior to D, Γ is the boundary of D B. where D is a bounded domain in R3 and D The hydrodynamics friction coefficient and the capacitance C of the domain D is defined as follows:   1 ∂u (y) 1 ∂u (y) C=− dS(y) = − ψ(y) dS(y) , where ψ(y) = ,y ∈ Γ. 4π ∂n y 4π ∂n y Γ

Γ

(11.25) Note that the capacitance of a sphere of radius a is known: C = a, so in our cal­ culations we evaluate the ratio C/a. The function ψ(y) is the charge density on the boundary. According to the Kupradze–Aleksidze scheme, we write down the Green formula   1 ∂ 1 B. u (x) = E(x, y)dS(y) − E(x, y)ψ(y)dS(y) , x ∈ D 4π ∂n y 4π Γ

Γ

Since we solve the exterior problem, we have for x ∈ D   1 ∂ 1 0= E(x, y)dS(y) − E(x, y)ψ(y)dS(y) . 4π ∂ny 4π Γ

(11.26)

Γ

Using the notation for the distance r(x, y) = ||x − y|| and the fundamental solu­ tion of the Laplace equation E(x, y) =

1 , r(x, y)

cos(n y , x − y) ∂ E(x, y) = , ∂n y r2 (x, y)

we rewrite equation (11.26) in the form   cos(n y , x − y) ψ (y ) dS ( y ) = dS(y) , r2 (x, y) r(x, y) Γ

x∈ D.

(11.27)

Γ

Thus, we get a boundary integral equation for the unknown density ψ(y) which can be solved by introducing collocation and source points and then applying the ran­ domized SVD method for solving the relevant system of linear algebraic equations. A natural way to derive this kind of linear algebraic equation is to construct a regular mesh of points {y i }ni=1 on the boundary and a set of regular points inside the domain D, ˜ which is at a distance h to Γ. This method can say, placing the points on a surface Γ be applied for 2D problems, but for 3D problems with nontrivial boundaries it is prac­ tically impossible. Therefore, we suggest the following approach which resembles the

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222 | 11 Stochastic boundary collocation and spectral methods Nyström method, with the difference that the integrals are approximated via random­ ization. Implementation of this method is very simple and convenient. So let p(y) be a probability density function defined on the boundary Γ. We rewrite the integral equation as follows:   cos(n y , x − y) ψ (y ) p(y)dS(y) = p(y)dS(y) , x ∈ D . (11.28) p(y)r2 (x, y) p(y)r(x, y) Γ

Γ

Let {yi }, i = 1, . . . , n be a set of random points on Γ sampled from the density p(y). Then from (11.28) we get the following approximation: n n 1 ψ (y i ) 1  cos(nyi , x − yi ) . = n i=1 p(yi )r2 (x, yi ) n i=1 p(yi )r(x, yi )

(11.29)

In our calculations, the points yi were sampled uniformly on the boundary: ˜ {y i }ni=1 on Γ and {xi }m i =1 on Γ. In this case, we obtain from (11.29) the following ap­ proximating system of linear algebraic equations Aw = b where A = {a i,j }, for n i = 1, . . . , m, j = 1, . . . , n, and b = {b i }m i =1 , w = { w j = ψ (y j } j =1 ,  cos(nyj , (xi − y j )) 1 , bi = . r (x i , y j ) r 2 (x i , y j ) j =1 n

a i,j =

Note that we deal generally with rectangular system where not necessarily n = m. In our calculations, we used the values n = 20 000 and m = 2000. The underdetermined system was solved by the randomized SVD with k = 750. First let us test the method and compare the results with the exact solution known for the domain D composed by two overlapped spheres, see [42]. Note that the capac­ #n itance approximation in our schemes has a simple representation C ≈ k=1 w i . We have made the test calculations for two cases: the overlapping angle α of two spheres of equal radius was taken as α = 30◦ and α = 45◦ . This angle α is defined as the half-angle of view of the overlapping cap from the center of the sphere. The exact re­ sults reported in [42] are: C/R = 1.36 for the first case and C/R = 1.28 for the second case. In our calculations we have obtained the values C/R = 1.370 and C/R = 1.293, respectively. In the next numerical experiment, we have studied the influence of the source distributions on the accuracy. We have varied the distance from the boundary Γ˜ to Γ where Γ˜ is the sphere of radius r lying inside the relevant sphere of the original domain composed by two overlapped spheres, see Figure 11.1. The two overlapped spheres have equal radius R = 8.0; the distance between their centers are  = 13.48. The radius of the inner spheres was varying from r = 5.818 to r = 0.727. In Figure 11.2 (left panel) we show the behavior of the spectral numbers of the matrix A. It can be seen that for small values of r the singular values decrease much faster than for r which are closer to R. From this we conclude that the smaller the value of r, the less number of terms can be taken in the SVD decomposition.

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11.6 Hydrodynamics friction and the capacitance of a chain of spheres | 223

Γ

R

r

Figure 11.1. Two overlapping spheres (shown a cross-section) with the boundary Γ and the inner ˜ with the point sources on it. boundary Γ 150

4.4

r=5.818 r=5.091 r=4.364 r=3.636 r=2.909 r=2.182 r=1.455 r=0.727

4 3.8

log ||wν||

100

r=5.818 r=5.091 r=4.364 r=3.636 r=2.909 r=2.182 r=1.455 r=0.727

4.2

50

3.6 3.4 3.2

3 2.8 2.6 0

5

10 15 20 25 30 The first 35 ordered singular values

−4

35

−2

0 2 4 log ||Awν−b||

6

8

Figure 11.2. Singular values (left panel) and the L-curves (right panel) for different values of r, the radii of the inner spheres.

A commonly used approach to balance the number of terms in the SVD to reach a reasonable estimator is the L-curve method [66]. The L-curve is defined as a set of points (log wν , log Awν − b) for a series of the regularization parameter ν. The natural choice of the optimal value of ν corresponds to the corner point of the L-curve. Thus the L-curve is a log–log plot of the norm of a regularized solution versus the norm of the corresponding residual norm. It is a convenient graphical tool for displaying the trade-off between the size of a regularized solution and its fit to the given data, as the regularization parameter varies [66]. In our case we deal with the vector wν =

ν  uT · b i

i =1

σi

vi ,

where {vi , i = 1, . . . , n}, {ui , i = 1, . . . , m} are the right and left singular vectors and {σ i , i = 1, . . . , min(m, n)} are the singular values of the matrix A. From the curves

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224 | 11 Stochastic boundary collocation and spectral methods 25

K = 10 = 20 = 30 = 40 = 50 = 60 = 70 = 80 = 90 = 100

Capacitance/R

20

15

10

K is the number of spheres in the chain

5

0

2

4

6

8

10

12

l

14

Figure 11.3. The ratio C / R for a chain of K spheres, as a function of l, the distance between the cen­ ters of two neighbor spheres in the chain.

series in Figure 11.2, left panel, we conclude that for small values of r it is easy to extract the corner point and the relevant value of the parameter ν. In the following calculations we have increased the number of overlapped spheres in the chain to K = 100 spheres each of radius R = 8.0 and the inner radius of r = 2.0. We have studied the dependence of the capacitance of the chain of spheres on the distance 0 <  < 2 R between the spheres while the number of the spheres in the chain was fixed to 10, 20, . . . , 100. The results are presented in Figure 11.3 where it is seen that the capacitance increases linearly with the distance , with different slope for chains of different length. In problems of polymer technology the distance between spheres may vary quite irregularly, so the following problem is of practical interest. Assume that we have a chain composed by a fixed number of overlapped spheres with a random distance uniformly distributed on (0, 2R). We calculate the mean capacity over 250 samples of random chains and compare it with the capacity of a deterministic chain with the fixed mean distance . All this has been done for chains with 5, 10, and 15 spheres of equal radii of R = 8.0, the inner radius was r = 2.0. The results of calculations are shown in Table 11.1. The relevant results for the de­ terministic chains differ from these values to about 1–2 %, so the latter can be taken as a good approximation for the case of irregular chains. It is possible to find an effective mean distance by choosing a deterministic chain with the same capacitance. It can be easily done by extracting the relevant capacitance value in they Table 11.2 where we give the capacitance for deterministic chains with the desired values of .

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Table 11.1. The mean ratio C/ R for chains with randomly distributed spheres for K = 5, 10, 15. K

C/ R

5

1.7216

10

2.4991

15

3.2073

Table 11.2. The ratio C/ R for the deterministic chains with a fixed values of the distance .

K=5 

K = 15

K = 10 C/ R

C/ R





C/ R

1.45455

1.0519

1.45455

1.1716

1.45455

1.3095

2.90909

1.1888

2.90909

1.4627

2.90909

1.7395

4.36364

1.3452

4.36364

1.7718

4.36364

2.1814

5.81818

1.4977

5.81818

2.0588

5.81818

2.5841

7.27273

1.6392

7.27273

2.3311

7.27273

2.9756

8.72727

1.7843

8.72727

2.6120

8.72727

3.3540

10.18182

1.9248

10.18182

2.8729

10.18182

3.7401

11.63636

2.0750

11.63636

3.1542

11.63636

4.1242

13.09091

2.2109

13.09091

3.4115

13.09091

4.4943

14.54545

2.3537

14.54545

3.6785

14.54545

4.8720

11.7 Lamé equation: plane elasticity problem Let us consider the plane elasticity problem in a bounded domain D μΔu(x) + (λ + μ ) grad div u(x) = 0 , u(y) = g(y) ,

x ∈ D, y ∈ ∂D ,

(11.30)

where u = (u 1 , u 2 )T is the displacement column vector which is prescribed on the boundary as a column vector g = (g1 , g2 )T , λ and μ are the elasticity constants. The problem can be solved by the standard MFS using the fundamental solution of the Lamè equation, a matrix with the entries (e.g. see [33]) ⎧ 7 8 λ + μ ( x − y )( x − y ) ⎨ λ+3μ log |x − y|−1 δ ij + λ+3μ i | xi−y|j2 j , i, j = 1, 2 , for R2 4πμ ( λ +2μ ) 7 8 [E(x)]ij = λ + μ ( x i − y i )( x j − y j ) 1 ⎩ λ+3μ δ ij + , i, j = 1, 2, 3 , for R3 . 3 8πμ ( λ +2μ )

|x−y|

λ +3μ

|x−y|

Here we extend the method, based on the Poisson integral formula, on the Lamé equation. Let x = reiθ be a point in the disk Disk(0, R), such that D ⊂ Disk(0, R) and let ρ = Rr . It is convenient to work in the polar coordinates, with the displacement

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226 | 11 Stochastic boundary collocation and spectral methods functions (u θ , u r ). The Poisson integral formula (6.26) presented in Section 6.1.2 reads

2π u r (r, θ) L11 (ρ; θ − φ) = u θ (r, θ) L21 (ρ; θ − φ) 0

L12 (ρ; θ − φ) L22 (ρ; θ − φ)



g r (Reiφ ) dφ . g θ (Reiφ )

(11.31)

This system is an equation with a separable matrix kernel; hence, we can apply the same approach we used in the case of Laplace equation. We work in polar coordinates and consider a disk Disk(0, R) such that D ⊂ Disk(0, R). Let (g r , g θ ) be the prescribed values of the displacement components in the Dirichlet boundary conditions. In any point (r, θ) ∈ Disk(0, R), the solution of the Lamè equation can be represented through the Poisson formula (6.26), so in particular, for any point on the boundary (r, θ) ∈ Γ we have

2π g r (r, θ) = g θ (r, θ)



0

L11 (ρ; θ − φ) L21 (ρ; θ − φ)

L12 (ρ; θ − φ) L22 (ρ; θ − φ)



f r (Reiφ ) dφ , f θ (Reiφ )

where (f r , f θ )T is the unknown vector function to be found. So we deal with the separable kernel, as the series representations (6.27) show. We take first m terms in these series and write the relevant approximations as follows: L11 =

2m +1 

V k11 (θ)W k (φ) ,

L12 =

k =1

L21 =

2m +1 

2m +1 

V k12 (θ)W k (φ) ,

k =1

V k21 (θ)W k (φ) ,

L22 =

k =1

2m +1 

V k22 (θ)W k (φ) ,

(11.32)

k =1

where we use the notations: ⎛

ρ 2π

⎞

⎟ ⎜ ii ⎜ B1 cos θ ⎟ ⎟ ⎜ ii ⎜ B1 sin θ ⎟ ⎟ ⎜ ⎟ ⎜ ii ⎜ B2 cos 2θ ⎟ ⎟ ⎜ ii ⎟ V ii (θ) = ⎜ ⎜ B2 sin 2θ ⎟ , ⎟ ⎜ ⎜ ··· ⎟ ⎟ ⎜ ⎜ ··· ⎟ ⎟ ⎜ ⎟ ⎜ ii ⎝B m cos mθ ⎠ B iim sin mθ

⎛

⎞ 0 ⎜ ⎟ ⎜ B1ij sin θ ⎟ ⎜ ⎟ ⎜ −B ij cos θ ⎟ ⎜ ⎟ 1 ⎜ ij ⎟ ⎜ B sin 2θ ⎟ ⎜ 2 ⎟ ⎜ ⎟ V ij (θ) = ⎜ −B2ij cos 2θ ⎟ , ⎜ ⎟ ⎜ ⎟ ··· ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ · · · ⎜ ij ⎟ ⎜ B sin mθ ⎟ ⎝ m ⎠ ij −B m cos mθ

Here we denote in the first column B iik = ij Bk

=

1 k π ρ λ ij ( ρ,

1 k π ρ λ ii ( ρ,

⎞ 1 ⎟ ⎜ ⎜ cos φ ⎟ ⎟ ⎜ ⎜ sin φ ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ cos 2φ ⎟ ⎟ ⎜ ⎟ W (φ) = ⎜ ⎜ sin 2φ ⎟ . ⎟ ⎜ ⎜ ··· ⎟ ⎟ ⎜ ⎜ ··· ⎟ ⎟ ⎜ ⎟ ⎜ ⎝cos mφ ⎠ sin mφ ⎛

k ), i = 1, 2, k = 1, . . . , m, and

k ), k = 1, . . . , m, for i = j, i, j = 1, 2 in the second column.

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11.7 Lamé equation: plane elasticity problem

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Now we substitute the approximations of the kernels (11.32) in the above second kind integral equation, this yields 2m +1 

2π

V i11 (θ)

W i (φ)f r (φ)dφ +

i =1

i =1

0

2m +1 

2m +1 

2π

V i12 (θ)

W i (φ)f θ (φ)dφ = g r (θ) , 0

2π 2π 2m +1  21 22 V i (θ) W i (φ)f r (φ)dφ + V i (θ) W i (φ)f θ (φ)dφ = g θ (θ) .

i =1

i =1

0

(11.33)

0

Note that here, to simplify the notations, we keep writing f r and f θ for the solutions of the approximating integral equations. Now we are ready to derive the explicit representation for the approximate solu­ tion. $ % b1 where the kth entries of the 2m + Let us define a 4m + 2-column vector b = b2  2π  2π 1-column vectors b 1 and b 2 are defined by 0 g r (ψ)W k (ψ)dψ and 0 g θ (ψ)W k (ψ)dψ, respectively. Let us prove that the approximations are given by the following representations: u r (ρ, θ) = u θ (ρ, θ) =

2m +1 

V k11 (ρ, θ)c1k +

2m +1 

k =1

k =1

2m +1 

2m +1 

V k21 (ρ, θ)c1k +

k =1

V k12 (ρ, θ)c2k , V k22 (ρ, θ)c2k ,

(11.34)

k =1

where the vectors of coefficients c1k and c2k are obtained from the system of 4m + 2 linear equations Ac = b where the 2 × 2-block matrix A consists of four (2m + 1) × (2m + 1)-blocks A kl , k, l = 1, 2, with the entries 2π

A kl ij

V ikl (ψ)W j (ψ)dψ ,

=

k, l = 1, 2 ,

i, j = 1, . . . , 2m + 1 ,

(11.35)

0

where c = (c1 , c2 )T . Indeed, we multiply equations (11.33) by W j , j = 1, . . . , 2m + 1 and integrate, this yields 2π 2m +1  

2π

V i11 (θ)W j (θ)dθ

i =1 0

+

W i (φ)f r (φ)dφ 0

2π 2m +1   i =1 0

2π

V i12 (θ)W j (θ)dθ

2π

W i (φ)f θ (φ)dφ = 0

g r (θ)W j (θ)dθ , 0

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228 | 11 Stochastic boundary collocation and spectral methods 2π 2m +1  

2π

V i21 (θ)W j (θ)dθ

i =1 0

+

W i (φ)f r (φ)dφ 0

2π 2m +1  

2π

V i22 (θ)W j (θ)dθ

i =1 0

2π

W i (φ)f θ (φ)dφ = 0

g θ (θ)W j (θ)dθ . 0

This system of equations has the form Ac = b, hence, in view of (11.33) and (11.35) the statement is proven. As mentioned above, the method based on the series expansion of the kernel in the Poisson formula is highly efficient yet quite technical. Therefore, we have sug­ gested above a direct numerical scheme, based on the randomized calculation of the surface integrals with the Poisson kernel involved. This scheme is quite general and is easily extended to the Lamè equation. Indeed, let us work with the Poisson formula (6.24). Here, there is no need to turn to the polar coordinates (u r , u θ ). Note that the kernel in this formula, K (ρ; θ − φ)B involves the Poisson kernel K (ρ; θ − φ), as the multiplier and the matrix B is given by (6.25). This can be used to construct the randomized evaluation of the surface integrals. So let us seek the unknown function f (φ) = (f1 , f2 )T in the form f (φ) =

n 

c j h j (φ) ,

j =1 (1)

(2)

where h j (φ) = (h j , h j )T is a set of complete functions on [0, 2π]. Substituting it in the Poisson formula (6.24) yields u (r, θ) =

n 

c j I j (r, θ) ,

j =1

where the 2 × 2 matrices I j (r, θ) are the surface integrals 2π

qI j (r, θ) =

K (r, θ − φ)Bh j (φ)dφ ,

j = 1, . . . , n .

0

These integrals can be calculated very fast, using its probabilistic interpretation, for all collocation points x i = (r i , θ i ) ∈ Γ, i = 1, . . . m. This leads to a system of linear equations n  ui = I j (r i , θ i )c j , i = 1, . . . m , j =1 (1) (u i ,

(2) u i )T

are the values of the boundary function g = (g1 , g2 )T in the where u i = collocation points x i ∈ Γ.

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11.8 SVD and randomized versions

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11.8 SVD and randomized versions 11.8.1 SVD background C In what follows, we will write simply uvT instead of u v to denote the direct prod­ C of a column vector u and a row vector vT . uct Let A be a rectangular m × n matrix with m rows and n columns, having rank r. From the fundamental theorem of linear algebra we know (e.g. see [204]) that the ma­ trix can be represented as a sum of r matrices of rank 1:

A=

r 

σ i u (i) v(i)T ,

(11.36)

i =1

where σ i ≥ σ 2 ≥ · · · ≥ σ r are the singular values and u (i) ∈ Rm , v(i) ∈ Rn , i = 1, . . . , r are its left and right singular column vectors, respectively. The families {u (i) }, {v(i) } are orthogonal sets of vectors: u (i)T · u (j) = δ ij and the same for {v(i) }. In the matrix form, the SVD representation (11.36) reads A = UΣV T , where U and V are orthonormal matrices with left and right singular vectors of A, respectively, and Σ is a diagonal matrix: Σ = diag(σ 1 , . . . , σ r ). Recall that U T U = I r×r and V T V = I r×r . The Frobenius norm AF and the spec­ tral norm A2 are defined by ⎞1/2 ⎛  AF = ⎝ a2ij ⎠ , A2 = max |Ax|2 = σ 1 . ij

| x | 2 =1

The following fundamental result is well known from linear algebra as Eckhart–Y­ oung theorem. Theorem 11.1 (Eckhart–Young theorem [49; 61]). The best approximation (in the norms #k  · F and  · 2 ) of A among all matrices D of rank k, is A k = i=1 σ i u (i) (v(i) )T , i.e. for all k rank matrices D, A − A k 2 ≤ A − D2 , A − A k F ≤ A − DF . The matrix A k admits the representation: A k = U k Σ k V kT = AV k V kT = U k U kT A , where U k , V k are submatrices of U and V which contain only the top left and right sin­ gular vectors, respectively. A matrix A has a good rank k approximation if A − A k  is small in Frobenius and 2-norms. To estimate the errors, one may use the well-known equalities: ⎛ ⎞1/2 r  2 A − A k  = ⎝ σ i ( A ) ⎠ ,  A − A k 2 = σ k + 1 ( A ) . i = k +1

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230 | 11 Stochastic boundary collocation and spectral methods

11.8.2 Randomized SVD algorithm Let us assume that the matrix A is large enough and we want to construct a randomized approximation of the first k right singular values and corresponding right singular vectors. The idea behind many versions of randomized algorithms for SVD is to sample randomly s rows of A, then to form an s × s matrix S and compute its right singular vectors. Let us give the following version presented in [44]. Let us choose a discrete probability distribution p1 , . . . , p m for sampling from the # rows A(1) , . . . , A(m) of A: m i =1 p i = 1.

Randomized SVD Algorithm 0. Fix an integer s such that s is much larger than k, say, s ≈ k /ε2 where ε is an error measure, but s ≤ m. 1. for j = 1 to s do: sample a random index {1, . . . , m} of the row of A according to the probability √ distribution {p}m j =1 and include A ( j ) / sp j as a row of S, 2. Compute SST and its SVD: SST =

s 

λ2j w(j) w(j)T

j =1

3.

Compute h(j) = ST w(j) /|ST w(j) | for j = 1, . . . , k. Construct H k as a matrix whose columns are the h(j) and λ1 , . . . λ k are our approximations to the first k singular values of A. Thus we get a rank (at most) k approximation to A as AH k H Tk .

Note that we could turn to sample columns of A instead of rows and compute approx­ imations of the left singular vectors, then, H k were a matrix RRT A where R is a m × k matrix containing approximations to the top k left singular vectors. Let us give the error estimators presented in [45]. Assume that we construct a k rank approximation AH k H Tk to our matrix A by the above algorithm where the sam­ pling of s random rows is carried out according to a probability distribution {p i }m i =1 satisfying the condition p i ≥ β |A(i) |2 /A2F for some positive β ≤ 1 and let ε > 0. If s ≥ 4k /βε2 then the following estimation of the mean is true: 7 8 (11.37) E A − AH k H Tk 2F ≤ A − A k 2F + εA2F . 9 Error estimation in probability is also possible. Let η = 1 + 8 log(1/δ)/β. If s ≥ 4kη2 /βε2 then with probability at least 1 − δ A − AH k H Tk 2F ≤ A − A k 2F + εA2F .

(11.38)

The same estimations in the spectral norm also hold true, with omitting the fac­ tor k in the conditions s ≥ 4k /βε2 and s ≥ 4kη2 /βε2 .

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11.8 SVD and randomized versions

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It should be mentioned that there exist many versions of randomized SVD, we also used the principal component analysis (PCA) version of [146]. The algorithm can be described as follows.

Randomized PCA algorithm Assume that we are given integer positive numbers i, k, m and n such that 2k < m ≤ n and a real-valued matrix of dimension m × n. Choose an integer l satisfying l > 2k and l ≤ m − k. The integer i can be chosen as i = 1, 2, or larger, depending on the desired accuracy (see the estimation (11.39) below). The algorithm is carried out in the following steps. 1. Generate an l × m matrix G whose entries are independent standard Gaussian random numbers and form the matrix R = G(AAT )i A. 2. Decompose R, using the QR algorithm: RT = QS where Q is a n × l matrix with orthogonal columns and the relevant l × l matrix S. 3. Calculate T = AQ which is an m × l matrix. ˜ ΣW ˜ T where U ˜ is a real-valued m × l matrix 4. Using SVD get the expansion T = U ˜ is a with orthogonal columns, W is l × l matrix with orthogonal columns and Σ l × l diagonal matrix with positive entries on its diagonal placed in nonincreasing order. ˜ = QW. 5. Form the n × l matrix V ˜ the leftmost n × k block V of V ˜ and the 6. Extract the leftmost m × k block U of U, ˜ leftmost uppermost k × k block Σ of Σ. The matrices U, V, and Σ are the desired approximations to the k-rank approximations U k , V k , and Σ k , respectively, of the matrix A. The error of this approximation has a probabilistic nature and with very high prob­ ability the following estimation in spectral norm holds [146]: A − UΣV T  ≤ Cm1/(4i+2) σ k+1 ,

(11.39)

where C is a constant which is not dependent on A and σ j is the jth singular value of A. The cost of the method, T, behaves like T = O((2il + k + l)mnl + l2 n). From the description of the Randomized SVD algorithms it is clear that the step 1 is crucial for the efficiency of the method. In step 1, we could of course use the uni­ form sampling which means, one call of the RAND generator will only be used, not depending on the dimension n. However this would work well only if the “weights” of the rows, |A(i) | are more or less equal for all i = 1, . . . , n. Generally, according to the estimates (11.37), (11.38), it is reasonable to sample the rows according to the probability distribution p i = β |A(i) |2 /A2F . In [47], the authors suggest to use the conventional sampling algorithm which needs about n log n operations. However, we can use Walker’s algorithm [213] (see the Fortran code in our recent paper [177]) which even in the general case needs only one call to RAND generator, not depending on the

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232 | 11 Stochastic boundary collocation and spectral methods dimension of the matrix. Out of the loop, we need only a preparation of two additional arrays of dimension n which are calculated in O(n) operations. This method works, of course, if we use the sampling of rows with replacement which is always the case since we deal with matrices of large dimension. Thus this sampling algorithm is practically equivalent in efficiency to the uniform sampling of rows. Let us turn to step 2, the computation of SVD for the sampled matrix SST . Here any efficient SVD algorithm can be used, since the dimension of the matrix, s, is not very large. In our calculations, typical values of s, the number of sampled rows, were of the order of log n/ε2 where n is the dimension of the original matrix and ε is the expected error.

11.8.3 Using SVD for the linear least squares solution The general formulation of a linear least squares problem for solving the equation Ac = b is the following: we have a set of columns {a1 , a2 , . . . , a n } of the matrix A, we wish to combine them linearly to provide the best possible approximation to the D vector b. So we seek coefficients c1 , c2 , . . . , c n which produce a minimal error Db − D #n D i =1 c i a i . Let the SVD of A be UΣV T (where U and V are square orthogonal matrices consist­ ing of orthogonal column vectors v i and u i , respectively), and Σ is diagonal, with the entries {Σ}ii = σ i , σ 1 > σ 2 > · · · σ r where r is the rank of the matrix A. Then we have Ac − b = UΣV T c − b = U (ΣV T c) − U (U T b ) = U (Σy − d) , where y = V T c and d = U T b. Note that U is an orthogonal matrix and so preserves lengths, i.e. U (Σy − d) = Σy − d and hence Ac − b  = Σy − d. This suggests a method for solving the least squares problem. First, determine the SVD of A and calculate d as the product of U T and b. Then, solve the least squares problem for Σ and d, i.e. find a vector y so that Σy − d is minimal which is obviously trivial since Σ is diagonal. Now, y = V T c so we can determine c as Vy. That gives the solution vector c as well as the magnitude of the error, Σy − d which can be written as c=

r  u Ti b vi . σi i =1

Regularization: if the right-hand side is given with the noise, b = b¯ + b , then

r r   u Ti b u Ti b¯ u Ti b vi = + vi . c= σi σi σi i =1 i =1 Regularization is usually in throwing away the terms with small values of σ i .

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11.9 Numerical experiments In this section, we present some results of numerical simulations to illustrate the per­ formance of the developed stochastic boundary methods. We present the results for three suggested methods: (1) MFS based on the simple layer kernel and the randomized SVD, (2) the method based on the Poisson integral formula presented in Section 11.5, and (3) a version of MFS based on the double-layer kernel. The linear system for the unknown coefficients was solved by the random­ ized SVD method. This method was in some examples compared against the Gaussian solver based on the LU-decomposition. Important issue in the numerical experiments was the size of the linear system and its condition number, as well as the accuracy of the methods. Note that the con­ ventional condition number is not well characterizing the condition quality of the lin­ ear systems in MFS, therefore, one suggests different versions of the condition num­ bers in the literature. For example, the authors [217] introduced an effective-conditionnumber (ECN) as a sensitivity measure for a linear system; it differs from the traditional condition-number in the sense that the ECN is the right-hand side vector dependent. In this chapter, we characterize the condition property by the spectrum of the first several singular values. It turns out that the decay of this spectrum is well character­ izing the condition property of the linear system. Let us start with the Laplace equation with Dirichlet boundary conditions: we have considered different 2D and 3D domains, in particular, a series of ellipses   x2 y2 Ω = (x, y) : 2 2 + 2 2 = 1 , k = 1, 2, . . . , 10 , k a k b for different values of the ratio ν = a/b where a and b are the semiaxes. The exact model solution was chosen as u (x, y) = exp(κx) cos(κy), for different values of the parameter κ. Here we deal with the following issue: how the accuracy of the MFS technique can be improved by an appropriate choice of the spatial distribution of the singu­ lar sources. We have compared two basically different schemes: (1) the sources were placed uniformly on the circle, (2) the sources were randomly distributed in an annu­ lus (see Figure 11.4). More precisely, in scheme 1 the ellipse has semiaxes 1 and 3, the values of the radius of the outer disk were taken as R2 = 3.05, . . . , 12.05. For these values of R2 , we have calculated the solution along the lines l1 = {(x, y) : y = 1.3} and l2 = {(x, y) : x = 0.7}. We have placed 400 singular sources uniformly on the circle and 900 collocation points were chosen on the boundary of the ellipse. The rel­ evant rectangular system of linear equations was solved by the stochastic SVD with rank k = 250 approximation. As discussed, it is expected that the best accuracy will be achieved for some value of R2 which is at some distance from the ellipse. The cal­ culations show that the minimal error is achieved at R2 ≈ 3, then, with the increase of

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Figure 11.4. Illustration to the random sampling of singular sources in the annulus around the el­ lipse. Black large points show schematically the uniform distribution of sources in the annulus, small red points correspond to the probability density (11.40).

R2 the error increases. This is clearly seen from the results given in Figures 11.5, 11.6, left panel, and Figure 11.8, left panel. In scheme 2, we have tried two different spatial distributions: in the first one, we have sampled the singular sources in the annulus with the radii R1 < R2 so that the ellipse lies centered in the disk of radius R1 . The random points in the annulus were sampled as follows: in polar coordinates, the angle was sampled uniformly, while the radius was sampled from a Gaussian distribution with the mean value R1 +0.5(R2 − R1 )  and variance σ = (R2 − R1 )/8. The results of calculations are shown in Figure 11.6. In the second version, we have changed the normal distribution with a distribution which concentrates the singular sources closer to the boundary of the ellipse. Thus the polar angle was isotropic and the radius was sampled at random on the interval [R1 , R2 ] from the density     R2 − r R2 − r exp π . (11.40) p(r) = α sin π R2 − R1 R2 − R1 For α = 2, see Figure 11.4 where the big black squares show the uniform distribution of sources in the annulus and small points are sampled according to the probability density (11.40). This choice improves the accuracy considerably. This shows us that any prior information about the character of the solution can be used to sample the sources appropriately. Two main conclusions can be made from these calculations. First, the scheme with the point sources distributed in the annulus according to the density p(r) has the minimum error (see Figures 11.7 and 11.8, left panel). Second, in the cases where

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11.9 Numerical experiments | 235

25 20 15

200 exact solution solution for radius = 7.05 solution for radius = 8.05 solution for radius = 9.05

150

exact solution solution for radius=10.05 solution for radius=11.05 solution for radius=12.05

100

10 50 5 0

0

−50

−5 −10 −1

−0.5

0 X

0.5

1

−100 −2.5 −2 −1.5 −1 −0.5 0

0.5

1

1.5

2

2.5

Y

Figure 11.5. Solutions of the Dirichlet problem obtained by MFS for a series of outer circles: plots along the line l 1 (left panel) and for the line l 2 (right panel). 100

10 exact solution solution for radius = 10.05 solution for radius = 11.05 solution for radius = 12.05

8

80

exact solution solution for radius = 12.05

60 40

6

20 4 0 2

−20 −40

0

−60 −2 −1

−0.5

0 X

0.5

1

−80 −2.5 −2 −1.5 −1 −0.5 0 Y

0.5

1

1.5

2

2.5

Figure 11.6. Solutions of the Dirichlet problem obtained by MFS with a normal distribution of sources in the annulus: plots along the line l 1 (left panel) and for the line l 2 (right panel). 10

100 exact solition solution for radius = 12.05

9

80

8

60

7

40

6

20

5

0

4 3

−20

2

−40

1

−60

0 −1

exact solution solution for radius = 12.05

−0.5

0 X

0.5

1

−80 −2.5 −2 −1.5 −1 −0.5 0 Y

0.5

1

1.5

2

2.5

Figure 11.7. Solutions of the Dirichlet problem obtained by MFS with the distribution of sources in the annulus according to the nonsymmetric density p given by (11.40): plots along the line l 1 (left panel) and for the line l 2 (right panel).

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2.4

103

2.2

annulus, sin−exp distribution

2.1

102 101 100

2 1.9 1.8

10−1

1.7

10−2 10−3 3

exact solution n=75 n=90 n=105 n=120

2.3

annulus, normal distribution

u(x,y0)

Max error on the line y=1.3

concentric circles

104

1.6 4

5

6 7 Ridius

8

9

10

1.5

−0.8 −0.6 −0.4 −0.2

0 x

0.2 0.4 0.6 0.8

Figure 11.8. Left: Comparison of the errors for three different distributions of the singular sources. Right: Solution of the Poisson equation by the method presented in Sect .11.5

the point sources are distributed in the annulus, the plotted curves are more smooth. This means the errors are less scattered for different points in the domain which is important from practical viewpoint. Let us now give an example of a nonhomogeneous Laplace equation Δu (x, y) = f (x, y) with Dirichlet boundary conditions solved in the ellipse Ω with half-axes a = 1, b = 3. We applied the method based on the Monte Carlo calculation of the integral in the Green function formula over the ellipse described in Section 11.5, the number of the random samples in the integration was 10 000, the Green function was calculated by the method based on the spectral inversion of the Poisson formula with n harmonics and the outer disk having radius R = 10. The source function was f (x, y) = 12.0x2 + 6.0y with the known exact solution u (x, y) = x4 + y3 + 5.0y and boundary function g = u |∂Ω = a4 (1 − y2 /b 2 )2 + y3 + 5.0y. In Figure 11.8, right panel, we compare the calculated results (with different number of harmonics n) against the exact solution shown on the line l = {(x, y) : y = 0.3}. It is seen that n = 120 was sufficient to obtain the solution with high accuracy. The Lamé equation was solved for the exact solution u r (r, θ) = ρ k sin kθ, u θ (r, θ) = ρ k cos kθ, for different values of k. In the MFS and MFS with double-layer kernel the number of collocation points was chosen to keep the length between two points independent of the diameter of the ellipse which means the number of points n was linearly increasing with the diameter. In the method based on the Poisson inte­ gral formula we use n to denote the number of terms in the series expansion. Thus in this method we deal with n × n matrices to be converted. In MFS and MFS with double-layer kernel we deal both with square matrices and overdetermined systems with rectangular matrices. It turns out that the case of overde­ termined systems becomes more efficient for problems with large number of colloca­ tion points. Note that this was also noticed in [198] and [104]. This is a crucial point: the randomized SVD methods are especially efficient when overdetermined systems hav­

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Table 11.3. Comparison of the randomized SVD and direct GE method. n, m

Accuracy, SVD

Accuracy, Gauss

Comp. time, SVD

Comp. time, Gauss

m = 1000, n = 1000

0.157e–02

0.72e–05

3.87

0.37

m = 5000, n = 1000

0.221e–02

0.99e–00

23.16

61.32

m = 1000, n = 5000

0.198e–02

0.99e–00

15.00

56.84

m = 5000, n = 5000

0.276e–02

0.21e–06

315.36

32.47

m = 6000, n = 1000

0.147e–02

0.387e–01

9.89

86.62

m = 11 000, n = 1000

0.844e–02

0.883e–01

17.30

418.31

m = 16 000, n = 1000

0.627e–02

0.226e–01

25.84

1334.15

m = 21 000, n = 1000

0.257e–02

0.616e–00

43.23

3471.98

ing good low-rank approximations are solved. Our calculations confirm the conclu­ sion made in [30] that in the case of relatively small square matrices the direct solvers like Gauss elimination (GE) are preferable compared to SVD. However for large-scale problems when the linear systems of MFS are very large and overdetermined, the ran­ domized SVD methods are well working while the direct GE solvers are not applicable anymore. To be more specific, let us consider the Dirichlet problem for an ellipse with halfaxes a = 1, b = 5. The exact solution was taken as u (x, y) = exp(3.75x) cos(3.75y). The sources were chosen uniformly on a circle of radius R = 7, centered at the center position of the ellipse. We calculated the solution along the line l1 = {(x, y) : x = 0.7}. In the randomized SVD, we calculated k = 750-rank approximations. In Table 11.1, we show the results of comparison for SVD and Gauss solver. From these results we conclude that for moderate dimensions, the Gauss solver outperforms the SVD, but for large dimensions, the SVD’s efficiency is much higher, especially in the case of overdetermined systems (e.g. see the last line in Table 11.1.). It should be noted that the accuracy of the method can be increased by increasing the number of collocation points, but this should be done carefully since the increase of the collocation points may not only increase the accuracy of the boundary func­ tion approximation, but also increase the error of the SVD. This is illustrated in Figure 11.9 where we show the error for the approximation on the boundary when solving the above problem for different number of collocation points. Further increase of the number of collocation points could not improve the results. Let us now describe the results obtained by the spectral method based on the inversion of the Poisson integral equation. As mentioned, the conditioning of this method is much better than the standard MFS. Note that here we hope to work with matrices which are considerably less than that for MFS. This is an important issue since with the dimension increase, the cost of calculation of the integrals in the spectral method is rapidly increasing. We illustrate this by the results shown in Fig­

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238 | 11 Stochastic boundary collocation and spectral methods 0.5 n=1000, m=1000 n=5000, m=1000 n=8000,m=1000

0.45 0.4 0.35

Error

0.3 0.25 0.2 0.15 0.1 0.05 0 0

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Boundary points

Figure 11.9. The error of the solution on the boundary in the MFS. 10

2

10

0

10

−2

10

−4

10

−6

10

−8

10

−10

20

40

60 Dimension

80

100

120

10

3

10

2

10

1

10

0

10

−1

20

40

60

80

100

120

Dimension

Figure 11.10. The error (left panel) and computer time (right panel) as functions of the dimension (the number of retained terms) in the spectral method.

ures 11.10–11.11. Here we have solved the same Dirichlet problem in the ellipse with a = 1, b = 5, k = 3, the exact solution u (x, y) = exp(x) cos(y) and the outer circle of the Poisson formula R = 25. The integrals were calculated by the simple Monte Carlo integration with 105 points sampled at random uniformly on the circle. Note that the error is rapidly decreasing with the dimension, but the computer time is also increasing fast. Since however the number of terms which guaranties the error about 10−2 is about 15, the method is considered as highly efficient. The spectral method was also applied to the following elastostatic problem. We solved the Lamé equation for the elastic ellipse with semiaxes a = 3.0, b = 9.0 and elastic constants λ = 1000.0, μ = 1.0, the exact solution is u r (ρ, θ) = ρ 3 cos 3θ, u θ (ρ, θ) = ρ 3 sin 3θ. In Figure 11.12 (left panel) we show the u 1 -component

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11.9 Numerical experiments | 239

Solution

0

exact solution n=2 5 8 10

1.5 1 Solution

0.2

−0.2 −0.4

0.5 0

−0.5

exact solution n=2 5 8 10

−0.6 −1 −0.8 −0.5

0 (x,y)ϵ l 1

−1.5

0.5

−2 −1.5 −1 −0.5 0 0.5 (x,y)ϵ l 2

1

1.5

2

Figure 11.11. The spectral method compared against the exact solution, for different values of the number of harmonics n = 2, . . . , 10. The solution is shown along the lines 1 (left panel) and 2 (right panel). 30

35 exact solution

exact solution n=1000 n=6000 n=11000 n=16000 n=21000

u1(x,y)

10

30

k=50

25

k=100 k=150

20 u1(x,y)

20

0

15 10

−10

5 −20 −30 −3

0 −2

−1

0 (x,y)ϵ l 1

1

2

3

−5 −3

−2

−1

0

1

2

3

(x,y)ϵ l 1

Figure 11.12. The u 1 -component of the displacement of the elastic ellipse calculated by the spectral method. The number of point sources m = 1000, the number of collocation points varies in the interval n = 1000, . . . , 21 000. The k = 100-rank approximation was used in the randomized SVD (left panel). Right panel: the same component obtained for different k-rank approximations k = 50, 100, 150, while n = 21 000 fixed.

of the solution calculated by the randomized SVD with k = 100-rank approximation, compared against the exact result. The number of the source points placed uniformly on the outer disk of radius 10 was 1000. The solution is plotted for the points of the line l1 = {(x, y) : y = 0.7}. It is seen that the number of collocation is optimal at about n = 16 000, the further increase of n leads to larger deviation if we do not appropriately increase the rank of the approximation. To confirm this we show the results for n = 16 000 with different k-rank approximations. It is seen (Figure 11.12, right panel) that k = 150 is sufficient to achieve the desired accuracy. To compare the condition properties for the three developed methods, we calcu­ lated the spectra of singular values by the SVD method. In Figure 11.13, we present

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240 | 11 Stochastic boundary collocation and spectral methods 10 4 n=200 300 400 500 600 700 800 900 1000 1100

Integral Poisson kernel

10 0 10 −1 10 −2

The value of the singular numbers

The value of the singular numbers

10 1

10 −3 10 −4 10 −5

2

4

6

8 10 12 singular number

14

16

18

Double layer Simple layer (MFS)

10 2

Integral Poisson kernel n=500

10 0 10 −2 10 −4 10 −6 10 −8

2

4

6

8 10 12 14 singular number

16

18

20

Figure 11.13. Left panel: the spectrum of the first 18 singular numbers of the new method based on the integral Poisson kernel (left panel) for different matrix sizes from n = 200 to n = 1100. Right panel: comparison of the spectra of singular numbers for the three methods, the matrix size n = 500.

the first several singular values for a series of ellipses with increasing diameters, with k = 1, 2, . . . , 10 and ν = 3. The number of points on the boundary of the smallest el­ lipse was taken equal to 100, for the next ellipse with doubled diameter the number of collocation points was 200, etc., the maximal ellipse was taken 10 times larger with 1000 collocation points. It is clearly seen that first the singular values are rapidly in­ creasing with the dimension n and second, for any fixed n, the decay of the singular values is close to exponential, the variation expands on 7–8 orders of magnitudes. The spectrum of singular values for the method based on the Poisson integral formula is shown in Figure 11.13, left panel. Here two remarkable properties can be seen: (1) the variation of singular values are practically the same for all dimensions, (2) the sin­ gular values decrease from 1 to about 0.5 × 10−4 . In the right panel of Figure 11.13 we compare the spectra of singular values of the three methods for the dimension n = 500. From these curves the advantage of the method based on the Poisson inte­ gral formula, from the view point of the condition property, is clearly seen. The most important conclusion is that for the method based on the Poisson integral formula the condition property is practically independent of the dimension of the linear system, so this implies, the method is well suited for large problems governing by matrices of very high dimension. In conclusion, we stress that in MFS-type methods, in practical problems with nontrivial boundary conditions the number of singular points can be taken, as a rule, much less than the number of collocation points, so we deal with large overdetermined systems with dimensions m >> n. The direct solver GE resolves these systems by the regularization method inverting m × m square systems AT Ac = AT b. Since m can be very large, these methods are not well suited for practical problems for large 3D domains. In contrast, the randomized SVD can be well applied for large problems since the cost of these methods is determined by n and is slowly dependent on m.

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12 Solution of 2D elasticity problems with random loads 12.1 Introduction In this chapter, we describe a stochastic simulation method for a numerical solution of the Lamé equation with random loads following the presentation given in [174]. The vector random field of loads that stands in the right-hand side of the system of elas­ ticity equations is simulated by the randomization spectral method presented in [113; 160] and recently revised and generalized in [98] and [163]. Comparative analysis of the Random Walk on Spheres method and an alternative direct evaluation of the correla­ tion tensor of the solution is made. We also derive a closed boundary value problem for the correlation tensor of the solution which is applicable in the case of inhomo­ geneous random loads. Calculations of the longitudinal and transverse correlations are presented for a domain which is a union of two arbitrarily overlapped disks. We also discuss a possibility to solve an inverse problem of determination of the elastic constants from the known longitudinal and transverse correlations of the loads and give some relevant numerical illustrations. It is well known that the boundary value problems with random parameters are very interesting models that become more and more popular in many fields of sci­ ence and technology, especially in problems where the data are highly irregular, for instance, such as in the case of turbulent transport [119], flows in porous medium [34], or evaluation of elastic properties of polymers and composites containing fibers which are randomly oriented in a plane [125; 84; 9]. In [102], a stochastic model that could realistically and accurately simulate wind loads that are generated by thunderstorm downbursts for transmission line design is developed. We also mention applications in physics of colloidal particles and polymers, liq­ uid crystals, planar hard dumbbell fluids (e.g. see [115; 142; 216; 9]) governed by di­ rect elasticity problems with random loads, which can be solved by the methods we present in this chapter, as well as elastography problems (e.g. see [130]) related to inverse problems of evaluation of elastic constants we discuss in our numerical sim­ ulations. In conventional deterministic numerical methods, these problems are solved as follows: first one constructs a synthesized sample of the input random parameter; then the obtained deterministic equation is solved numerically, say, by finite element method and give the solution in all points of the grid domain. These two steps are re­ peated many times, so that the obtained statistics is sufficient to calculate the desired averages accurate enough. This approach is used in stochastic finite element methods (e.g. see [59; 207; 125; 6]). Obviously, this technique is generally time consuming and to

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242 | 12 Solution of 2D elasticity problems with random loads solve practically interesting problems one needs supercomputers to extract sufficient statistical information. In the Monte Carlo approach, the algorithms are designed so that the solution is calculated only in the desired set of points without constructing the solution in the whole domain (e.g. see [160; 155; 171]). To evaluate different statistical characteris­ tics of random boundary value problems we use the Double Randomization technique (e.g. see [160]). This approach is possible if the desired statistical characteristics (e.g. the mean or the correlation tensor) can be represented in the form of a double expec­ tation over the input random parameters and over the trajectories of a Markov process used in a stochastic estimator for solving the deterministic equation. The advantage of this method is that we do not have to solve the equation many times; hence, the cost of this method is drastically decreased compared to the stochastic finite element method. The well-known drawback of stochastic simulation methods should be men­ tioned: the error behaves like ε ∼ O(N −1/2 ), where N is the number of samples; hence, it is reasonable to apply the Monte Carlo methods if the desired accuracy is not too high. So for realistic applied problems the typical Monte Carlo accuracy lies in the range of 0.1% to several percents. The basic idea behind the Double Randomization can be explained by the follow­ ing very simple example. Assume that we have to evaluate an integral  J (x; ω) = f (x, y; ω)dy , D

where f (x, y; ω) is a random function indexed through x, y, defined on a probabil­ ity space, ω being the relevant random element. Obviously of interest are statistical characteristics of the random process J (x; ω) such as the expectation  J (x; ω) and the covariance  J (x1 ; ω)J (x2 ; ω). In deterministic methods, to calculate the expectation  J (x; ω), first one has to construct a sample of the random function f , say, f (x, y; ω1 ) and then calculate the integral J (x; ω1 ) by one of the quadrature formulae. This is then repeated N times, N large enough to guarantee that the average over N samples provides a good approx­ imation to  J (x; ω). Thus in short, you have to solve your deterministic problem (in this case, evaluation of the integral) N times, N being the number of samples. The Double Randomization is based on the representation of the desired func­ tional as a double expectation. Indeed, we choose an arbitrary probability density function p(y), y ∈ D, arbitrary enough so that p(y) = 0 for y where f (x, y; ω) = 0 for all x and ω. Then we can write ' & J (x; ω) = E p f (x, ξ ; ω)/p(ξ ) , where E p stands for the average over the random points ξ distributed in D according to the density p. Therefore, ' ' & &  J (x; ω) =  E p f (x, ξ ; ω)/p(ξ )  = E(ω,p) f (x, ξ ; ω)/p(ξ ) ,

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12.1 Introduction

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where E(ω,ξ ) stands for averaging over random elements (ω, ξ ). This statement is just the Fubini theorem and it shows that the desired result can be obtained by averaging over random samples of ω and ξ . This approach also works when the deterministic problem is not simply an inte­ gral evaluation, but a PDE, or an integral equation, or even nonlinear equations. The main challenging problem is here to transform the solution of the original problem to the evaluation of an expectation over relevant stochastic elements, often in functional spaces. In our case expectations are constructed over Markov chains. The approach we use in the present chapter is based on the Poisson-type inte­ gral formula written for each disk of a domain consisting of a family of overlapping disks. We call this method a Random Walk on Fixed Spheres (RWFS) algorithm. The original differential boundary value problem is equivalently reformulated in the form of a system of integral equations defined on the intersection surfaces (arches, in 2D, and caps, if generalized to 3D spheres). To solve the obtained system of integral equa­ tions, a Random Walk procedure is constructed where the random walks are living on the intersection surfaces. Since the disks (spheres) are fixed, it is also convenient to construct discrete random walk methods for solving the system of linear equations approximating the system of integral equations. In [172], we have concluded that in the case of classical potential theory, the RWFS considerably improves the convergence rate of the standard Random Walk on Spheres method. More interesting, we succeeded there to extend the algorithm to the system of Lamé equations which cannot be solved by the conventional Random Walk on Spheres method. Therefore, we are able to use this method in our numerical analysis of the Lamé equation with random loads. The 2D vector random load is assumed to be in­ compressible, isotropic, and Gaussian, with a given form of the spectral tensor which in the studied case is uniquely defined by one scalar function, the energy spectrum. To simulate this vector random field we apply the Randomization Spectral Method presented in detail in [98], see also [163]. The RWFS can be quite convenient for solving elasticity problems for domains that can be approximated by a set of overlapping disks (or spheres, in 3D). An ideal case is of course when the domain itself is represented as a family of overlapping disks, such as in the case of polymers and macromolecules (e.g. see [142; 115; 125; 9]). In this case there is no need to introduce any grid system and the Random Walk method can be competitive with the best deterministic methods. When random fields are involved, such as the random loads or boundary conditions (e.g. see [179]), then the advantage of the stochastic algorithms is clear, because via double randomization, one avoids to solve the boundary value problem many times, in contrast to the conventional deter­ ministic methods. The chapter is organized as follows. In this chapter, we deal with random loads, so a generalization of RWFS algorithm to the Lamé equation with nonzero body forces is needed which is done in Section 12.2. Here we present, in detail, the Direct and Adjoint stochastic algorithms.

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244 | 12 Solution of 2D elasticity problems with random loads In Section 12.3 we describe the random loads and the Randomization Spectral method for simulation these vector random fields. Section 12.4 deals with an alterna­ tive method of calculation of the correlation tensor of the solution of Lamé equation based on its closed representation in the form of an iterated integral Green formula. The relevant boundary value problem for the correlation tensor is also derived. Numerical results are presented in Section 12.5. In the first subsection we test the random field simulation method by comparing calculations with exact results. The RWFS method for the case of nonzero body forces is tested in the second subsection by solving a test problem with known exact solution. The main numerical results are presented in the third subsection where we give the results of numerical calculations of the longitudinal and transverse correlations of the displacement vector under ran­ dom loads.

12.2 Lamé equation with nonzero body forces Now we extend the RWFS method described in Chapter 11 to the case when there are nonzero body forces. The state of the body with fixed boundary is then governed by the following inhomogeneous Lamé equation: Δu(x) + α grad div u(x) + f (x) = 0 ,

x∈ D,

u|Γ = 0 .

(12.1)

Here f (x) = g(x)/μ where g(x) describes the body forces. The solution to the problem (12.1) can be represented through the Green tensor as follows:  u(x) = G(x, y)f (y)dy . (12.2) D

Here and in what follows, we use the notation y for points in the domain D. We take the Green tensor G(x, y) in the form G(x, y) = Γ(x − y) + W (x, y) ,

(12.3)

where the tensor Γ(x − y) is the fundamental solution of the Lamé equation which in 2D case is given by [33] $ % β¯ 1 ¯ log |x − y|)I − Q . (12.4) Γ (x − y ) = (α xy 8π |x − y|2 Here Qxy is a 2D-matrix with the entries Q ij = (x i − y i )(x j − y j ), i, j = 1, 2, I is an identity matrix and the constants α¯ , β¯ are expressed through the constant α as follows: α¯ =

1+α , 0.5 + α

β¯ =

α . 0.5 + α

Let us introduce some notations. Let L = Δ + α grad div be the Lamé operator which is defined in (12.1) as a differential operator acting on a column-vector function

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12.2 Lamé equation with nonzero body forces

u(x). If W (x) is a 2 × 2-matrix with the first column W(1) and second column W(2) , then we define LW (x) as a matrix whose first column is LW(1) (x) and the second column ˆ that acts on the rows of is LW(2) (x). Analogously we define an “adjoint” operator L T ˆ W, which actually implies that LW (x) = LW (x). Let us now consider the column vectors consisting of the relevant entries of the matrices Γ and W in (12.3): Γ(i) = (Γ i1 , Γ i2 )T , and W(i) = (W i1 , W i2 )T , i = 1, 2. From (12.3) it follows that the vector functions W(i) (x, y), i = 1, 2 solve the follow­ ing pair of boundary value problems: ΔW(i) (x, y) + α grad div W(i) (x, y) = 0 , (i)

x∈ D, (i)

W (x, y)|x→z∈∂D = −Γ (z − y) ,

i = 1, 2 .

(12.5)

The integral representation (12.2) can be very conveniently used in the Monte Carlo calculations. Let us describe the method that we call a Double Randomization tech­ nique (for details, see [160]). First, we rewrite the integral (12.2) in the form of an expectation. To this end, we introduce in the domain D an arbitrary probability density function π(y) satisfying the condition that π(y) = 0 for all points y ∈ D where G(x, y)f (y) = 0. ˜ from π(y) In particular, we could choose a uniform density so that the samples y are uniformly distributed in D. Let us denote by E π the expectation taken according to the distribution density π(y). Then (12.2) can be written as follows: $ %  ˜ )f (y ˜) G(x, y)f (y) G(x, y π(y)dy = E π , (12.6) u(x) = ˜) π (y ) π (y D

˜ are sampled in D from the density π(y). In view of (12.3) we get where the points y $ % ) f (y ( ˜) ˜) ˜ ) + W (x, y , (12.7) u(x) = E π Γ(x − y ˜) π (y where the entries of the matrix W (x, y) solve the boundary value problem (12.5). This problem is solved by our RWFS method, which conveniently finds the solutions on the arches 𝛾1 and 𝛾2 without calculating the solution in the whole domain. We can use this feature to construct very efficient method based on representation (12.7) rewritten as follows. Indeed, using the spherical mean representation for the vector functions W(i) defined in (12.5) and assuming that they are already found on the arches 𝛾1 and 𝛾2 we arrive at the following probabilistic representation: u(x) = Eπ

$

E E F F ˜ ) − B(x, z˜)Γ(z˜ − y ˜ )|y ˜ ˜z∈Γ i + B(x, z˜)W (z˜ , y ˜ )|y ˜ z˜∈𝛾 Γ (x − y j



(12.8) % ˜) f (y . ˜) π (y

˜ )| The matrix B is defined in Theorem 9.3. The conditional expectation B(x, z˜)Γ(z˜ − y ˜ ˜z is taken according to a distribution of random points ˜z on the relevant circle, the y

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246 | 12 Solution of 2D elasticity problems with random loads distribution density being p(z; x) from ((9.4)) under the condition that the random ˜ (sampled in D from π(y)) is fixed: so if y ˜ ∈ K (O i , R i ), then the sampled ran­ point y dom point ˜z lies on the circle S(O i , R i ), i = 1, 2. Let us rewrite (12.8) in a form more convenient for practical calculations: % $  f (y  E F ˜) ˜ ) + B(x, z˜)V (z˜ , y ˜ )|y ˜ ˜z∈S(O i ,R i ) Γ (x − y , (12.9) u(x) = E π ˜) π (y ˜ ) if the sampled point z˜ lies on the external where the matrix V (x, z˜) equals to −Γ(z˜ − y boundaries Γ1 or Γ2 ; if z˜ ∈ 𝛾, then the matrix V (x, z˜) is set to be the known values of ˜ ) calculated by the Random Walk algorithm as explained above. W (z˜ , y Note that the points ˜z can be sampled uniformly on the circles, then, instead of (12.9), the representation has the form $ %   f (y E F ˜) ˜ ) + 2πp(z˜; x)B(x, ˜z)V (z˜ , y ˜ )|y ˜ ˜z∈S(O i ,R i ) u(x) = E π Γ (x − y . (12.10) ˜) π (y It is clear that for calculations, (12.9) is more attractive, since the singularity of the integral equation is here included into the density p(z; x); in addition, the simulation algorithm according to this density is very simple and efficient (see [171]). However in some cases, it is desirable to use a distribution which is the same for all points x, like the uniform distribution in (12.10). Thus, let us describe a direct randomized calculation of the expectations in (12.9), which gives an approximate value of u(x) at a fixed point x. A direct randomized evaluation of the expectations in (12.9) can be designed in a form of a simple and efficient algorithm which we call here Direct algorithm. To present it in a detailed scheme, we first write down a Direct unbiased estimator: & ' f (y ˜) ˜ , z˜) = Γ(x − y ˜ ) + B(x, z˜)V (z˜ , y ˜) ξ (x, y , ˜) π (y

where

⎧ ⎨−Γ(z ˜−y ˜) , ˜) = V (z˜ , y ⎩ W (z ˜, y ˜) ,

if z˜ ∈ Γ i , if z˜ ∈ 𝛾i

(i = 1, 2) .

(12.11)

(12.12)

˜ ) is known on 𝛾1 and 𝛾2 , then in view of (2.20) we So if we now assume that W (z˜ , y conclude that the estimator (12.11) is unbiased: ˜ , z˜) . u(x) = Ey˜ ,˜z ξ (x, y ˜) solve the problem (12.5), hence they can be precalculated The entries of W (z˜ , y on 𝛾 as mentioned above, by solving the linear algebraic equation (9.22) which approx­ imates the integral equation (9.35). To solve (9.22), one of the following methods can be used: (1) SOR method, (2) direct inversion of the matrix A, (3) an SOR-based Ran­ dom Walk. We note that (1) and (2) give the solution at all the grid points on 𝛾 = 𝛾1 ∪ 𝛾2 while the method (3) gives the solution at an arbitrary fixed point on 𝛾. Clearly, all

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12.2 Lamé equation with nonzero body forces |

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these three methods produce a bias in the estimator ξ . This bias can be made smaller by increasing the number of nodes on 𝛾 in the methods (1) and (2) or by increasing the number of Random Walks in the method (3). ˜ ) is calculated by one of these three methods, we Thus assuming that W (z˜, y present the Direct algorithm as follows:

Direct algorithm 0. Put the initial score as zero: Ξ(x) = 0. ˜ in our domain D according to a density π(y) (say, uni­ 1. Sample a random point y formly in D) and suppose the sampled point lies in the disk K (O i , R i ) (i = 1, 2). 2. On the circle S(O i , R i ) one samples a random point ˜z according to the density p R i (z; x). 3. If the sampled point z˜ belongs to the external boundary, i.e. if z˜ ∈ Γ i , then calcu­ late & ' f (y ˜) ˜ , z˜) = Γ(x − y ˜) − B(x, z˜)Γ(z˜ − y ˜) ξ (x, y . (12.13) ˜) π (y ˜ ) should be replaced in (12.13) with −W (z˜ , y ˜ ) precalculated If ˜z ∈ 𝛾, then Γ(z˜ − y as described above. Note that when applying for this purpose the Random Walk ˜ ) by the relevant random estimator along one path method, we can replace W (z˜ , y of the Random Walk. ˜ , z˜)/N . 4. Ξ(x) := Ξ(x) + ξ (x, y 5. Repeating the steps 1–4 N times, N sufficiently large, we get the approximate so­ lution as u(x) ≈ Ξ(x). Note that in this algorithm we actually have to resolve the homogeneous boundary ˜ j ), j = 1, . . . , N. Hence value problem (12.5) for N different boundary functions −Γ(z −y if N is large, it may be time consuming. But to achieve high accuracy, N should be large enough. To improve the efficiency, we use the symmetry property of the Green tensor G(x, y) in accordance with the so-called global Monte Carlo algorithm shortly de­ scribed in Section 11.5.2. The symmetry property G(x, y) = GT (y, x) implies that the evaluation of W(i) (x, y), i = 1, 2 at the point x considered as the solution of the prob­ lem (12.5) with the boundary function −Γ(i) (·, y) (i = 1, 2, y fixed) is equivalent to evaluation of W(i) (x, y), i = 1, 2 in point y with the boundary function −Γ(i) (x, ·) (i = 1, 2, x fixed). ˜j, In this method, when calculating the Green function G(x, y) in many points y j = 1, . . . , N we have to solve only one boundary value problem with the boundary function −Γ(i) (x, ·) (i = 1, 2, x fixed) but the solution is to be calculated in many ˜ j , j = 1, . . . , N. points y ˜ j , j = 1, . . . , N of the Formally written this means, we have to get the solution in y following two (i = 1, 2) boundary value problems (differential operators do here act

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248 | 12 Solution of 2D elasticity problems with random loads on the variable y, while x is fixed) Δy W(i) (y, x) + α grad div W(i) (y, x) = 0, (i)

x∈ D, (i)

W (y, x)|y→z∈∂D = −Γ (z, x) .

(12.14)

Hence, we can write down an unbiased adjoint estimator.

Adjoint estimator

where

& ' f (y ˜) ˜ , z˜) = Γ(x − y ˜ ) + B(y, z˜)V (z˜ , x) , η(x, y ˜) π (y ⎧ ⎨ − Γ (z ˜ − x) ,

V (z˜ , x) = ⎩

W (z˜ , x) ,

if z˜ ∈ Γ i , if z˜ ∈ 𝛾i

(i = 1, 2) .

(12.15)

(12.16)

From this we arrive at the adjoint algorithm.

Adjoint algorithm 0. Put the initial score as zero: Ξ∗ (x) = 0. ˜ in our domain D ac­ 1. The same as in the Direct algorithm, sample a random point y cording to a density π(y) and suppose the sampled point lies in the disk K (O i , R i ) (i = 1, 2). ˜ , one samples on the circle S(O i , R i ) a random point z˜ according 2. For this value of y ˜ ). to the density p R i (z; y 3. If the sampled point ˜z belongs to the external boundary, i.e. if z˜ ∈ Γ i , then calcu­ late & ' f (y ˜) ˜ , ˜z) = Γ(x − y ˜ ) − B(y ˜ , ˜z)Γ(z˜ − x) η(x, y . (12.17) ˜) π (y If z˜ ∈ 𝛾i , then Γ(z˜ − x) should be replaced in (12.17) with −W (z˜ , x) precalculated as described above. Here too, when applying for this purpose the Random Walk method, we can replace W (z˜ , x) by the relevant random estimator along one path of the Random Walk. ˜ , z˜)/N . 4. Ξ∗ (x) := Ξ∗ (x) + η(x, y 5. Repeating the steps 1–4 N times, N sufficiently large, we get the approximate so­ lution as u(x) ≈ Ξ∗ (x). The main difference between the direct and adjoint algorithms can be explained as follows. In the direct algorithm, the solution at a point x is obtained as an average over N solutions of N boundary value problems with N random boundary functions ˜ sampled in D, evaluated at the fixed point x. ˜ ) generated by N random points y Γ(z˜ − y In the adjoint algorithm, a solution of only one boundary value problem with bound­ ˜ sampled in D with the ary function Γ(z˜ − x) is calculated, but for N random points y density π(y) and then one takes the spatial average over these N points.

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Note that if we consider the Green tensor G(x, y) = Γ(x, y) + W (x, y) (as well as ˜ , ·)) as a random field generated by the random boundary the direct estimator ξ (x, y function −Γ(z − y) (in turn, generated by the distribution π(y)) then this field obeys an ergodic property in the sense that the average of G(x, y)f (y)/π(y) over the ensem­ ˜ ) evaluated at a fixed point x equals the average of ble of boundary functions Γ(z˜ − y G(y, x)f (y)/π(y) taken over spatial points y distributed in D with the density π(y). Remark 12.1. The Adjoint algorithm is obviously efficient if the number of points x where the solution is calculated is not large, say, it is desired to calculate the solu­ tion in several fixed points. This is the case, when one calculates correlation functions along a chosen direction, for instance as in our case, the longitudinal and transverse correlation functions. The Direct algorithm is more efficient if the number of points where the solution is constructed is large while the volume forces f are concentrated in a small subregion of D, or they are presented as a set of point sources. This is a general heuristic comparative characteristics of the two methods. To com­ pare their efficiencies, one also needs to compare the costs of calculations of the direct and adjoint estimators and their variances. Indeed, assume the variance of the adjoint estimator (12.15) is much larger than that of (12.11) due to larger dispersion of the so­ lution in the domain compared to the variance of the boundary functions. This may discard the advantages of the adjoint method.

12.3 Random loads The random loads f (x) are considered as an isotropic vector Gaussian random field with a given spectral tensor S ij (k) defined as a Fourier transform of the correlation tensor B ij (r)  1 S ij (k) = B ij (r)e−i(r·k) dr , i = 1, 2 , (12.18) (2π )2 R2

hence

G H  B ij (r) = f i (x)f j (x + r) = S ij (k)e i(r·k) dk .

(12.19)

R2

The spectral tensor of the isotropic random field f (x) has the following general struc­ ture [119]: S ij (k) = [S LL (k ) − S NN (k )]

ki kj + S NN (k )δ ij , k2

k = |k| .

(12.20)

Here S LL and S NN are longitudinal and transversal spectra, respectively. We consider incompressible random field f (x), for which div f (x) = 0. In this case, S LL (k ) = 0 and

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250 | 12 Solution of 2D elasticity problems with random loads S ij (k) = S NN (k )P(k), where P(k) = (δ ij − as the usual Kronecker delta symbol. Then

ki kj k2 )

is a projection tensor with δ ij defined

S NN (k ) = E(k )/(2πk ) , where E(k ) is an energy spectrum. We choose √ 2 2 L − Lk 2 , E(k) = σ √ e π

(12.21)

k = |k| ,

here L is a positive parameter characterizing the correlation length. For modeling the random field f (x), the Randomization spectral method for Gaus­ sian fields with the given energy spectrum E(k ) can be applied. We use the following simulation formulae for the incompressible random vector field [160]: m & ' σ  f (x) ≈ fm (x) = √ P(k l ) ξ l cos(k l · x) + η l sin(k l · x) , m l =1

(12.22)

∞ where σ 2 = 0 E(k )dk, k l = k l ω l , and k l are independent random wave numbers are sampled according to the density p = E(k )/σ 2 and ω l , l = 1, . . . , m is a family of mutually independent random vectors distributed uniformly on the unit sphere in 2D; ξ l , η l l = 1, . . . , m, are standard Gaussian random vectors mutually independent and independent of k l . To test the code written according to the simulation procedure described above we calculated the correlation tensor B ij . Since we deal with the isotropic case, the correlation tensor is defined uniquely by two scalar functions, namely B LL and B NN , the longitudinal and transverse correlation functions, respectively:

B ij (r) = [B LL (r) − B NN (r)]

ri rj + B NN (r)δ ij , r2

r = |r| .

(12.23)

So it is enough to calculate the correlation tensor as a function of r1 for r2 = 0. Due to (12.23), along the axes r1 the vectors f1 and f2 are uncorrelated, so we calculate the functions B ii (r1 ) =  f i (0)f i (r1 ), i = 1, 2. These functions can be evaluated explicitly. Indeed, from (12.19), (12.20), and (12.21) we obtain using some standard table in­ tegrals from [63] that  S ij (k)e i(r·k) dk B ij (r1 ) = R R



√ ∞ 2π sin2 (φ) − cos(φ) sin(φ) σ2 L − Lk 2 ikr 1 cos( φ ) = √ e dk e dφ − cos(φ) sin(φ) cos2 (φ) π π 0 0

0 σ 2 −r21 /8L I0 (r21 /8L) + I1 (r21 /8L) = (12.24) e 0 I0 (r21 /8L) − I1 (r21 /8L) , 2

where I0 and I1 are the modified Bessel functions of the first kind.

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12.4 Random Walk methods and Double Randomization |

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Note that (12.24) does not imply that the correlation tensor B ij is diagonal; it is diagonal only along the chosen direction x = r1 . From (12.23) and (12.24), B11 = B LL , B22 = B NN , so we will also call B11 and B22 longitudinal and transverse correlation functions, respectively. Note that B LL (0) = B NN (0) = |f |2 /2 in accordance with the theory of isotropic random fields [221].

12.4 Random Walk methods and Double Randomization 12.4.1 General description Assume that we have to solve a PDE that includes a random field σ, say in the right-­ hand side, in coefficients, or in the boundary conditions: Lu (x, σ ) = f (x, σ ) ,

u |Γ = u 𝛾 .

To solve this problem directly by constructing the ensemble of solutions via con­ ventional numerical methods like finite elements or finite difference schemes is a hard task, which is not realistic for most practical problems. If however one of the Ran­ dom Walk Methods can be applied, then a technique we call a Double Randomization Method is very useful. Let us describe it shortly. Suppose we have constructed a stochastic method for solving this problem, for a fixed sample of σ. This implies, e.g. that an unbiased random estimator ξ (x|σ ) is defined so that for a fixed σ, u (x, σ ) =  ξ (x|σ ) , where · stands for averaging over the random trajectories of the stochastic method (e.g. a diffusion process, a Random Walk on Spheres, or a Random Walk on Boundary, e.g. see [160]). Let us denote by E σ the average over the distribution of σ. The double randomization method is based on the equality E σ u (x, σ ) = E σ  ξ (x|σ ) . The algorithm for evaluation of E σ u (x, σ ) then reads 1. Choose a sample of the random field σ. 2. Construct a random walk along which the random estimator ξ (x|σ ) is calculated. 3. Repeat steps 1 and 2 N times and take the arithmetic mean. Suppose one needs to evaluate the covariance of the solution. Let us denote the ran­ dom trajectory by ω. It is not difficult to show [160] that  u (x, σ )u (y, σ ) = E(ω1 ,ω2 ,σ )[ξ ω1 (x|σ )ξ ω2 (y|σ )] .

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252 | 12 Solution of 2D elasticity problems with random loads The algorithm for calculation of  u (x, σ )u (y, σ ) follows from this relation: 1. Choose a sample of the random field σ. 2. Having fixed this sample, construct two conditionally independent trajectories ω1 and ω2 , starting at x and y, respectively, and evaluate ξ ω1 (x|σ )ξ ω2 (y|σ ). 3. Repeat steps 1 and 2 N times and take the arithmetic mean.

12.4.2 Green-tensor integral representation for the correlations Note that for the correlation function we can derive a closed equation. Indeed, assume that we have a linear scalar PDE with random right-hand side and zero boundary val­ ues Lu = f , x ∈ D , u |Γ = 0 , where the random field f (not necessarily homogeneous) has B f (x, y) as its correlation function. The solution u can be represented through the Green formula  u (x) = G(x, y)f (y)dy , D

where G(x, y) is the Green function for the domain D. Under certain smoothness conditions, we can prove that the correlation function of the solution B u (x, y) =  u (x)u (y) and the input correlation function B f (x, y) =  f (x)f (y) are related by the iterated equation Lx Ly B u (x, y) = B f (x, y)

(12.25)

with boundary conditions B u |y∈Γ = 0, Ly B(x, y)|x∈Γ = 0. Here Lx implies that the operator L acts with respect to the variable x, for fixed y. This can be derived as follows. First, starting from the definition B u (x, y) =  u (x)u (y), we use the above Green formula to evaluate this product for the points x and y and change the product of integrals by the double integrals over the domain D; then we take the expectation under the double integral sign. This leads us to the representation  G(x, y )G(y, y )B f (y , y ) dy dy . (12.26) B u (x, y) =  u (x)u (y) = D D

It now remains to notice that the same expression is obtained by applying the Green formula successively to the iterated equation (12.25).

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12.4 Random Walk methods and Double Randomization | 253

These arguments work only in the case of scalar equations, e.g. in the case of Laplace operator L = Δ, the boundary value problem (12.25) is Δx Δy B u (x, y) = B f (x, y) with boundary conditions B u |y∈Γ = 0, Δy B(x, y)|x∈Γ = 0. For systems of PDEs the relevant expressions are more complicated. Let us con­ sider our system of Lamé equations (12.1). In the notations used in (12.1)–(12.4) we consider the correlation tensor of the solution B(u) (x, y) = u(x)uT (y) , and the correlation tensor of the body forces B(f ) (x, y) = f (x)f T (y). Let L = Δ + α grad div be the Lamé operator. In accordance with (12.14), the Lamé operator L acts on a matrix W column-wise. This means, the matrix equation LW = B (B is a ma­ trix) is a pair of Lamé equations written for the relevant first and second columns of matrices W and B. Now we are in a position to present the generalization of (12.26) to the system of Lamé equations. Analogously to the derivation of (12.26), after direct evaluations we arrive at  (u) B (x, y) = G(x, y )B(f ) (y , y )GT (y, y )dy dy . (12.27) D D

It is also possible to write down a differential relation between the input matrix B(f ) (y , y ) and the correlation matrix of the solution B(u) (x, y). Indeed, introduce a tensor V (x, y) and write the following system of coupled systems: L x B(u) (x, y) = V T (x, y) ,

B(u) (x, y)|x∈Γ = 0 ,

L y V (x, y) = [B(f ) (x, y)]T ,

V (x, y)|y∈Γ = 0 .

(12.28)

To prove that (12.27) solves the system (12.28), it is enough to notice that repre­ sentation (12.27) can be obtained by a successive application of the Green formula representation of the solutions to (12.28). The system of equations (12.28) can be written as one system of fourth order. In­ ˆ y to both sides of (12.28). ˆ = LV T , we apply the operator L deed, using the definition LV This yields ˆ y L x B(u) (x, y) = [B(f ) (x, y)]T L with boundary conditions B(u) (x, y)|x∈Γ = 0 ,

L x B(u) (x, y)|y∈Γ = 0 .

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254 | 12 Solution of 2D elasticity problems with random loads

12.5 Simulation results In this section, we present the results of numerical experiments and start with testing the used simulation procedure based on the Randomized spectral method.

12.5.1 Testing the simulation procedure for random loads Now we can compare the results obtained by direct Monte Carlo calculations based on simulation formula (12.22) with the explicit functions (12.24). In Figure 12.1, we present the transverse and longitudinal correlations obtained by using 105 samples (circles) compared with the exact values evaluated by (12.24) (solid lines). It is seen that the accuracy is very high so that the calculated and exact results are practically coinci­ dent in these figures. Thus the simulation formula (12.22) with the chosen number of harmonics is accurate enough to use it in our numerical calculations.

12.5.2 Testing the Random Walk algorithm for nonzero body forces To test the Random Walk algorithm described above in Section 12.3, we solved the inhomogeneous problem (12.1) for two overlapped disks of unit radii, with the body force function with components f1 (x1 , x2 ) = 16(x21 + x22 − x1 ) − 4 + α (12x21 + 4x22 − 12x1 − 2) , f2 (x1 , x2 ) = 4αx2 (2x1 − 1) , the exact solution being u 1 = (x21 + x22 )((x1 − 1)2 + x22 − 1), u 2 = 0. 1.2

1.2 σ=1, L=0.2

1 0.8

0.8 Longitudinal corr. B 11(x)

0.6

0.6

0.4

0.4

0.2

0.2

0 1

2

3

4

5 x

Longitudinal corr. B 11(x)

0

Transverse corr. B22(x) −0.2 0

σ=1, L=1

1

6

7

8

9

10

Transverse corr. B22(x) 0

5

10 x

15

20

Figure 12.1. Transverse and longitudinal correlations: L = 0.2 (left panel) and L = 1 (right panel). In both cases, σ = 1, the number of harmonics m = 100 and N = 105 samples of random loads.

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12.5 Simulation results | 255

0.6 0.4 0.2 0 −0.2 u1

−0.4 −0.6 −0.8 −1 −1.2 −1

−0.5

0

0.5 x

1

1.5

2

Figure 12.2. Comparison of an approximate solution obtained by the Random Walk algorithm (dots) with the exact result (solid line); α = 2, the number of points sampled in the domain: N = 100 000. Maximal error less than 2.5%.

In Figure 12.2, we compare the function u 1 obtained by the Random Walk algo­ rithm against the exact results. The errors are less than 2.5%.

12.5.3 Calculation of correlations for the displacement vector The calculations were carried out for the two equal overlapped disks of radius R = 1; the first disk is centered at zero (0, 0) and the second disk is centered at the point (1, 0). Thus the longitudinal coordinate x = r1 varies from x = −1 to x = 2. In the nu­ merical experiments we calculated the transverse and longitudinal correlations of the displacement vector between the point x0 = −0.5 and the current point x varying be­ tween −0.5 and 1.9. The elastic constant μ was set to 1, so the calculations were made for different values of α = λ + 1. The random loads f1 and f2 (having zero mean values) were simulated by (12.22), with the intensity of fluctuations σ = 1 and different corre­ lation length L. Note that L = 1 implies that the loads are strongly correlated since the radii of the disks are equal to 1, while L = 0.001 corresponds to an approximation of Gaussian white noise. The cases L = 0.1–0.5 are intermediate. In all calculations, we used the Adjoint algorithm. In Figure 12.3, we present the transverse and longitudinal correlations for the dis­ placement vector, normalized by the variance at the point x0 = −0.5: (u)

B11 (x) =  u 1 (x0 )u 1 (x)/ u 21 (x0 )

(12.29)

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256 | 12 Solution of 2D elasticity problems with random loads 1.8

1.8

Longitudinal corr. B11(u)(x)

1.6 1.4

1.4

1.2

1.2

1

σ=1, L=0.1

0.4

σ=1, L=1

1

0.8 0.6

Longitudinal corr. B11(u)(x)

1.6

0.8 (u) Transverse corr. B22 (x)

0.6

0.2

0.4

0

0.2

−0.2 −0.5

0

0.5

1

1.5

2

0 −0.5

(u) Transverse corr. B22 (x)

0

x

0.5

1

1.5

2

x

Figure 12.3. Transverse and longitudinal correlations: L = 0.1 (left panel) and L = 1 (right panel). In both cases, σ = 1, the number of harmonics m = 100 and N = 105 samples of random loads.

for the longitudinal correlation and (u)

B22 (x) =  u 2 (x0 )u 2 (x)/ u 22 (x0 )

(12.30)

for the transverse correlations. It is seen from Figure 12.3 (right panel) that in the case when the correlation length is L = 1, the longitudinal correlation function has one maximum at a position about x = 0.55 and the transverse correlation function is positive. For L = 0.1 (left panel) this function becomes negative after x = 1.4, while the maximum of the longitudinal correlation function is attained at x ≈ 0.4. Note that for larger value of L (equal to 0.5) the negative part disappears (see Figure 12.4, left panel), while for small values of L (equal to 0.001) the negative part begins already after x = 0.2 (right panel). It should be noted that in the latter case we have taken α = 10 which also makes a con­ siderable contribution to the increasing of the negative part of the transverse correla­ tions. Indeed, for comparison, we present in Figure 12.5 the case of small correlation (L = 0.001) for α = 2 (left panel) and α = 100 (right panel) which clearly shows when compared to the figures in the right panel of Figure 12.4 that the larger the value of α, the larger the negative part of the transverse correlations.

Comparison of the direct Monte Carlo with a method based on representation (12.27) To test our method, we used to obtain the results presented in Figures 12.3–12.5 we compared it with the method based on representation (12.27) and described above in Section 12.4. In Figure 12.6, we compare the longitudinal and transverse correlations obtained by these two methods: the solid lines are the results obtained by the direct Monte Carlo algorithm and dots are the results obtained by the method of Section 12.4. The difference between the results is less than 1% .

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1.4

1.8 Longitudinal corr. B11(u)(x)

1.6

1.2 1

1.4

Longitudinal corr. B11(u)(x)

0.8

1.2 1

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0.6 0.4

(u) Transverse corr. B22 (x)

0.8

(u) Transverse corr. B22 (x)

0.2

0.6

0

0.4

σ=1, L=0.5; α=2

0.2 0 −0.5

0

0.5

σ=1, L=0.001; α=10

−0.2

1

1.5

2

−0.4 −0.6 −0.5

0

0.5

x

1

1.5

2

x

Figure 12.4. Transverse and longitudinal correlations: L = 0.5 (left panel) and L = 0.001 (right panel). In both cases, σ = 1, the number of harmonics m = 100 and N = 105 samples of random loads. 1.6

1.2

1.4

1

(u) 11

Longitudinal corr. B (x)

1.2

0.8

1

0.6

0.8

0.4 σ=1, L=0.001; α=2

0.6

σ=1, L=0.001; α=100

0.2

0.4 0.2

Longitudinal corr. B11(u)(x)

0 −0.2

(u) 22

Transverse corr. B (x)

0 −0.2 −0.5

(u) Transverse corr. B22 (x)

−0.4 0

0.5

1 x

1.5

2

−0.6 −0.5

0

0.5

1

1.5

2

x

Figure 12.5. The same as in Figure 12.4, but for L = 0.001 (left panel) and α = 100 (right panel).

We also applied these two methods for calculation of the mean displacements under the same random loads f1 and f2 but having nonzero mean values: the curves of mean displacements presented in the left panel of Figure 12.7 correspond to the mean loads  f1  = 1,  f2  = 1, while the curves in the right panel were obtained for  f1  = 1,  f2  = 0. Note that in the latter case, the mean value of the second component is zero along the axis x1 . The results are in a good agreement and the relatively large difference in the transverse curves (right panel) can be decreased by increasing the number of trials. The presented correlation functions can be informative enough to solve an in­ verse problem of finding the elastic constant α assuming the relevant correlations are known. In Figure 12.8, we show the longitudinal and transverse correlation functions for two values of α: α = 1 and α = 2 (left panel) and α = 10 and α = 11 (right panel). It

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258 | 12 Solution of 2D elasticity problems with random loads 1.8 Longitudinal corr. B (u) 11 (x)

1.6 1.4 1.2 (u) Transverse corr. B22 (x)

1 0.8 0.6

σ=1, L=1; α=2

0.4 0.2 0 −0.5

0

0.5

1

1.5

2

x

Figure 12.6. Comparison of direct Monte Carlo calculations (solid lines) with the results obtained by a numerical solution of the equation governing the correlation functions (dots). The errors in calculations were everywhere less than 1%. 0

0

−0.05

−0.05

−0.1

Mean displacements, second component

−0.1

−0.15

−0.15

−0.2

−0.2

Mean first component

First component

−0.25 −0.5

0

0.5

1 x

1.5

2

−0.25 −0.5

0

0.5

1

1.5

2

x

Figure 12.7. The vector of mean displacements. Left panel: dots – solution of the deterministic equa­ tion for the mean displacements under the mean loads f1 = 1, f2 = 1; solid lines – the mean dis­ placements obtained by numerical simulation, for random loads with the same mean loads and Gaussian fluctuations with L = 1, σ = 1. Right panel: the same as in the left panel, but for the mean loads f1 = 1, f2 = 0; in this case the second component (u 2 ) is zero along the axis x 1 (not shown).

is seen from the curves of the left panel that the longitudinal correlation function is as almost unchanged, while the transverse correlation function is informative enough to see a clear difference between the cases α = 1 and α = 2. For larger values of α, the longitudinal correlation function is more informative (see the right panel) but the difference between α = 10 and α = 11 is smaller. Note that this sensitivity analysis is made for the case of “white noise,” i.e. when L = 0.001. We have repeated these calculations changing the value of L as L = 1, see the results in Figure 12.9. It is seen

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12.5 Simulation results

1.6

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1.4

1.4

α=10

1.2

Longitudinal corr. B (u) 11 (x)

Longitudinal corr. B (u) 11 (x)

1

1.2

α=11

0.8

1 α=1

0.8

0.6 0.4

0.6

α=2

0.2

0.4

0

0.2 0

i

Transverse corr. B (x)

−0.4

σ=1, L=0.001

−0.2 −0.5

0

(u) Transverse corr. B22 (x)

−0.2

(u) 22

0.5

1

1.5

2

σ=1, L=0.001

−0.6 −0.5

0

0.5

x

1

1.5

2

x

Figure 12.8. Sensitivity to the elasticity constant α: the correlation functions for α = 1 (solid line) and α = 2 (dots) are shown in the left panel and the case α = 10 and α = 11 is presented in the right panel. 2

1.8 Longitudinal corr. B11(u)(x)

1.6

Longitudinal corr. B11(u)(x)

α=1

1.5

1.4

α=11

1.2

α=2

1

1 0.8

0.5

α=10

0.6 (u) 22

(u) 22

Transverse corr. B (x)

0.4 0.2

Transverse corr. B (x)

0 σ=1, L=1

σ=1, L=1

0 −0.5

0

0.5

1 x

1.5

2

−0.5 −0.5

0

α=11

0.5

1

1.5

2

x

Figure 12.9. Sensitivity to the elasticity constant α: the same as in Figure 12.8, but for L = 1, the correlation functions for α = 1 (solid line) and α = 2 (dots) are shown in the left panel and the case α = 10 and α = 11 is presented in the right panel.

that this leads to more expressed difference of curves for different values of α with the exception that the longitudinal correlation functions for α = 10 and α = 11 almost coincide.

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13 Boundary value problems for some elliptic PDEs with random boundary conditions 13.1 Introduction Estimating the random perturbations caused by different stochastically fluctuating parameters in the solutions of PDEs is of high interest in many fields of science and technology. The random excitations can be considered both as a natural source of stochastic fluctuations and as a model to describe extremely complicated irregulari­ ties and uncertainties (e.g. see [220; 203; 150]). Classical examples are the Navier–S­ tokes equation with a stochastic forcing, the Darcy equation with a random hydraulic conductivity (e.g. see [119; 53; 34; 59]), localization for random perturbations of Schrödinger equation [85], porous and composite material studies [145], etc. In electri­ cal impedance tomography [79] the important problem is to evaluate a global response to random boundary excitations and to estimate local fluctuations of the solution fields. Similar analysis is made in the inverse problems of elastography [130; 18; 180], recognition technology [60], acoustic scattering from rough surfaces [219], and reac­ tion–diffusion equations with white noise boundary perturbations [203]. Homogeneous Gaussian random fields are efficiently simulated by conventional and randomized spectral models, they are easy in implementation, as can be seen from the examples given in the previous chapter (e.g. see also [196; 113; 160; 92]). The main method in inhomogeneous random fields simulation is the Karhunen–Loève (K-L) expansion (e.g. see [221; 208; 106; 136]). It is computational demanding because it requires us to solve numerically eigenvalue problems of high dimension. However, in some practically interesting applications, models with the analytically solvable eigen­ value problem for the correlation operator can be constructed. This gives then a very efficient numerical method because the K-L expansions are fast convergent. The solu­ tions of such problems are partially homogeneous random fields and we construct K-L expansions that are used to calculate different statistical characteristics of the solu­ tion. Further extensions of these results were given for the viscous flows [156], elastic half-plane [173; 184], elastic half-space [193], and the fractional Laplacian equation governing the density correlations in the Universe [162]. In this chapter, we deal with elliptic boundary value problems with random boundary conditions. We follow here mainly the study [179] where we have given explicit solutions of a series of boundary value problems for elliptic equations with random boundary conditions. Solutions to these problems are inhomogeneous ran­ dom fields that can be represented as series expansions involving a complete set of deterministic functions with corresponding random coefficients. We construct the K-L series expansion which is based on the eigen-decomposition of the covariance operator. It can be applied to simulate both homogeneous and inhomogeneous ran­

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dom fields. We study the correlation structure of solutions to some classical elliptic equations in response to random excitations of functions prescribed on the boundary. We analyze the stochastic solutions for Dirichlet and Neumann boundary conditions to the Laplace equation, biharmonic equation, and to the Lamé system of elasticity equations. Explicit formulae for the correlation tensors of the generalized solutions are obtained when the boundary function is a white noise, or a homogeneous random field on a circle, a sphere, and a half-space. These exact results may serve as an ex­ cellent benchmark for developing numerical methods, e.g. Monte Carlo simulations, stochastic volume, and boundary element methods. More details about the subject can be found in [185; 194]. First let us mention some problems of the classical potential theory dealing with random boundary conditions and sources [35] where the Monte Carlo methods are very efficient (e.g. see [179; 154; 172]), [174]). In electrical impedance tomography [79] the important problem is to evaluate a global response to random boundary excita­ tions and estimate local fluctuations of the solution fields. Similar analysis is made in the inverse problems of elastography [130; 180], recognition technology [60], acoustic scattering from rough surfaces [219], fluid dynamics [7], and reaction–diffusion equa­ tions with white noise boundary perturbations [203]. In this chapter, we construct exact proper orthogonal decomposition for some classical boundary value problems, for a disk, ball, and a half-plane, with a Dirichlet and Neumann boundary conditions, where the boundary functions are white noise or homogeneous (2π-periodic) random processes. In case the boundary function is a white noise, the solutions are treated as generalized random fields with the conver­ gence in the proper spaces and relevant generalized treatment of boundary condi­ tions, e.g. see [149; 150; 195]. The chapter is organized as follows. After a short description of the spectral and K-L expansions, we consider in Section 13.2 the 2D Laplace equation, with Dirichlet and Neumann boundary conditions, for a disk and a half-plane. Generalizations to a three-dimensional case are given in Section 13.3. In Section 13.4, we analyze the bihar­ monic equation for a disk. The plane elasticity problem is presented in Section 13.5. For all these boundary value problems, we explicitly find the correlation functions and give the K-L expansion of the relevant random fields.

13.1.1 Spectral representations Let us first consider a real-valued zero mean homogeneous Gaussian one-dimensional vector random field u(x) = (u 1 (x), . . . , u l (x))T , x ∈ Rd with a given covariance ten­ sor B(r) with entries B ij (r) =  u i (x + r)u j (x) ,

i, j = 1, . . . l ,

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262 | 13 Boundary value problems with random boundary conditions or with the corresponding spectral tensor F:   F ij (k) = e−i2πk·r B ij (r)dr , B ij (r) = ei2πr·k F ij (k)dk . Rd

(13.1)

Rd

We also call B ij a correlation tensor that is equivalent since we assume without loss of generality that the random fields have zero means.  # Often it is reasonable to assume [98] that the condition Rd lj=1 |B jj (r)|dr < ∞ is satisfied which ensures that the spectral functions F ij are uniformly continuous with respect to k. Note that a weaker assumption that B is squared integrable guarantees only the existence of the spectral tensor in the space L2 . ¯ (k ) . Let Q(k) be an l × n-matrix defined by Q(k)Q∗ (k) = F (k), Q(−k) = Q Here the star stands for the complex conjugate transpose which is equivalent to taking two operations, the transpose T and the complex conjugation of each entry. Then the spectral representation of the random field is written as follows (e.g. see [221])  u(x) = ei 2π kx Q(k)Z(dk) , (13.2) Rd

where the column vector Z = (Z1 , . . . Z n )T is a complex-valued homogeneous n-di­ mensional white noise on Rd with a unit variance and zero mean: Z(dk) = 0,

¯ j (dk2 ) = δ ij δ(k1 − k2 )dk1 dk2 ,  Z i (dk1 )Z

¯ (dk) . Z(−dk) = Z

Note that in the literature, different forms of the Fourier transform between the correlation and spectral tensors are used. Along with (13.1), we will mainly use   1 −ik·r e B ( r ) dr , B ( r ) = eir·k F ij (k)dk , i, j = 1, . . . l . F ij (k) = ij ij (2π )d Rd

Rd

The spectral representation (13.2) is used in different numerical simulation meth­ ods, through a deterministic or randomized evaluation of the stochastic integral in (13.2), see for instance [179; 92; 98]. A straightforward evaluation of the stochastic integral (13.2) is based on the Rie­ mann sums calculation with fixed cells (see, e.g. [196]). The integral is approximated by a finite sum u(x) ≈

n  & ' cos(2πk i · x)ξ i + sin(2πk i · x)η i , i =1

where k i are deterministic nodes in the Fourier space, and ξ i and η i are Gaussian ran­ dom vectors with zero mean and relevant covariance. The efficient calculation of the above sum is usually carried out by the fast Fourier transform which assumes that the nodes are chosen uniformly. It should be mentioned that this scheme suffers from an artificially periodicity in the scale of 1/Δk where Δk is the integration step in the Fourier space. In Randomized models, the nodes are chosen at random, with an appro­

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13.1 Introduction |

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priate probability distribution so that the model has the desired correlation structure (see, e.g. [179; 92]). Partially homogeneous random fields present an important class of random fields where this approach can be efficiently used. Let x = (y, z), y ∈ Rn , z ∈ Rm , and let V(x) = (v1 (x), . . . , v l (x))T . Assume that the random field V(y, z) is homogeneous with respect to the variable y, i.e. V(y1 , z1 )V∗ (y2 , z2 ) = B(y1 − y2 , z1 , z2 ) .

Random fields with this property are called partially homogeneous random fields [179]. The partial spectral tensor is defined by  1 f (λ, z1 , z2 ) = B(ρ, z1 , z2 ) exp {−i(λ, ρ)} dρ . (2π )n Rn

It is not difficult to verify that for a general complex-valued random field V(x), which is partially homogeneous, V(y, z) =

1 exp {i(λ, y)}ξ λ (z) [p(λ)]1/2

its correlation tensor is equal to B(ρ, z1 , z2 ), if λ is distributed according to a proba­ bility density p(λ ) which can be chosen quite arbitrarily, and ξ λ (λ fixed) is a homo­ geneous one-dimensional complex-valued random field with the correlation tensor f (λ, z1 , z2 ). A rigorous proof of this statement is given in [179].

13.1.2 Karhunen–Loève expansion Let us now consider a real-valued inhomogeneous random field u (x), x ∈ G defined on a probability space (Ω, A, P) and indexed on a bounded domain G. The case of unbounded domains can also be treated, in particular, if the covariance tensor be­ longs to a class A defined in [23], for which the corresponding covariance operator is compact and trace class. This important generalization is based on the result due to Novitsky [127] (see also [24]). In the fourth subsection of Section 13.2 we deal with an unbounded domain when analyzing the Dirichlet problem for the half-plane. To simplify the notations, we will not use here and in what follows the boldface charac­ ters to denote the vectors if not otherwise indicated. They will be essentially used in Section 13.5 for the vector solution to the Lamé equation. Assume (without loss of generality) that the field has a zero mean and a variance Eu 2 (x) that is bounded for all x ∈ G. The K-L expansion has the form [221] ∞ 9  λ k ξ k h k (x) , u(x) = k =1

where λ k and h k (x) are the eigenvalues and eigenfunctions of the covariance function B(x1 , x2 ) =  u (x1 )u (x2 ), and ξ k is a family of random variables.

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264 | 13 Boundary value problems with random boundary conditions By definition, B(x1 , x2 ) is bounded, symmetric, and positive definite. For such kernels, the Hilbert–Schmidt theory says that the following spectral representation is valid: ∞  B(x1 , x2 ) = λ k h k (x1 )h k (x2 ) , k =1

where the eigenvalues and eigenfunctions are the solutions of the following eigen­ value problem for the correlation operator:  B(x1 , x2 )h k (x1 )dx1 = λ k h k (x2 ) . G

 The eigenfunctions form a complete orthogonal set G h i (x)h j (x)dx = δ ij where δ ij is the Kronecker delta-function. The family {ξ k } is a set of uncorrelated random variables which are obviously related to h k by  1  u (x)h k (x)dx , Eξ k = 0, Eξ i ξ j = δ ij . ξk = λk G

We also mention that the assumptions of the Hilbert–Schmidt can be weakened as it is done in Mercer’s theorem. This will be discussed in Section 13.2. It is well known that the K-L expansion presents an optimal (in the mean square sense) convergence for any distribution of u (x). If u (x) is a zero mean Gaussian ran­ dom field, then {ξ k } is a family of standard Gaussian random variables. Some gener­ alizations to non-Gaussian random fields are reported in [136]. Now consider a case when G is unbounded, e.g. a homogeneous random process u (x) is defined on the whole real line R. The eigenvalue problem reads  B(x2 − x1 )h k (x1 )dx1 = λ k h k (x2 ) , −∞ < x2 < ∞ . (13.3) R

Note that we can take h(x) = eiωx ; then from (13.3) we get ∞

λ=

B(x2 − x1 )e−iω(x2−x1 ) dx1 ≡ S(ω) .

−∞

To make further considerations more rigorous, we assume that G is large but finite and u is periodic (see, e.g. [106; 208]). Then, we may develop B(x2 − x1 ) in a Fourier series,  B(x − x ) = λ k ei2πk(x−x ) . (13.4) k

The eigenvalue problem can then be solved via the unique representation  B(x − x ) = λ k ei2πkx e−i2πkx ,

(13.5)

k

which implies that ei2πkx are the eigenfunctions with eigenvalues λ k = S(ω k ). And conversely, if the eigenfunctions are Fourier modes we can write equality (13.5) which leads to (13.4).

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13.2 Stochastic boundary value problems for the 2D Laplace equation |

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Thus the correlation function B depends on the difference x − x if and only if the eigenfunctions of the correlation operator are Fourier modes. In our considerations this fact will be used in two-dimensional regions, when G is a disk, a ball, or a half-plane. The correlation function of a zero mean random process has the form B(x, x ) = B(x, y; x y ). Suppose that our random process is homoge­ neous with respect to one coordinate, say, B = B(x − x ; y, y ). Then we can perform the above procedure over the x-direction and get a 1D eigenvalue problem for every Fourier wavenumber. It means we then work with the partial spectral density. Assume that we deal with a homogeneous real-valued process on the whole line. Then it is possible to cutoff the integration in the eigenvalue problem, i.e. we have to solve the eigenvalue problem a

B(x2 − x1 )h(x1 )dx1 = λ k h k (x2 ) , −a

where a is sufficiently large. Then it is possible to show (see, e.g. [208]) that λ k ≈ S(ω k ) = S(πk /a) ,

h k (x) ≈

1 i(πkx/a) , e 2π

which yields an approximation ˜ a (x1 , x2 ) = B(x1 , x2 ) ≈ B

  ∞  1 πk πk (x2 − x1 ) , S( ) cos a a a k =1

and the K-L expansion approaches in this case to the spectral representation . ∞  ) 1 πk 1/2 ( ξ k cos[πkx/a] + η k sin[πkx/a] . S( ) u (x) ≈ u˜ a (x) = 2a a k =1 The rate of convergence of the K-L expansion is closely related to the smoothness of the correlation kernel and to the ratio between the length a and L, the correlation length of the process. For example, in [106] it is reported that for the particular case B(x1 , x2 ) = σe−| x2−x1 |/L , an upper bound for the relative error in variance ε of the process represented by its K-L expansion is given by ε ≤ π42 n1 aL where n is the number of retained terms.

13.2 Stochastic boundary value problems for the 2D Laplace equation Let us start with the two-dimensional boundary value problems for the Laplace equa­ tion. We are interested in the statistical structure of the solution when the solution (Dirichlet boundary conditions) or the normal derivative (Neumann boundary condi­ tions) is the homogeneous random functions (g(y)) on the boundary. The basic idea

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266 | 13 Boundary value problems with random boundary conditions is first to establish the K-L expansion for the case when the boundary function g is a white noise; therefore, the solutions are considered as generalized random fields. This expansion gives a smooth representation for the solution and the correlation func­ tion inside the open disk, and the case of general homogeneous boundary functions is immediately obtained from this expansion by a simple substitution of the spectral expansion of the boundary random function g(x). Before we start with the details for the Laplace equation, let us outline shortly the general scheme. Assume that we are given a stochastic Dirichlet boundary value problem for a linear elliptic equation in a domain D with a boundary Γ = ∂D: Lu (x) = 0 ,

x ∈ D,

u (x)|x→y∈Γ = g(y) ,

where g(y) is a random field with zero mean and covariance function B g (y1 , y2 ) =  g(y1 )g(y2 ). We are interested in the covariance of the solution, B u (x1 , x2 ) =  u (x1 ) u (x2 ). Suppose that with probability 1, there exists a continuous normal derivative of the Green function on the boundary, ∂G ∂n , so that the solution is represented by the Green formula  ∂G u(x) = (x, y)g(y)dS(y) . ∂n Γ

Using the Green formula representation for the solution in points x1 and x2 we obtain  ∂G ∂G (x1 , y1 ) (x2 , y2 )B g (y1 , y2 ) dS(y1 )dS(y2 ) . (13.6) B u (x1 , x2 ) = ∂n ∂n Γ Γ

If g is a white noise, B g (y1 , y2 ) = δ(y1 − y2 ), and we obtain formally from (13.6) that  ∂G ∂G B u (x1 , x2 ) = (x1 , y) (x2 , y)dS(y) . (13.7) ∂n ∂n Γ

This representation shows that the covariance function B u (x, x2 ) solves the boundary value problem L x B(x, x2 ) = 0 , B(x, x2 )|x→y∈Γ =

∂G (x2 , y)|y∈Γ , ∂n

x, x2 ∈ D , (13.8)

so that the solution of this problem at any point x = x1 ∈ D yields B u (x1 , x2 ) for any fixed x2 ∈ D which defines well the covariance function for any two points x1 and x2 inside the domain D. These formal considerations leave open the singularity problem of the correlation function when both points tend to one point on the boundary, but the weak convergence to the delta-function can be given in the framework of generalized solutions (see, e.g. [149; 150; 195]).

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13.2 Stochastic boundary value problems for the 2D Laplace equation | 267

13.2.1 Dirichlet problem for a 2D disk: white noise excitations Let us consider the Dirichlet boundary value problem for the Laplace equation Δu (x) = 0 ,

x ∈ D,

u (y) = g(y)

y ∈ Γ = ∂D ,

(13.9)

where the domain D is a disk K (x0 , R) centered at O = x0 , bounded by the circle ¯ ( x 0 , R ) = K ( x 0 , R ) ∪ S ( x 0 , R ). Γ = S(x0 , R). We denote the closed disk by K The regular solution to the harmonic equation is represented by the Poisson inte­ gral formula:  g(y)dS y R2 − r2 u(x) = , 2πR |x − y|2 S ( x 0 ,R )

for any point x ∈ K (x0 , R), where r = |x − x0 |. We suppose that the boundary function g(y) is a zero mean Gaussian ran­ dom field, homogeneous or not, defined by its correlation function B g (y1 , y2 ) =  g(y1 )g(y2 ). In case g is homogeneous, it is alternatively defined by its spectral den­ sity function f (k ) related to the correlation function B g (y), y = y2 − y1 , by the Fourier transform   1 f (k) = B g (y)e−i(yk) dy , B g (y) = f (k )ei(yk) dk . 2π When dealing with the homogeneous random processes g (φ) on the circle, we assume throughout the chapter that they are 2π-periodic, so the spectra are discrete, and the Fourier integral transforms become Fourier series. Let us start with the case when the prescribed boundary function g is a Gaussian white noise, B g (y, y ) = δ(y − y ); thus we deal in this chapter with generalized ran­ dom solutions which however are smooth in the open domain (in a disk, ball, and a half-plane). The generalized treatment of the convergence to the boundary functions can be explicitly described (see, e.g. see [149]) in more general cases. We introduce polar coordinates centered at x0 , so that a point x is specified by (r, θ); hence, for two points, x1 = (r1 , θ1 ), x2 = (r2 , θ2 ), and ρ 1 = r1 /R, ρ 2 = r2 /R. It is then convenient to rewrite the Poisson formula as follows: u (r, θ) =

1 − ρ2 2π

2π 0

g(φ)dφ , 1 − 2ρ cos(θ − φ) + ρ 2

(13.10)

where ρ = r/R. Theorem 13.1. The solution of the Dirichlet problem (13.9) in a disk K (x0 , R) with the white noise boundary function g(y) is an inhomogeneous 2D Gaussian random field uniquely defined by its correlation function  u (r1 , θ1 )u (r2 , θ2 ) = B u (ρ 1 , θ1 ; ρ 2 , θ2 ) =

1 1 − ρ 21 ρ 22 , (13.11) 2π 1 − 2ρ 1 ρ 2 cos(θ2 − θ1 ) + ρ 21 ρ 22

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268 | 13 Boundary value problems with random boundary conditions which is harmonic and it depends only on the angular difference θ2 − θ1 and the product of radial coordinates ρ 1 ρ 2 = r1 r2 /R2 . The random field u (r, θ) is thus homogeneous with respect to the angular coordinate θ and its partial discrete spectral density has the form f θ (0) = 1/2π, f θ (k ) = (ρ 1 ρ 2 )k /π, k = 1, . . . . Proof. We start by simple evaluations: B u =  u (r1 , θ1 )u (r2 , θ2 ) (13.12) J I 2π 2π   1 (R2 − r1 2 ) g(φ) dφ 1 (R2 − r2 2 ) g(φ) dφ = · 2π R2 − 2 Rr1 cos(θ1 − φ) + r1 2 2π R2 − 2 Rr2 cos(θ2 − φ) + r2 2 0

1 = (2π )2 =

1 (2π )2

0

2π 2π 0 0

2π 0

E

F ((R2 − r21 )(R2 − r22 ) g(φ )g(φ ) dφ dφ [R2 − 2Rr1 cos(θ1 − φ ) + r21 ][R2 − 2Rr2 cos(θ2 − φ ) + r22 ]

1 − ρ1 2 1 − ρ2 2 · dφ . 2 1 − 2 ρ 1 cos(θ1 − φ) + ρ 1 1 − 2 ρ 2 cos(θ2 − φ) + ρ 2 2

Here we used the property of the white noise  g(φ )g(φ ) = δ(φ − φ ). This integral can be evaluated explicitly and the result is given in (13.11). However, we will obtain it using the Fourier series expansion which not only presents a simple deriva­ tion of (13.11), but also yields the spectrum of our random field, and the K-L expansion. Indeed, we start with the expansion 1 1 − ρ2 · 2π 1 − 2ρ cos(θ − φ) + ρ 2 ∞ 1 1  k = + ρ cos[k (θ − φ)] 2π π k =1

K (ρ; θ − φ) ≡

(13.13)

and proceed (13.12) as follows: ⎫⎧ ⎫ ⎧ 2π ⎨ ∞ ∞ ⎬⎨ 1 ⎬ 1 1  k 1  k Bu = ⎩ + ρ 1 cos[k (θ1 − φ)]⎭ ⎩ + ρ 2 cos[k (θ2 − φ)]⎭ dφ 2π π k =1 2π π k =1 0

 ∞ 1 1  k = + ρ 1 cos[k (θ1 − φ)]K (ρ 2 ; θ2 − φ)dφ 2π π k=1 2π

(13.14)

0

 ∞ 1 1  k + ρ 1 [cos kθ1 cos kφ + sin kθ1 sin φ]K (ρ 2 ; θ2 − φ)dφ 2π π k=1 2π

=

0

∞ 1 1  k k 1 1 − ρ 21 ρ 22 = + ρ 1 ρ 2 cos[k (θ1 − θ2 )] = · . 2π π k=1 2π 1 − 2ρ 1 ρ 2 cos(θ2 − θ1 ) + ρ 21 ρ 22

Here we used the nice property of the integral operator with the kernel K (ρ; θ − φ) that it has the following system of eigenvalues {λ k } and the corresponding orthonor­

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13.2 Stochastic boundary value problems for the 2D Laplace equation | 269

mal eigenfunctions {h k (φ)} complete in L2 (0, 2π): 1 h0 = √ , 2π = π −1/2 cos(kθ) ;

λ2k−1 = λ2k = ρ k ,

λ0 = 1 , h2k−1

h2k = π−1/2 sin(kθ) ,

k = 1, 2, . . . .

(13.15)

This can be verified by a direct substitution of the series expansion (13.13) into the eigenvalue problem 1 2π

2π 0

(1 − ρ 2 )h k (φ)dφ = λ k h k (θ) . 1 − 2ρ cos(θ − φ) + ρ 2

So it remains to prove that our random field u (ρ, θ) has a discrete partial spectral density, f θ (0) = 1/2π, and 1 f θ (k) = 2π

2π

B u (ρ 1 , θ1 ; ρ 2 , θ2 )e−ik(θ 2−θ 1 ) d(θ2 − θ1 ) = (ρ 1 ρ 2 )k /π ,

k = 1, . . . .

0

Actually this can be easily seen from the arguments given in (13.14). A direct proof follows from the Fourier transform property for convolutions. Indeed, representation (13.14) shows that the correlation function B u is written in the form of a convolution, i.e. B u = K (ρ 1 ; ψ) ∗ K (ρ 2 ; ψ − (θ1 − θ2 )) =

1 (2π )2

2π 0

1 − ρ 21 1 − ρ 22 dψ . 2 · (1 − 2ρ 1 cos(ψ) + ρ 1 ) (1 − 2ρ 2 cos(ψ − (θ2 − θ1 )) + ρ 22 )

Now we take the inverse Fourier transform of both parts and use the Fourier trans­ form property for convolutions. This yields f θ (0) = 1/2π ,

f θ (k ) = ρ 1k

ρ 2k /π ,

which is the desired result. Here we used the property [63] 1 2π

2π 0

(1 − ρ 2 ) cos(kx)dx = ρk 1 − 2ρ cos x + ρ 2

while the sin-transform is zero. Finally, the covariance function B u (x1 , x2 ) is harmonic with respect to both of its coordinates which follows from the general representation (13.8). The proof is complete.

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270 | 13 Boundary value problems with random boundary conditions 1.6

0.45 ρ1=0.5, ρ2=0.5

0.4 0.35

Bu(x1,x)

Bu

x1= − 4

1.2

ρ1=0.5, ρ2=0.2

0.3 0.25

x1= − 3 x1= − 2

1

x1= − 1

0.8

0.2

0.6

0.15

0.4

0.1

0.2

0.05

x1= − 4.2

1.4

ρ1=0.9, ρ2=0.5

x1= 0

0 0

10

20 30 40 angular section k, θ=k 2π/50

50

0

10

20 30 radial section k, x=k 2R/50

40

50

Figure 13.1. Laplace equation, Dirichlet boundary conditions: angular correlations B u , for three different values of the ratio ρ i = r i / R (left panel), and radial correlations B( x 1 , x ), for six different values of the starting point x 1 , R = 5 (right panel).

Remark 13.1. The angular behavior of the correlation function shows thus that the random field is partially homogeneous. The radial behavior is also interesting. Let us 2 x1 x2 fix a direction, say the line y = 0, then, B(x1 , x2 ) = 2π1 · RR2 + − x 1 x 2 , where x 1 and x 2 vary between −R and R. This shows that if one of the points, x1 , x2 is in the center of the disk, the covariance is equal to a constant value, 1/2π. For illustration, in Figure 13.1 we show the angular (left panel) and radial (right panel) behavior of the correlation function B u . The angular and radial functions are both plot­ ted versus the section number k, the number of sections being 50, so that θ = k2π/50 (angular behavior, left panel), and x = k2R/50 (radial behavior, right panel). The an­ gular behavior in the left panel is shown for three different choices of the radii ρ 1 and ρ 2 . The radial behavior is given for six different values of the value x1 , the radius of the disk was 5, see the right panel in Figure 13.1. As expected, a low number of eigen­ modes in the K-L expansion is enough to have a good approximation; in Figure 13.2 we compare the K-L approximation against the exact result, taking M = 5 and M = 10 terms (left panel), and M = 2 and M = 5 terms (right panel). Theorem 13.2. The Gaussian random field described in Theorem 13.1 has the following K-L-type expansion: ∞ ' ξ0 1  k& u (r, θ) = √ + √ ρ ξ k cos(kθ) + η k sin(kθ) , π k =1 2π

(13.16)

where {ξ k }, {η k } are sets of mutually independent standard Gaussian random vari­ ables. Proof. The idea of the proof appeals to Mercer’s theorem which states the following (see, e.g. [81]). Let U be a compact set in Rd and let K (s, t) be a symmetric L2 (U )-kernel

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13.2 Stochastic boundary value problems for the 2D Laplace equation | 271

1 0.9

0.45

exact M=10 M=5

0.8

0.35

0.7 ρ 1=0.8, ρ 2=0.9

0.6

ρ 1=0.7, ρ 2=0.7

0.3 B u 0.25

B u 0.5 0.4

0.2

0.3

0.15

0.2

0.1

0.1 0

M=5 M=2 exact

0.4

0.05 0

100

200 300 400 angular section k, x=k 2π/500

500

600

0

100

200 300 400 angular section k, x=k 2π/500

500

600

Figure 13.2. Laplace equation, Dirichlet boundary conditions: angular correlations B u , for two different values of the retained number of terms M, for ρ1 = 0.8, ρ2 = 0.9 (left panel), and ρ1 = 0.7, ρ2 = 0.7 (right panel). The number of angular sections equals 500.

with eigenvalues {λ n } and eigenfunctions h k (t):  K (s, t)h k (t)dt = λ k h k (x) ,

k = 1, 2 . . . .

Theorem 13.3 (Mercer’s theorem). If a nonnull, symmetric L2 (U )-kernel K (s, t) is quasidefinite (i.e. when all but a finite number of eigenvalues are of one sign) and # continuous, then the series ∞ n =0 λ n is convergent, and K (s, t) =

∞ 

λ n h n (s)h¯ n (t) ,

n =0

where h¯ k (t) be the complex conjugate of h k (t) and the series converges absolutely and uniformly in U × U. From this theorem, the K-L expansion can be obtained (see, e.g. [221]): Let v(x) be a real-valued, zero mean, Gaussian random field with the continuous covariance function K (x, y) which has Mercer’s expansion  K (x, y) = λ k h k (x)h k (y) . k

Then, under some regularity conditions, v(x) =

∞ 9 

λ k h k (x)ξ k ,

(13.17)

k =0

in L2 and a.s., where {ξ k }k∈N is a sequence of independent and identically standard normally distributed random variables.

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272 | 13 Boundary value problems with random boundary conditions Note that although our correlation function (13.14) is continuous everywhere in­ side the disk, it increases infinitely as both points approach a point on the boundary, i.e. when θ1 = θ2 , and ρ 1 → 1, ρ 2 → 1. ¯ 0 ), for each However our kernel, the covariance function (13.14), belongs to L2 (K ¯ disk K0 (x0 , ρ 0 ) ⊂ K (x0 , 1), and we find from the expansion (13.14) that  B u dxdy < ∞ , ¯ 0 ( x 0 ,ρ0 ) K

and so the weak convergence as ρ 0 → 1 can be proven. Now we consider the eigenvalue problem for the covariance function B u : 1

2π

dρ 1 0

0

1 (1 − ρ 21 ρ 22 )h k (ρ 1 , θ1 )dθ1 = λ k h k (ρ 2 , θ2 ) . 2π 1 − 2ρ 1 ρ 2 cos(θ2 − θ1 ) + ρ 21 ρ 22

Using expansion (13.14), we find the eigenfunctions and eigenvalues: 1 1 h0 = √ ; λ2k−1 = λ2k = ; 2k + 1 2π   cos(kθ) sin((θ) ; h2k (ρ, φ) = 2k + 1ρ k , h2k−1 (ρ, φ) = 2k + 1ρ k π 1 /2 π 1 /2 k = 1, 2, 3, . . . , λ0 = 1 ,

where the eigenfunctions are orthonormal to one another: 1 2π

h n (ρ, θ)h m (ρ, θ)dρdθ = δ nm . 0 0

Thus the K-L expansion (13.16) follows from the representation u (r, θ) =

∞ 

9 ζ k λ k h k (ρ, θ) ,

k =1

where ζ is a family of standard independent Gaussian random variables. The proof of Theorem 13.2 is complete. The explicit representation of our random field (13.16) is very convenient in practical simulations, as well as in analytical evaluations of different statistical functionals. Note that since our random field is homogeneous with respect to the angular vari­ able, we can also write down the relevant randomized spectral representation when ρ = ρ1 = ρ2 . Indeed, we now let the discrete wave numbers k be randomly distributed with the distribution 1 − ρ 2 2k ρ , k = 1, 2, . . . . pk = ρ2

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Then the random field ' & ξ0 ρ ξ cos(kθ) + η sin(kθ) u (r, θ) = √ + 9 2π π (1 − ρ 2 )

(13.18)

has the desired correlation function (13.11). Here ξ0 , ξ , and η are standard indepen­ dent Gaussian variables. Further, to make the distributions close to Gaussian, in the spectral models one usually takes independent sums of models (13.18) (see, e.g. [179]).

13.2.2 General homogeneous boundary excitations Assume now that a zero mean real-valued Gaussian random process g is defined on the circle by its spectrum f k so that the covariance function reads B g (φ − φ ) =

∞ f0 1  + f k cos k (φ − φ ) . 2π π k=1

Substituting this in (13.12) and using the series expansion of the kernel K (ρ; θ − φ), we arrive at the following series expansion for the covariance function B u : B u (ρ 1 , θ1 ; ρ 2 , θ2 ) =

∞ f0 1  + f k ρ 1k ρ 2k cos k (θ2 − θ1 ) . 2π π k =1

(13.19)

Thus the generalization of the random filed representation (13.17) has the form  ∞ ' f0 ξ0 1 9 k& +√ f k ρ ξ k cos(kθ) + η k sin(kθ) . (13.20) u (r, θ) = √ π 2π k =1 The result (13.19) is an indication that there should be a simple relation between the correlation function B u and the correlation function B g of the homogeneous process g defined on the boundary. Indeed, we present this relation below in Theo­ rem 13.4. The correlation function of the solution in the case when g is a white noise, is given in (13.11). It depends on the difference ψ = θ2 − θ1 and on the product ρ 1 ρ 2 . Thus in the notation of the Poisson kernel given in (13.13) the correlation function (13.11) reads 1 1 − ρ 21 ρ 22 B u = K (ρ1 ρ2 ; ψ) = . (13.21) 2π 1 − 2ρ 1 ρ 2 cos(ψ) + ρ 21 ρ 22 Now we can give the desired relation between the correlation functions. Theorem 13.4. Assume the boundary function g in the Dirichlet problem (13.9) is a ho­ mogeneous random process with a continuous correlation function B g (ψ). Then the so­ lution of the problem (13.9) is partially homogeneous with respect to the angular coor­ dinate and its correlation function B u (ρ 1 , θ1 ; ρ 2 , θ2 ) depends on the angular difference

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274 | 13 Boundary value problems with random boundary conditions ψ = θ2 − θ1 and the product ρ1 ρ 2 , and is explicitly given by the convolution B u = K ∗ B g , i.e. by the Poisson formula B u (ρ1 ρ2 ; ψ) =

1 2π

2π

K (ρ 1 ρ 2 ; ψ − ψ )B g (ψ )dψ ,

(13.22)

0

which implies that the correlation function B u (ρ, θ) is harmonic in the unit disk and it is the unique solution of the Dirichlet boundary value problem ΔB u = 0,

B u | ρ →1 = B g .

(13.23)

Proof. To obtain (13.22), we turn to the proof of Theorem 13.1 and use in the double integral in (13.12) the change of variable ψ = φ − φ , use there the series expansions for the both Poisson kernels and perform the integration over the variable φ . This yields (13.22). Remark 13.2. From the proof it is clear that the same convolution relation result re­ mains true if two homogeneous and homogeneously correlated stochastic processes are given on the boundary. Indeed, let g1 and g2 be two homogeneous processes on the circle with zero mean and a cross-correlation B g 1 g 2 (θ2 − θ1 ). Then the correspond­ ing solutions u 1 and u 2 are also homogeneously correlated and the cross-correlation function B u1 u2 is related to B g 1 g 2 by the same convolution formula with the kernel K as in Theorem 13.4: B u1 u2 = K ∗ B g 1 g 2 . Finally, we note that from (13.20) we can derive the expressions for B u x and B u y , the correlation functions for the derivatives u x and u y which is in our case remarkably coincide: ∞ 1  Bux = Buy = f k k 2 ρ k−1 cos [(k − 1)θ] . π k =1

13.2.3 Neumann boundary conditions Let us study the case when on the boundary, the normal derivative is prescribed, i.e. we consider the inner problem for the disk D = K (x0 , R): Δu (x) = 0 ,

x ∈ D,

∂u (y) = g(y) ∂n

y ∈ Γ = ∂D ,

(13.24)

where n is the external normal vector. The Poisson-type formula in polar coordinates centered at x0 has the form [210] 1 u (r, θ) = − 2π

2π

ln(1 − 2ρ cos(θ − φ) + ρ 2 )g(φ)dφ + const , 0

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where ρ = r/R and const is an arbitrary constant which we further take equal to zero. As in the Dirichlet problem, here the eigenvalue property of the kernel (see (13.15) plays the crucial role. By direct evaluations we can prove that −

1 2π

2π

ln(1 − 2ρ cos(θ − φ) + ρ 2 )h k (φ)dφ = λ k h k (θ) ,

(13.25)

0

where ρk ; h k = π−1/2 cos(kθ) ; k k = 1, 2, 3, . . . .

λ2k−1 = λ2k =

h2k = π−1/2 sin(kθ) ,

This can be easily proved by substituting the expansion [63] ln(1 − 2ρ cos(θ − φ) + ρ 2 ) = −2

∞  ρk cos[k (θ − φ)] k k =1

in (13.25). From this, we can derive the following result which is a counterpart of Theo­ rem 13.1. Theorem 13.5. The solution of the Neumann problem (13.24) in a disk K (x0 , R) with the Gaussian white noise boundary function g(y) is an inhomogeneous 2D Gaussian random field uniquely defined by the correlation function B u (ρ 1 , θ1 ; ρ 2 , θ2 ) = KNeum (ρ 1 ρ 2 ; θ2 − θ1 ) =

∞ 1  ρ 1k ρ 2k cos k (θ2 − θ1 ) . π k =1 k 2

(13.26)

The random field u (r, θ) is homogeneous with respect to the angular coordinate θ, ρk ρk

1 2 k = 1, . . . . More­ and its respective discrete spectral density has the form f θ (k ) = πk 2 over, if g is a homogeneous random process with a correlation function B g (ψ ) then the correlation function of the solution is related to B g by the convolution

1 B u (ρ1 ρ2 ; ψ) = 2π

2π

KNeum (ρ 1 ρ 2 ; ψ − ψ )B g (ψ )dψ .

(13.27)

0

Proof. The proof of (13.26) is essentially the same as in the case of the Dirichlet prob­ lem. The correlation function B u is written in the form of a convolution, i.e. B u = K1 (ρ 1 ; ψ) ∗ K1 (ρ 2 ; ψ − (θ1 − θ2 )) , where K1 (ρ; ψ) = ln(1 − 2ρ cos(ψ) + ρ 2 ). Then we take the Fourier transform of both parts and use the Fourier transform property for convolutions. This yields f θ (k ) = ρ 1k ρ 2k /πk 2 which is the desired result. The proof of (13.27) follows basically the same scheme and repeats the scheme given in the proof of Theorem 13.4.

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276 | 13 Boundary value problems with random boundary conditions 0.3

0.4

Laplace,ρ 1=0.9, ρ 2=0.5 Laplace,ρ 1=0.5, ρ 2=0.5 Neumann,ρ 1=0.9, ρ 2 = 0.5 Neumann,ρ 1=0.5, ρ 2=0.5

0.3

Neumann problem

0.2 0.1

Bu

B u(x 1,x)

0.2 0.1

ρ 1=0.9 ρ 1=0.8 ρ 1=0.5 ρ 1=0.2

0

−0.1 0

−0.2 −0.1

−0.3 0

10

20 30 40 angular section k, x=k 2π/50

50

5

10

15 20 25 30 35 40 45 50 radial section k, x=k 2R/50

Figure 13.3. Comparison of angular correlations for Laplace and Neumann boundary conditions, for two different values of the radii (left panel). Radial correlation function for the Neumann boundary conditions (right panel).

From these considerations, we can find the eigenvalues and eigenfunctions of the correlation function. These are λ2k−1 = λ2k =

1

k = 1, 2, 3, . . .

; + 1)  cos(kθ) h2k−1 (ρ, φ) = 2k + 1ρ k ; π 1 /2 k 2 (2k

 sin(kθ) h2k (ρ, φ) = 2k + 1ρ k . π 1 /2

This leads to the K-L expansion (13.26). The random filed is therefore written as follows: ∞ ' 1  ρk & u (r, θ) = √ ξ k cos(kθ) + η k sin(kθ) . π k =1 k We compare in Figure 13.3 the angular correlations for the Laplace and Neumann boundary conditions (left panel) and show the radial behavior of the correlation func­ tion for the solution of the Neumann problem (right panel).

13.2.4 Upper half-plane Let us consider the Dirichlet problem in the half-plane: Δu (x) = 0 ,

x ∈ D+ ,

u (y) = g(y)

y ∈ Γ = ∂D+ ,

(13.28)

where the domain D+ is the upper half-plane with the boundary Γ = {(x, y) : y = 0}. The Poisson formula reads [210] y u(x) = π

∞ −∞

g(ξ )dξ . ( x − ξ )2 + y 2

(13.29)

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Direct evaluation of the covariance function B(x1 , y1 ; x2 , y2 ) even in the case when the function g(ξ ) is a white noise meets some technical difference in comparison to the disk, ∞ y1 y2 dξ B(x1 , y1 ; x2 , y2 ) = . 2 2 ( π )2 [(x1 − ξ ) + y1 ][(x2 − ξ )2 + y22 ] −∞

Therefore, we use the Fourier transform technique. Let us introduce the notation for the kernel y K p (η, y) = 2 η + y2 so that the covariance is written in the form of convolution B(x1 , y1 ; x2 , y2 ) = K p (η, y1 ) ∗ K p (η − (x2 − x1 ), y2 ) , and the Fourier transform yields FB = FK (· ,y1) · FK (· ,y2) . Since [63] y FK (· ,y) = π

∞ −∞

cos(kx)dx = e−| k| y , y2 + x2

we get FB = e−| k|(y1+y2 ) . The inverse Fourier transform finally yields [63] B(x1 , y1 ; x2 , y2 ) = F −1 (e−| k|(y1+y2 ) ) =

1 y1 + y2 . π ( y 1 + y 2 )2 + ( x 1 − x 2 )2

(13.30)

Now we need to solve the eigenvalue problem for the covariance operator: ∞

∞

dy2 0

−∞

y1 + y2 h k (x2 , y2 )dx2 = λ k h k (x1 , y1 ) . π ( y 1 + y 2 )2 + ( x 1 − x 2 )2

(13.31)

Here we cannot apply the classical Hilbert–Schmidt theory since the process is defined on the unbounded domain D+ . Therefore, we can apply the cutoff approach described in Section 13.1. Indeed, the correlation function (13.30) is partially homoge­ neous with respect to the horizontal coordinate x. Through a Fourier analysis we find that the partial spectrum is S(k ) = exp(−k (y1 + y2 ) and the eigenfunctions are the Fourier modes. Thus as discussed in Section 13.1, λ k ≈ S(ω k ) = S(πk /a) = exp{−πk (y1 + y2 )/a} , and the spectral approximations are ˜ a (x1 , y1 ; x2 , y2 ) = Bu ≈ B u ≈ u˜ a (x, y) =

. ∞  1 − πk (y1 +y2 ) πk (x2 − x1 ) cos , e a a a k =1

(13.32)

∞  & ' 1 √ exp{−πky/a } ξ k cos [πkx/a)] + η k sin[πkx/a) , (13.33) a k =1

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278 | 13 Boundary value problems with random boundary conditions 0.25 exact K−L, a=40 K−L, a=10 K−L, a=100

0.2 0.15

Bu(x)

y1=0.5, y2=0.9

0.1 0.05 0 0

100

200

300

400

500

section k, x=kΔx Figure 13.4. Comparison of the approximate correlation function with the exact result, for three different values of the cutoff parameter a, Δx = 0.01.

where ξ k and η k are two mutually independent families of standard Gaussian random variables. Thus introducing the cutoff we can find the orthonormal set of eigenfunc­ tions of the eigenvalue problem for the correlation operator (13.31): K K cos(πkx/a) 2πk − πky sin(πkx/a) 2πk − πky a √ √ , h2k (x, y) = e e a , h2k−1 (x, y) = a a a a a , k = 1, 2, . . . , . λ2k−1 = λ2k = 2πk Note that the cutoff parameter a should be chosen large enough. For illustration, we present in Figure 13.4 the approximation (13.32) for three dif­ ferent values of a compared to the exact representation (13.30). The numerical con­ vergence is clearly seen as the cutoff parameter increases. Obviously, as mentioned at the end of the second subsection of the Introduction to this chapter, the larger a, the larger the number of retained terms n, so that n ∼ a/ε where ε is the approximation error. Finally, we notice that Theorem 13.4 proved above for the case of a disk also holds for the half-plane where the kernel K (ρ 1 ρ 2 ; θ2 − θ1 ) should be replaced by the kernel (13.30). Thus if the random function g defined on the axis x, {(x, y) : y = 0} is a homogeneous random process with the correlation function B g , then the correlation function of the solution B u (x1 , y1 ; x2 , y2 ) = B u (x2 − x1 , y2 + y1 ) depends on x = x2 − x1 and y = y2 + y1 , so B u (x, y) is harmonic in D+ , with the boundary conditions B u |y→0 = B g . We will now show that this is true indeed for a half-space in any dimension. So let us give the result in more detail. Here it is convenient to use the boldface character x for the horizontal coordinates and y for the vertical one. Theorem 13.6. Let u (x, y) and x = (x1 , . . . x n−1 ) be a random field defined in the halfn space D+ = R+ as a harmonic function with the boundary condition u |y=0 = g, where g

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13.3 3D Laplace equation | 279

is a zero mean homogeneous random field on the boundary {y = 0} with the correlation function B g (x) which is bounded in dimension n = 2, or tends to zero as |x| → ∞ if n > 2. Then B u (x, y) = B u (x2 − x1 , y1 + y2 ), the correlation function of the solution, is n , and is related to B g by the Poisson-type formula: a harmonic function in R+  (y1 + y2 )B g (x )dS(x ) Γ(n/2) B u (x2 − x1 , y1 + y2 ) = . (13.34) n / 2 π [(x − (x2 − x1 ))2 + (y1 + y2 )2 ]n/2 ∂D +

The proof is obtained by the same Fourier transform technique as we used above. Remark 13.3. We remark that exactly as in the case of a disk as discussed in Re­ mark 13.2 to Theorem 13.3, the same convolution relation (13.34) is true for the crosscorrelation functions, we need to write it only for the kernel K p : B u1 u2 = K p ∗ B g 1 g 2 . Note that in the n-dimensional case, K p has the form of the kernel given in (13.34). In practice, it is often important to know the statistical structure of the gradient of the solution. Let us denote by B u xi (x, y), i = 1, . . . , n − 1, and B u y (x, y) the correlation functions of the partial derivatives of the solution u. They also obviously depend only on x = x2 − x1 and y = y1 + y2 by the assumption that g is homogeneous. Direct evaluation gives B u xi = −

∂2 B u , ∂x i

i = 1, . . . , n − 1 ,

Buy =

∂2 B u . ∂2 y

Note that since the correlation function B u is harmonic, this implies the following re­ # markable property: B u y = ni=−11 B u xi . So in dimension 2, B u y = B u x .

13.3 3D Laplace equation For a ball in 3D, all considerations are quite similar, where the eigenfunctions involved are the spherical harmonics. The regular solution to the harmonic equation in a 3D ball D(x0 , R) of radius R centered at a point x0 is represented by the Poisson integral formula as an integral over the sphere S(x0 , R) = ∂D(x0 , R) [210]:  g(y)dS y R2 − r2 u(x) = 4πR |x − y|3/2 S ( x 0 ,R )

for any point x ∈ D(x0 , R), where r = |x − x0 |. In spherical coordinates centered at x0 the Poisson formula reads (see Theorem 3.2 in Chapter 3): 1 − ρ2 u (r, θ, φ) = 4π

2π π 0 0

sin(θ )g(θ , φ )dθ dφ , [1 − 2ρ cos(ψ) + ρ 2 ]3/2

(13.35)

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280 | 13 Boundary value problems with random boundary conditions where ρ = r/R and ψ are the angle between the vectors s and s defined by the relevant unit directions, which implies, cos(ψ) = cos θ cos θ + sin θ sin θ cos(φ − φ ) .

(13.36)

Let g be a zero mean random field defined on the sphere S(x0 , R). It is called isotropic if its correlation function B g (s, s ) depends only on the angular distance be­ tween s and s , i.e. only on the angle ψ as defined in (13.36). We say that a random field defined in a ball D(x0 , R) is partially isotropic in the ball if it is isotropic with respect to the angular coordinates. The statement of Theorem 13.1 for the 3D case can be formulated as follows. Theorem 13.7. The solution of the Dirichlet problem in the ball D(x0 , R) with the white noise boundary function g(y) is an inhomogeneous 3D Gaussian random field uniquely defined by the correlation function B u = K3 (ρ 1 ρ 2 ; ψ12 ) ≡

1 1 − ρ 21 ρ 22 , 4π [1 − 2ρ 1 ρ 2 cos(ψ1,2 ) + ρ 21 ρ 22 ]3/2

(13.37)

where cos(ψ1,2 ) = cos θ1 cos θ2 + sin θ1 sin θ2 cos(φ1 − φ2 ) . The random field u (r, θ, φ) is partially isotropic in the ball and its respective discrete spectral density has the form f θ (0) = 1/4π, f θ (k ) = (ρ 1 ρ 2 )k /2π, k = 1, . . . . Generally, if g is an isotropic ran­ dom field on the sphere the correlation function of the solutions B u is related to B g by B u (ρ 1 ρ 2 ; ψ) = K3 (ρ 1 ρ 2 ; ψ − ψ ) ∗ B g (ψ ) which implies that the solution u is partially isotropic in the ball, and the correlation function B u is harmonic, with the prescribed boundary function B g . Proof. We also use here a series expansion method. Let us recall some definitions. The Legendre polynomials we denote by Pl (cos θ), – recall that these functions are defined on (−1, 1) as follows: Pl (μ) =

1 dl (μ 2 − 1)l , 2l l! dμ l

l = 0, 1, . . . .

The associated Legendre polynomials Plm (μ ), l = 0, 1, . . . ; m = 0, 1, . . . , l are defined via the (m)-derivatives of Pl (μ ) as follows: Plm (μ ) = (1 − μ 2 )m/2 P (m) (μ ) ,

l = 0, 1, . . . ;

m = 0, 1, . . . , l .

Then, the system of spherical harmonics functions {Y lm (θ, φ)}, l = 0, 1, . . . ; m = 0, ±1, ±2, . . . , ±l is defined as follows: Y lm (θ, φ) = Plm (cos θ) cos(mφ), Y lm (θ,

φ) =

Plm (cos θ) sin(|mφ|) ,

m = 0, 1, 2, . . . , l ; m = −1, −2, . . . .

(13.38)

It is well known that this is a system of orthogonal functions complete in L2 (S) and Y lm 2

π 2π 1 + δ0m (l + |m|)! = [Y lm (θ, φ)]2 sin θdθdφ = 2π . 2l + 1 (l − |m|)! 0 0

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13.3 3D Laplace equation | 281

The following expansion is well known (see, e.g. [210; 33]): K (ρ, ψ) ≡

∞  1 − ρ2 = 1 + ρ k (2k + 1)Pk (cos(ψ)) . [1 − 2ρ cos(ψ) + ρ 2 ]3/2 k =1

(13.39)

For brevity, let us introduce the notation for the unit vectors, s , s1 , and s2 defined by their direction angles (θ , φ ), (θ1 , φ1 ), and (θ2 , φ2 ), respectively, and let (s , s1 ) = cos(ψ1 ) = cos θ1 cos θ + sin θ1 sin θ cos(φ1 − φ ) (s , s2 ) = cos(ψ2 ) = cos θ2 cos θ + sin θ2 sin θ cos(φ2 − φ ) .

In what follows, we will sometimes use a shorter notation for the integration over a surface measure ds on a unit sphere: 

2π π

ds =

sin(θ )dθ dφ .

0 0

We use expansion (13.39) in the following explicit evaluations: B u ( ρ 1 , s1 ; ρ 2 , s2 ) 1 = (4π )2

2π π

(1 − ρ 21 )(1 − ρ 22 ) sin(θ )dθ dφ [1 − 2ρ 1 cos(ψ1 ) + ρ 21 ]3/2 [1 − 2ρ 2 ρ 2 cos(ψ2 ) + ρ 22 ]3/2 0 0 ⎤⎡ ⎤ ⎡  ∞ ∞   ⎣1 + ρ 1k (2k + 1)Pk ((s , s1 ))⎦ ⎣1 + ρ 2k (2k + 1)Pk ((s , s2 ))⎦ ds

1 (4π )2 k =1 ⎡ ⎤ ∞  1 ⎣ k 2 P k (( s1 , s2 )) ⎦ = 1 + (ρ 1 ρ 2 ) (2k + 1) . 4π 2k + 1 k =1 =

k =1

Here we used the following property:  1 Pk ((s1 , s2 )) Pl ((s, s1 ))Pk ((s, s2 ))ds = δ kl . 4π 2k + 1 This can be derived from  1 1 Pk ((s, s ))Y lm (s )ds = Y m (s)δ lk , 4π 2l + 1 l

(13.40)

(13.41)

(13.42)

which in turn follows from Pl ((s, s )) =

l 

κ lm Y lm (s)Y lm (s ) .

(13.43)

m =− l

Here the coefficients are given by κ lm =

(l − |m|)! 2 . (1 + δ0m ) (l + |m|)!

(13.44)

Thus the last line of (13.40) gives the desired result (13.37) and the proof is complete.

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282 | 13 Boundary value problems with random boundary conditions Now we use the series representation (13.40) to solve the eigenproblem for the corre­ lation function 1



dρ 2

B u (ρ 1 , s1 ; ρ 2 , s2 )h l (ρ 2 , s2 )ds2 = λ l h l (ρ 1 , s1 ) .

(13.45)

0

The next assertion follows immediately from properties (13.41)–(13.43). Theorem 13.8. The eigenvalue problem (13.45) has a complete set of orthonormal eigen­ functions and the relevant eigenvalues (l = 0, 1, . . . , m = −l, . . . , l): K  1 κ lm (2l + 1) (13.46) λl = , h l (ρ, s) = Y lm · 2l + 1ρ l . 2l + 1 4π The K-L expansions of the correlation function and the random field are given by ⎫ ⎧ ∞ k ⎬ ⎨  1 1  m m B u ( ρ 1 , s1 ; ρ 2 , s2 ) = + (2k + 1)ρ 1k ρ 2k κ km Y k ( s1 ) Y k ( s2 ) , ⎭ ⎩ 4π 4π k=1 m =− k ⎫ ⎧ ∞ k ⎬ ⎨  √ ξ0 1  u (r, s) = √ + √ 2k + 1ρ k ⎩ ξ km κ km Y km (s)⎭ , 4π 4π k=1 m =− k where ξ0 , {ξ km } are independent standard Gaussian random variables.

13.4 Biharmonic equation Let us consider the following problem for a biharmonic equation in a disk D = K (x0 , R), governing a slow viscous motion inside a circular cylinder of radius R [99]: Δu 2 (x) = 0 ,

x ∈ D,

u (y) = g0 (y) ,

∂u (y) = g n (y) ∂n

y ∈ Γ = ∂D ,

(13.47)

where n is the external normal vector. In polar coordinates centered at x0 with ρ = r/R the Poisson-type integral formula reads u (r, θ) =

(1 − ρ 2 )2 2π

2π  − 0

(1 − ρ 2 )2 + 2π

2π 0

R 2[1 − 2ρ cos(θ − φ) + ρ 2 ]



g n (φ)dφ

(13.48)

[1 − ρ cos(θ − φ)] g0 (φ)dφ . [1 − 2ρ cos(θ − φ) + ρ 2 ]2

Assuming that the random white noise excitations g0 and g n are independent, we de­ compose the random field into two independent components: u = u (1) + u (2) . Then,

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13.4 Biharmonic equation | 283

the covariance of u is the sum of covariances of u (1) and u (2) . From (13.48) we obtain B u =  u (r1 , θ1 )u (r2 , θ2 ) = 1 + (2π )2

2π 0

R2 (1 − ρ 21 )(1 − ρ 22 )B Δ (ρ 1 , θ1 ; ρ 2 , θ2 ) 4

(1 − ρ 21 )2 (1 − ρ 1 cos(θ1 − φ)) (1 − ρ 22 )2 (1 − ρ 2 cos(θ2 − φ)) · dφ , [1 + ρ 21 − 2ρ 1 cos(θ1 − φ)]2 [1 + ρ 22 − 2ρ 2 cos(θ2 − φ)]2

(13.49) where B Δ is the covariance of the solution of the Dirichlet problem for the Laplace equation given in Theorem 13.1. To tackle the second term that represents the covariance of the second compo­ nent, u (2) , we first remark that 1 2π

2π 0

. (1 − ρ 2 )2 (1 − ρ cos(θ − φ)) k 2 cos(kφ)dφ = 1 + ((1 − ρ ) ρ k cos(kθ) [1 + ρ 2 − 2ρ cos(θ − φ)]2 2

(13.50) 1 (2π )

2π 0

. (1 − ρ 2 )2 (1 − ρ cos(θ − φ)) k 2 sin ( kφ ) dφ = 1 + (( 1 − ρ ) ρ k sin(kθ) . [1 + ρ 2 − 2ρ cos(θ − φ)]2 2

(13.51) This can be shown as follows. First, we note that by differentiating with respect to ρ we obtain the following useful equality: ⎞ ⎛ ∞ ∞   −4ρ 2 + 2ρ 3 cos θ + 2ρ cos θ k k ⎝ 2 kρ cos(kθ) = ρ 2 ρ cos(kθ)⎠ = . [1 + ρ 2 − 2ρ cos θ]2 k =1 k =1 Now, combining with expansion (13.13) we find that the kernel in the eigenvalue prob­ lem (13.50), (13.51) is represented as the following series: 1 (1 − ρ 2 )2 (1 − ρ cos(θ − φ)) 2π [1 + ρ 2 − 2ρ cos(θ − φ)]2 ∞ ∞ 1 − ρ2  k 1 1  k = kρ cos[k (θ − φ)] + + ρ cos[k (θ − φ)] 2π k=1 2π π k=1 % ∞ $ 1 1  (1 − ρ 2 ) k + + 1 ρ k cos[k (θ − φ)] . (13.52) = 2π π k =1 2

K (ρ; θ − φ) =

Substituting this representation in the eigenvalue problem we arrive at (13.50), (13.51). The covariance can be evaluated by substituting the series expansion (13.52) in (13.49). This yields $ %$ % ∞ 1 1  (1 − ρ 21 )k (1 − ρ 22 )k + +1 + 1 ρ 1k ρ 2k cos[k (θ1 − θ2 )] . B u (2 ) = 2π π k =1 2 2

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284 | 13 Boundary value problems with random boundary conditions Analogously to the case of Laplace equation, we consider the eigenvalue problem for the covariance kernel: 2π

1

dρ 1 B u(2) (ρ 1 , θ1 ; ρ 2 , θ2 )h k (ρ 1 , θ1 ) = λ k h k (ρ 2 , θ2 ) .

dθ1 0

0

Let us introduce the notation: 1 $

Δk =

%2 (1 − ρ 2 ) k + 1 ρ 2k dρ . 2

0

Using the series expansion of the kernel, it is not difficult to find that the eigen­ value problem has the following system of eigenfunctions and eigenvalues: 1 λ0 = 1 , h0 = √ ; λ2k−1 = λ2k = Δ k ; 2π % $ 2 ρk cos[k (θ)] (1 − ρ ) k +1  · ; h2k−1 (ρ, φ) = 2 π 1 /2 Δk $ % (1 − ρ 2 ) k ρk sin[k (θ)] +1  · ; k = 1, 2, . . . h2k (ρ, φ) = 2 π 1 /2 Δk where the eigenfunctions are orthonormal to one another: 1 2π

h n (ρ, θ)h m (ρ, θ)dρdθ = δ nm . 0 0

0.45 Biharmonic, ρ 1=0.5, ρ 2=0.5

0.4

Laplace, ρ 1=0.5, ρ 2=0.5

0.35 0.3 B u 0.25 0.2 0.15 0.1 0.05

0

10

20 30 40 angular section k, θ=k 2π/50

50

Figure 13.5. Angular correlation function for the biharmonic equation, compared to the correlation function for the Laplace equation with Dirichlet boundary conditions.

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13.5 Lamé equation: plane elasticity problem

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From this we finally arrive at the K-L expansion $ % ∞ & ' ξ0 1  (1 − ρ 2 ) k (2) + 1 ρ k ξ k cos kθ + η k sin kθ . u (r, θ) = √ + √ π k =1 2 2π R (1− ρ2 )

u (r, θ), where The first component is obviously represented as u (1) (r, θ) = 2 u (r, θ) is modeled by the K-L expansion given in (13.16). In Figure 13.5, we show the angular behavior of the correlation function of the so­ lution to the biharmonic equation compared to the correlation function for the Laplace equation. In both cases, the correlations are plotted for the fixed values of ρ taken equal to 0.5 and R = 1.

13.5 Lamé equation: plane elasticity problem 13.5.1 White noise excitations Let us consider the plane elasticity problem in the disk K (x0 , R): μΔu(x) + (λ + μ ) grad div u(x) = 0 , u(y) = g(y) ,

x ∈ K ( x0 , R) , y ∈ S ( x0 , R) ,

(13.53)

where u = (u 1 , u 2 )T is the displacement column vector which is prescribed on the boundary as a column vector g = (g1 , g2 )T . Let us work in polar coordinates centered at x0 , so that the point x is re iθ , and on the boundary, y = Re iφ , and as everywhere above, ρ = r/R. Let us recall that the kernel in the Poisson integral formula (13.10) for the Laplace equation given explicitly by (13.13), has in the polar coordinates the form K (ρ; θ − φ) =

1 2π

1 − ρ2 . 1 − 2ρ cos(θ − φ) + ρ 2

(13.54)

Here it is convenient to work in the polar coordinates, with the displacement func­ tions (u θ , u r ). The Poisson integral formula (6.26) presented in Section 6.1.2 reads

2π u r (r, θ) = u θ (r, θ) 0



L11 (ρ; θ − φ) L21 (ρ; θ − φ)

L12 (ρ; θ − φ) L22 (ρ; θ − φ)



g r (Reiφ ) dφ . g θ (Reiφ )

(13.55)

We are now in a position to formulate and solve the eigenvalue problem for the integral operator with the matrix-kernel L of the Poisson-type integral (6.26) L(ρ; θ − φ) =

1 K (ρ; θ − φ)G(ρ; θ − φ) . λ + 3μ

(13.56)

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286 | 13 Boundary value problems with random boundary conditions The eigenvalue problem is written as the following system:

2π

L(ρ; θ − φ) 0

h1 (φ) h1 (θ) dφ = λ . h2 (φ) h2(θ)

(13.57)

Theorem 13.9. The eigenvalue problem (13.57) has the following system of eigenvalues and eigenfunctions (k = 1, 2, . . . ):







h1,2k−1 sin kθ h1,2k − cos kθ k −1 λ2k−1 = λ2k = ρ , , , = = h2,2k−1 h2,2k cos kθ sin kθ and for the case k = 2, for λ3 = λ4 = ρ, there is a third eigenfunction h1,3 1 . = h 2,3 1 Proof. In the proof, we use the expansion of the matrix kernel in the Fourier series L11 =

∞ ρ 1  + λ11 (ρ, k )ρ k cos[k (θ − φ)] , 2π π k=1

L12 =

∞ 1  λ12 (ρ, k )ρ k sin[k (θ − φ)] , π k =1

L21 =

∞ 1  λ21 (ρ, k )ρ k sin[k (θ − φ)] , π k =1

L22 =

∞ ρ 1  + λ22 (ρ, k )ρ k cos[k (θ − φ)] , 2π π k=1

(13.58)

where we use the notations

% $ 1 2(λ + 2μ ) k (λ + μ )(1 − ρ 2 ) 2μρ + + , λ11 (ρ, k ) = 2(λ + 3μ ) ρ ρ % $ 1 2μ k (λ + μ )(1 − ρ 2 ) − 2μρ − , λ12 (ρ, k ) = 2(λ + 3μ ) ρ ρ % $ 1 2(λ + 2μ ) k (λ + μ )(1 − ρ 2 ) 2(λ + 2μ )ρ − − , λ21 (ρ, k ) = 2(λ + 3μ ) ρ ρ % $ 1 2μ k (λ + μ )(1 − ρ 2 ) 2(λ + 2μ )ρ + − . (13.59) λ22 (ρ, k ) = 2(λ + 3μ ) ρ ρ

Note that each of these series could be written in the form of a power series a1 ρ + a2 ρ 2 + a3 ρ 3 + · · · ; however, as we will see below, the form (13.58) is very convenient when solving the eigenvalue problem for the correlation operator. From expansions (13.58) we find that



2π sin kφ sin kθ k L11 (ρ; θ − φ) dφ = λ11 (ρ, k )ρ , cos kφ cos kθ 0

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13.5 Lamé equation: plane elasticity problem

2π 0

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cos kφ sin kθ k L12 (ρ; θ − φ) dφ = λ12 (ρ, k )ρ , − cos kθ sin kφ



2π

L21 (ρ; θ − φ)



− cos kθ sin kφ dφ = λ21 (ρ, k )ρ k , cos kφ sin kθ

0

2π





cos kφ cos kθ k L22 (ρ; θ − φ) dφ = λ22 (ρ, k )ρ . sin kφ sin kθ

(13.60)

0

Now, by substituting these equalities in the eigenvalue problem (13.57) and taking into account that λ11 (ρ, k ) + λ12 (ρ, k ) = ρ −1 , −λ21 (ρ, k ) + λ22 (ρ, k ) = ρ −1 , we find the solution of the eigenvalue problem for k = 1, 2 . . . . The existence of the eigenfunction (1, 1)T for λ3 = ρ follows from the properties 2π

2π

L11 (ρ; θ − φ) · 1dφ = ρ ,

L22 (ρ; θ − φ) · 1dφ = ρ ,

0

0

2π

2π

L12 (ρ; θ − φ) · 1dφ = 0 , 0

L21 (ρ; θ − φ) · 1dφ = 0 . 0

The proof is complete. We now turn to the derivation of the correlation tensor of the solution

I J u r ( r1 , θ1 ) ( u r ( r2 , θ2 ) , u θ ( r2 , θ2 ) B u (ρ 1 , θ1 ; ρ 2 , θ2 ) = u(r1 , θ1 ) ⊗ u(r2 , θ2 ) ≡ u θ ( r1 , θ1 ) (13.61) assuming that the boundary random vector function g has a Gaussian distribution specified by the zero mean and covariance tensor I J

g r (φ1 ) B g (φ1 , φ2 ) = (g r (φ2 ) , g θ (φ2 ) . g θ (φ1 ) We use here and in what follows the following notation for v ⊗ u, a tensor product of two vectors: v ⊗ u = vuT . Substituting the Poisson integral formula (10.17) in (13.61) and changing the rel­ evant product of integral expressions by double integrals, we arrive at the following representation: 2π 2π

B u (ρ 1 , θ1 ; ρ 2 , θ2 ) =

L(ρ 1 ; θ1 − φ )B g (φ , φ )LT (ρ 2 ; θ2 − φ )dφ dφ . (13.62)

0 0

Let us first again consider the case when the boundary vector function g is a white noise, namely, assume that

δ(φ1 − φ2 ) 0 . (13.63) B g (φ1 , φ2 ) = 0 δ(φ1 − φ2 )

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288 | 13 Boundary value problems with random boundary conditions Note that this property then also holds in rectangular coordinates (see (13.74) below). Then, from (13.62) we obtain 2π

L(ρ 1 ; θ1 − φ)LT (ρ 2 ; θ2 − φ)dφ .

B u (ρ 1 , θ1 ; ρ 2 , θ2 ) =

(13.64)

0

Theorem 13.10. The exact K-L representations for the covariance tensor and the random field (u r , u θ )T which solves the Lamé equation under the boundary white noise excita­ tions with the covariance tensor (13.63) are given by B u (ρ 1 , θ1 ; ρ 2 , θ2 ) ⎛ # =⎝

ρ1 ρ2 2π

+ π1 1 #∞

π

k =1

∞ k =1

1 π

Λ11 ρ1k ρ2k cos[ k( θ2 − θ1 )]

Λ21 ρ1k ρ2k sin[ k( θ2 − θ1 )]

ρ1 ρ2 2π

#∞

+

(13.65) ⎞

k k k=1 Λ 12 ρ 1 ρ 2 sin[ k ( θ 2 − θ 1 )] ⎠ # 1 ∞ k k k=1 Λ 22 ρ 1 ρ 2 cos[ k ( θ 2 − θ 1 )] π

Λ11 = λ11 (ρ 1 , k )λ11 (ρ 2 , k ) + λ12 (ρ 1 , k )λ12 (ρ 2 , k ) , Λ12 = λ11 (ρ 1 , k )λ21 (ρ 2 , k ) − λ12 (ρ 1 , k )λ22 (ρ 2 , k ) , Λ21 = λ22 (ρ 1 , k )λ12 (ρ 2 , k ) − λ21 (ρ 1 , k )λ11 (ρ 2 , k ) , Λ22 = λ22 (ρ 1 , k )λ22 (ρ 2 , k ) + λ21 (ρ 1 , k )λ21 (ρ 2 , k ) , and u r (r, θ) =

∞ & ' ξ0 ρ 1  + λ11 ρ k ξ k cos kθ + η k sin kθ 2π π k =1

+

u θ (r, θ) =

∞ & ' 1  λ12 ρ k −η k cos kθ + ξ k sin kθ , π k =1

(13.66)

∞ & ' ξ0 ρ 1  + λ21 ρ k −η k cos kθ + ξ k sin kθ 2π π k =1

+

∞ & ' 1  λ22 ρ k ξ k cos kθ + η k sin kθ , π k =1

(13.67)

where {ξ k , η k } and {ξ k , η k }, k = 0, 1, 2, . . . , are two independent families of stan­ dard independent Gaussian random variables. Thus the random field is homoge­ neous with respect to the angular variable and the respective partial spectra are: S mm (k ) = π1 Λ mm ρ 1k ρ 2k , S mm (0) = ρ 1 ρ 2 /2π, and for n = m the spectrum is pure imagi­ nary: S mn (k ) = i π1 Λ mn ρ 1k ρ 2k . Proof. To get the expansion of the correlation tensor (13.65), we substitute expansions (13.58) in (13.64) and use the eigenfunction properties (13.60). To construct the explicit simulation formula (13.66), (13.67) for our random field, we first split it into two independent Gaussian random fields: u(r, θ) = V1 (r, θ) + V2 (r, θ) .

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We will now show that for each of these random fields we can obtain a K-L expansion. We introduce four single mode vector functions

λ11 (ρ, k ) cos kθ λ11 (ρ, k ) sin kθ ˜ , h1k (ρ, θ) = , (13.68) h1k (ρ, θ) = −λ21 (ρ, k ) cos kθ λ21 (ρ, k ) sin kθ

−λ12 (ρ, k ) cos kθ ˜ 2k (ρ, θ) = λ12 (ρ, k ) sin kθ . h2k (ρ, θ) = , h (13.69) λ22 (ρ, k ) sin kθ λ22 (ρ, k ) cos kθ Here the modes are indexed by k = 1, 2 . . . , while the subindexes 1 and 2 stand for the first and second series of eigenfunctions. Note that these vectors are pairwise orthogonal: 1

2π

dρ 0

1

dθ

˜ 1k = 0 , h1k · h

0

2π

dρ 0

dθ

˜ 2k = 0 , h2k · h

0

also as the following two vectors are orthogonal:

ρ √ 0 2π ˜ . h0 = √ρ . h0 = 0 2π It is now a matter of technical evaluations to find that the correlation tensor can be represented in the form B u (ρ 1 , θ1 ; ρ 2 , θ2 ) = h0 (ρ 1 ) · hT0 (ρ 2 ) +

1 π2

∞ 

(13.70)

˜ 1k (ρ 1 , θ1 )h ˜ T1k (ρ 2 , θ2 )}ρ 1k ρ 2k {h1k (ρ 1 , θ1 )hT1k (ρ 2 , θ2 ) + h

k =1

˜ 0 (ρ1 ) · h ˜ T0 (ρ 2 ) +h +

∞ 1  ˜ 2k (ρ 1 , θ1 )h ˜ T2k (ρ 2 , θ2 )}ρ 1k ρ 2k . {h2k (ρ 1 , θ1 )hT2k (ρ 2 , θ2 ) + h π 2 k =1

This follows from the easily verified representation ˜ 1k (ρ 1 , θ1 ) h ˜ T1k (ρ 2 , θ2 ) h1k (ρ 1 , θ1 )hT1k (ρ 2 , θ2 ) + h ⎛ =⎝

⎞

λ 11 ( ρ1 , θ1 ) λ 11 ( ρ2 , θ2 ) cos[ k( θ2 − θ1 )]

λ 11 ( ρ1 , θ1 ) λ 21 ( ρ2 , θ2 ) sin[ k( θ2 − θ1 )]

−λ 21 ( ρ1 , θ1 ) λ 11 ( ρ2 , θ2 ) sin[ k( θ2 − θ1 )]

λ 21 ( ρ1 , θ1 ) λ 22 ( ρ2 , θ2 ) cos[ k( θ2 − θ1 )]

⎠

and ˜ 2k (ρ 1 , θ1 )h ˜ T2k (ρ 2 , θ2 ) h2k (ρ 1 , θ1 )hT2k (ρ 2 , θ2 ) + h ⎛ =⎝

λ 12 ( ρ1 , θ1 ) λ 12 ( ρ2 , θ2 ) cos[ k( θ2 − θ1 )]

−λ 12 ( ρ1 , θ1 ) λ 22 ( ρ2 , θ2 ) sin[ k( θ2 − θ1 )]

λ 22 ( ρ1 , θ1 ) λ 12 ( ρ2 , θ2 ) sin[ k( θ2 − θ1 )]

λ 22 ( ρ1 , θ1 ) λ 22 ( ρ2 , θ2 ) cos[ k( θ2 − θ1 )]

⎞ ⎠.

So we can see from (13.70) that the first and the second pairs of lines present the covariances of the first and second vectors in our splitting, respectively: B u = u(r1 , θ1 ) · uT (r2 , θ2 ) = V1 (r1 , θ1 ) · VT1 (r2 , θ2 ) + V2 (r1 , θ1 )VT2 (r2 , θ2 ) ,

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290 | 13 Boundary value problems with random boundary conditions thus, B V1 = h0 (ρ 1 ) · hT0 (ρ 2 ) +

∞ 

˜ 1k (ρ 1 , θ1 )h ˜ T1k (ρ 2 , θ2 )}ρ 1k ρ 2k , {h1k (ρ 1 , θ1 )hT1k (ρ 2 , θ2 ) h

k =1

˜ 0 (ρ1 ) · h ˜ T0 (ρ 2 ) B V2 = h +

∞ 

˜ 2k (ρ 1 , θ1 )h ˜ T2k (ρ 2 , θ2 )}ρ 1k ρ 2k , {h2k (ρ 1 , θ1 )hT2k (ρ 2 , θ2 ) + h

k =1

where B V1 = V1 (r1 , θ1 ) · VT1 (r2 , θ2 )

B V2 = V2 (r1 , θ1 )VT2 (r2 , θ2 ) .

Note that each part, i.e. B V1 and B V2 , is represented as an orthogonal-mode expan­ sion. Therefore, we can construct a K-L expansion for our random fields V1 and V2 . We have not yet normalized the eigenfunctions. We can do it through dividing the 1 √ angular modes by π and the radial modes by Δ1 (k ) = 0 (λ211 + λ221 )ρ 2k dρ, the first 1 2 family of eigenfunctions (13.68), and by Δ2 (k ) = 0 (λ12 + λ222 )ρ 2k dρ , the second family of eigenfunctions (13.69). We then collect the orthonormal eigenmodes in one family: (1) H2k−1 = 

1 h1k (ρ, θ) , Δ1 ( k ) π

(1) H2k = 

1 ˜ 1k (ρ, θ) , h Δ1 ( k ) π

k = 1, 2, . . .

(2) H2k−1 = 

1 h2k (ρ, θ), Δ2 ( k ) π

(2) H2k = 

1 ˜ 2k (ρ, θ) , h Δ2 ( k ) π

k = 1, 2, . . .

and

(1)

(2)

Then, the orthonormal functions Hk and Hk are eigenfunctions of the covari­ ance tensors B V1 and B V2 , respectively, with the corresponding eigenvalues Δ1 (k ) and Δ 2 ( k ): 1 2π

(m)

B V m · Hk

(m)

(ρ 2 , θ2 )dρ 2 dθ2 = Δ m (k )Hk

(ρ 1 , θ1 ) ,

m = 1, 2 .

0 0

We can now construct a K-L expansion for the random field V1 (r, θ) in the form V1 (r, θ) =

∞ 

(1)

ζ k Hk (ρ, θ) ,

k =1

where ζ k are Gaussian random variables such that  ζ k ζ j  = Δ1 (k )δ jk ,

and the same for V2 (r, θ).

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13.5 Lamé equation: plane elasticity problem |

1.2

1

3

α = 10 α = 10 α=2 α = 1.1 α =1/3

0.8 0.6 0.4

291

α = 10 3 α = 10 α=2 α = 1.1 α = 1/3

0.8 0.6 B22 0.4

0.2

0.2

0 −0.2

0

−0.4

−0.2

−0.6 0

i

1.2

1

B 11

i

10

20

30

40

angular section k, θ=k 2π/50

50

−0.4

0

10

20

30

40

50

angular section k, θ=k 2π/50

Figure 13.6. Correlations B11 (left panel) and B22 (right panel) for the Lamé equation, for different values of the elasticity parameter α; ρ1 = ρ2 = 0.3.

Putting these expansions together we finally arrive at the desired representation u r (r, θ) =

∞ & ' ξ0 ρ 1  + λ11 ρ k ξ k cos kθ + η k sin kθ 2π π k =1

+

u θ (r, θ) =

∞ & ' 1  λ12 ρ k −η k cos kθ + ξ k sin kθ , π k =1

∞ & ' ξ0 ρ 1  + λ21 ρ k −η k cos kθ + ξ k sin kθ 2π π k =1

+

∞ & ' 1  λ22 ρ k ξ k cos kθ + η k sin kθ , π k =1

where {ξ k , η k } and {ξ k , η k }, k = 0, 1, 2, . . . are two independent families of standard independent Gaussian random variables. Finally, note that the spectra given in the theorem are obtained immediately from representation (13.65). This completes the proof of Theorem 13.9. It is interesting to note that we could obtain these expressions by substituting formally a generalized representation of the boundary white noises on the circle g1 (φ) =

∞ ' ξ0 1 & + ξ k cos kφ + η k sin kφ 2π π k=1

g2 (φ) =

∞ ' ξ0 1 & + ξ k cos kφ + η k sin kφ 2π π k=1

into the Poisson formula (6.26) with the kernels given by the series expansions (13.58). But the justification would then need to work with generalized stochastic processes.

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292 | 13 Boundary value problems with random boundary conditions 0.8

0.8 α = 10 3 α = 10 α=2 α = 1.1 α = 1/3

0.6 0.4 0.2 B 12

0.6

B 12

0.2

0

0

−0.2

−0.2

−0.4

−0.4

−0.6 −0.8

B21

0.4

−0.6 0

10

20 30 40 angular section k, θ=k2π/50

−0.8 60 0

50

10

20

30

40

50

60

Figure 13.7. Correlations B12 (left panel), for different values of the elasticity parameter α. The same curves are shown in the right panel, superimposed by the relevant correlations B21 ; ρ1 = ρ2 = 0.3. 3

7 6

ρ 1=0.9

5

ρ 1=0.6

1.5

ρ 1=0.2

2

ρ=0.1

1

B22(x)

ρ 1=0.3

3 B 11(x)

2

ρ 1=0.5

4

1 0.5

0

0

−1

−0.5

−2

−1

−3

−1.5

0

10

20

30

radial section k, x=k 2R/50

40

50

ρ 1=0.95 ρ 1=0.9 ρ 1=0.85 ρ 1=0.75 ρ 1=0.6 ρ 1=0.3 ρ 1=0.2 ρ 1=0.1

2.5

0

10

20

30

40

50

radial section k, x=k 2R/50

Figure 13.8. Radial correlations B11 (left panel), and B22 (right panel), for different values of the start­ ing point ρ1 . Lamé equation, λ = 2222, μ = 2.

In Figures 13.6–13.10, we show the longitudinal correlation function B11 , the trans­ verse correlation function B22 , and the cross-correlation functions B12 and B21 , in po­ lar coordinates, as well as in rectangular coordinates. Figure 13.6 presents the angular behavior of B11 for five different values of the elasticity constant α (left panel) and the same for B22 (right panel). The relevant cross-correlations are shown in Figure 13.7. The radial behavior of B11 and B22 is shown in Figure 13.8. As is clearly seen from all these curves, the angular behavior is periodic. When plotting these functions in rectangular coordinates, we get a complicated behavior shown in Figures 13.9 and 13.10, where the correlations depend on the starting angle θ; we present the curves for different values of θ, see Figures 13.9 and 13.10.

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13.5 Lamé equation: plane elasticity problem

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1.2

1.4

1.1

3

θ=0, α=10 θ=0, α=2 3 θ=2, α=10 3 θ=1, α=10

1.2 1

θ=0 θ=1 θ=2

1 0.9

α=10 3

0.8

0.8 rec B 11

rec 0.7 B 22

0.6

0.6 0.5

0.4

0.4 0.2 0

0.3 0.2 0

10

20

30

40

50

0

60

10

20

30

40

50

60

angular section k, θ=k 2π/50

angular section k, θ=k 2π/50

rec Figure 13.9. Angular correlations in rectangular coordinates, Brec 11 (left panel), and B 22 (right panel), for different values of the starting angle θ; ρ1 = ρ2 = 0.3.

0.4

0.4

0.3

0.3

0.2

0.2

0.1 B

rec 12

0.1 B rec 21

0

−0.1

0

−0.1

−0.2 −0.3

−0.3

−0.4 0

10

20

30

40

angular section k, θ=k 2π/50

θ=0 θ=1 θ=2 θ=4

−0.2

θ=0 θ=1 θ=2 θ=4

50

60

−0.4

0

10

20

30

40

50

60

angular section k, θ=k 2π/50

rec Figure 13.10. Angular correlations in rectangular coordinates, Brec 12 (left panel), and B 21 (right panel), for different values of the starting angle θ; ρ1 = ρ2 = 0.3.

13.5.2 General case of homogeneous excitations We have so far considered the case when the boundary functions g1 and g2 are two independent white noise processes. We will now see that the general case when g1 and g2 are some arbitrary dependent homogeneous processes is basically derived from the white noise case. Thus assume that we are given two homogeneous zero mean processes g1 and g2 with the correlation tensor B g (φ2 − φ2 ), with the entries B g,ij, i, j = 1, 2. As shown above, the correlation tensor of the solution B u is related to B g by the double inte­ gral representation (13.62). Changing the integration variable φ to a new integration

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294 | 13 Boundary value problems with random boundary conditions variable ψ by φ − φ = ψ we obtain from (13.62) for u = (u ρ , u θ )T : 2π 2π

B u (ρ 1 , θ1 ; ρ 2 , θ2 ) =

L(ρ 1 ; θ1 − φ )B g (ψ)LT (ρ 2 ; θ2 − ψ − φ )dφ dψ .

(13.71)

0 0

The idea is now to evaluate explicitly the inner integral with respect to φ using the series expansions for the kernel L(ρ, θ) given above in (13.58). We now rewrite relation (13.71) in a different form. ˆ u as fol­ We construct from the correlation tensor B u a column-vector function B T ˆ u = (B u,11 , B u,12 , B u,21 , B u,22) . Analogously, we use the notation B ˆ g for the col­ lows B umn vector ˆ g = (B g,11 , B g,12 , B g,21 , B g,22)T . B Using this notation, we can rewrite (13.71) as follows: 2π 2π

ˆ u (ρ 1 , θ1 ; ρ 2 , θ2 ) = B

ˆ g (ψ)dφ dψ . L ( ρ 1 ; θ 1 − φ ) ⊗ L ( ρ 2 ; θ 2 − ψ − φ )B

(13.72)

0 0

Here we denote by ⊗ a tensor product of two matrices which is defined in our case as a 4 × 4 matrix, represented as a 2 × 2-block matrix each block being a 2 × 2 matrix of the form L ij (ρ 1 ; θ1 − φ )L(ρ 2 ; θ2 − ψ − φ ), i, j = 1, 2. We will now evaluate explicitly all the 16 entries a ij of the matrix 2π

A=

L(ρ 1 ; θ1 − φ ) ⊗ L(ρ 2 ; θ2 − ψ − φ )dφ .

(13.73)

0

Substituting the series representation of the matrix L given by (13.58) in (13.73) we obtain after a long but simple calculations a11 =

∞ ρ1 ρ2 1  + λ11 (ρ 1 , k )λ11 (ρ 2 , k )ρ 1k ρ 2k cos [k (θ2 − θ1 − ψ)] 2π π k =1

a12 =

∞ 1  λ11 (ρ 1 , k )λ12 (ρ 2 , k )ρ 1k ρ 2k sin [k (θ2 − θ1 − ψ)] π k =1

a13 = −

∞ 1  λ12 (ρ 1 , k )λ11 (ρ 2 , k )ρ 1k ρ 2k sin [k (θ2 − θ1 − ψ)] π k =1

a14 =

∞ 1  λ12 (ρ 1 , k )λ12 (ρ 2 , k )ρ 1k ρ 2k cos [k (θ2 − θ1 − ψ)] π k =1

a21 =

∞ 1  λ11 (ρ 1 , k )λ21 (ρ 2 , k )ρ 1k ρ 2k sin [k (θ2 − θ1 − ψ)] π k =1

a22 =

∞ ρ1 ρ2 1  + λ11 (ρ 1 , k )λ22 (ρ 2 , k )ρ 1k ρ 2k cos [k (θ2 − θ1 − ψ)] 2π π k =1

a23 =

∞ 1  λ12 (ρ 1 , k )λ21 (ρ 2 , k )ρ 1k ρ 2k cos [k (θ2 − θ1 − ψ)] π k =1

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13.5 Lamé equation: plane elasticity problem

a24 = −

∞ 1  λ12 (ρ 1 , k )λ22 (ρ 2 , k )ρ 1k ρ 2k sin [k (θ2 − θ1 − ψ)] π k =1

a31 = −

∞ 1  λ21 (ρ 1 , k )λ11 (ρ 2 , k )ρ 1k ρ 2k sin [k (θ2 − θ1 − ψ)] π k =1

a32 =

∞ 1  λ21 (ρ 1 , k )λ12 (ρ 2 , k )ρ 1k ρ 2k cos [k (θ2 − θ1 − ψ)] π k =1

a33 =

∞ ρ1 ρ2 1  + λ22 (ρ 1 , k )λ11 (ρ 2 , k )ρ 1k ρ 2k cos [k (θ2 − θ1 − ψ)] 2π π k =1

a34 =

∞ 1  λ22 (ρ 1 , k )λ12 (ρ 2 , k )ρ 1k ρ 2k sin [k (θ2 − θ1 − ψ)] π k =1

a41 =

∞ 1  λ21 (ρ 1 , k )λ21 (ρ 2 , k )ρ 1k ρ 2k cos [k (θ2 − θ1 − ψ)] π k =1

a42 = −

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∞ 1  λ21 (ρ 1 , k )λ22 (ρ 2 , k )ρ 1k ρ 2k sin [k (θ2 − θ1 − ψ)] π k =1

a43 =

∞ 1  λ22 (ρ 1 , k )λ21 (ρ 2 , k )ρ 1k ρ 2k sin [k (θ2 − θ1 − ψ)] π k =1

a44 =

∞ ρ1 ρ2 1  + λ22 (ρ 1 , k )λ22 (ρ 2 , k )ρ 1k ρ 2k cos [k (θ2 − θ1 − ψ)] . 2π π k =1

Thus we see from these formulae that the entries of the matrix A depend on the difference θ = θ2 − θ1 ; hence the correlation tensor B u also depends on θ = θ2 − θ1 , and from (13.72), (13.73) we arrive at the desired convolution representation 2π

ˆ u (ρ 1 , ρ 2 ; θ) = B

ˆ g (ψ)dψ . A ( ρ 1 , ρ 2 ; θ − ψ )B 0

Note that if the boundary correlation tensor B g is given by its spectral expansion, we can express the correlation tensor of the solution through the spectra. For instance, assuming that the spectral tensor is real valued, so that B g,ij(φ − φ) =

∞ f ij (0) 1  + f ij (k ) cos k (φ − φ ) , 2π π k =1

i, j = 1, 2 ,

we can derive a general formula for the covariance tensor by substituting this expan­ sion in (13.71). After routine evaluations we obtain the general formulae B11 =

∞  f11 (0)ρ 1 ρ 2 1  k k c s + ρ ρ Λ cos[k (θ2 − θ1 )] + Λ11 sin[k (θ2 − θ1 )] , 2π π k=1 1 2 11

B12 =

∞  1  k k c s ρ ρ Λ cos[k (θ2 − θ1 )] + Λ12 sin[k (θ2 − θ1 )] , π k=1 1 2 12

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296 | 13 Boundary value problems with random boundary conditions

B21 =

∞  1  k k c s ρ ρ Λ cos[k (θ2 − θ1 )] + Λ21 sin[k (θ2 − θ1 )] , π k=1 1 2 21

B22 =

∞  f22 (0)ρ 1 ρ 2 1  k k c s + ρ ρ Λ cos[k (θ2 − θ1 )] + Λ22 sin[k (θ2 − θ1 )] , 2π π k=1 1 2 22

where c = f11 λ111 λ211 + f22 λ112 λ212 , Λ11

s Λ11 = f21 λ111 λ212 − f12 λ112 λ211 ,

c = f21 λ112 λ221 + f12 λ111 λ222 , Λ12

s Λ12 = f11 λ111 λ221 − f22 λ112 λ222 ,

c Λ21 = f21 λ122 λ211 + f12 λ121 λ212 ,

s Λ21 = f22 λ122 λ212 − f11 λ121 λ211 ,

c Λ22 = f22 λ122 λ222 + f11 λ121 λ221 ,

s Λ22 = f21 λ122 λ221 − f12 λ121 λ222 .

Here we use the notations λ m ij = λ ij ( ρ m , k ), m = 1, 2. Remark 13.4. Note that using the relation between the vectors in polar and rectangu­ lar coordinates,



u 1 (r, θ) u ρ (r, θ) = Rθ u 2 (r, θ) u θ (r, θ , we can easily relate the desired statistical characteristics in these two coordinate sys­ tems. For example, the covariance tensors are related as follows: B(u1 ,u2) (ρ 1 , ρ 2 ; θ1 , θ2 ) = Rθ 1 B(u r ,u θ ) (ρ 1 , ρ 2 ; θ1 , θ2 )RTθ 2

(13.74)

The K-L expansion in the rectangular coordinates is also obtained directly from the K-L expansion of the random field in the polar coordinates on the basis that the ˜ polar . ˜ rectangular = Rθ h eigenfunctions are related by hrectangular = Rθ hpolar and h Let us write down here relation (13.74) in detail. We denote the entries of the co­ pol variance matrix B(u1 ,u2) by Brec ij and the entries of the covariance matrix B ( u r ,u θ ) by B ij . From (13.74) we obtain pol

pol

pol

pol

pol

pol

pol

pol

Brec 11 = cos θ 1 cos θ 2 B 11 − cos θ 1 sin θ 2 B 12 − sin θ 1 cos θ 2 B 21 + sin θ 1 sin θ 2 B 22 , Brec 12 = cos θ 1 sin θ 2 B 11 + cos θ 1 cos θ 2 B 12 − sin θ 1 sin θ 2 B 21 − sin θ 1 cos θ 2 B 22 , pol

pol

pol

pol

pol

pol

pol

pol

Brec 21 = sin θ 1 cos θ 2 B 11 − sin θ 1 sin θ 2 B 12 + cos θ 1 cos θ 2 B 21 − cos θ 1 sin θ 2 B 22 , Brec 22 = sin θ 1 sin θ 2 B 11 + sin θ 1 cos θ 2 B 12 + sin θ 2 cos θ 1 B 21 + cos θ 1 cos θ 2 B 22 . This representation clearly shows that the property that the covariance functions pol B ij all depend only on the angle difference θ2 − θ1 does not generally hold for the rec covariance functions Brec ij . It is however seen that B ij will depend only on θ 2 − θ 1 if pol

pol

(u r , u θ ) is homogeneous, and B11 = B22 .

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13.6 Response of an elastic 3D half-space to random excitations

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13.6 Response of an elastic 3D half-space to random correlated displacement perturbations on the boundary 13.6.1 Introduction In this chapter, we deal with the random boundary value problem for the Lamé equa­ tion governing an elastic deformation of a half-space under random perturbations of the displacements on the free boundary plane under the condition of no shearing forces. It should be noted that for the half-space, the Poisson formula is known for the case when all the forces are prescribed on the boundary (e.g. see [5]), or all the dis­ placements (e.g. see [96]). Here we deal with mixed boundary conditions when shear­ ing forces and the normal displacements are prescribed on the boundary. It should be mentioned that the explicit representations of the elastic displacement fields through the Green functions is a difficult problem which attracts much attention. In particular, we refer to [214] where an isotropic elastic half-space was treated, but the Green func­ tions were expressed through Fourier integrals which are hard for numerical calcula­ tions even in the isotropic case. Nevertheless, the author was able to give an analysis of the case when a periodic force distribution was applied. We also mention an applica­ tion of explicit formulae for displacements, strains, and tilts due to inclined shear and tensile faults in a half-space for both points and finite rectangular sources [129]. This analysis can be extended to stochastically distributed sources using our approach. We analyze two practically interesting cases: (1) white noise perturbations and (2) ex­ ponential–cosine perturbations along the longitudinal direction. In both cases, we succeeded to derive exact representations of the correlation tensor of the displace­ ments u (r). The case of excitations with the exponential–cosine correlation function is often used in the statistical physics when dealing with a disorder in crystals; in par­ ticular, we mention the x-diffraction peak analysis of dislocations in crystals [25; 77]. Here we stress the following important point. The technique developed in this chapter provides not only exact representations for the correlation tensor of the displacement field, but also a simple and constructive representation for the K-L expansion of the so­ lution to the considered boundary value problem. Using this representation, it is pos­ sible to calculate, by the Monte Carlo method, arbitrary functionals of the solutions. As an important example we mention the average exp[i(u (r1 ) − u (r2 ))] which rep­ resents the X-ray diffraction signals from correlated dislocations in crystals, see [77] and [78].

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13.6.2 System of Lamé equations governing an elastic half-space with no tangential surface forces Let us consider an elastic half-space with a free surface Γ = {(x, y, z) : z = 0}. We are interested in the response of the displacement vector u to random excitations of the displacement component u 3 on the surface Γ provided there are no tangential surface forces. This problem can be treated mathematically as a third boundary value problem of elastostatics governed by the system of Lamé equations in the domain D+ ⊂ R3 , the upper half-space with the boundary Γ [96]: μΔu(x) + (λ + μ ) grad div u(x) = 0 , with boundary conditions   ∂u 1 ∂u 3   + = 0, σ 13 |z=0 = μ ∂z ∂x z=0 u 3 (x ) = g3 (x ),



σ 23 |z=0 = μ

x ∈ D+ ,

∂u 2 ∂u 3 + ∂z ∂y

(13.75)

    

x ∈ Γ = ∂D+ ,

= 0 , (13.76) z =0

(13.77)

where u(x) = (u 1 (x, y, z), . . . , u 3 (x, y, z)) is a column vector of displacements, σ ij are the strain components, and g3 (x ) = g3 (x , y ) is the third component of dis­ placement vector prescribed on the boundary. The coefficients λ and μ are the Lamé constants of elasticity. Thus there are no shearing loads, i.e. the tangential forces on the boundary vanish (the boundary conditions (13.76)), and the normal displacement u 3 is prescribed on the boundary (the boundary condition (13.77)). It should be noted that for the half-space, an explicit integral formula is known for the case when all the forces are prescribed on the boundary (e.g. see [5]), or all the displacements (e.g. see [96]). Here we deal with the mixed boundary conditions when the tangential forces and the normal dis­ placement are prescribed on the boundary. The integral representation of the solution (Poisson formula) is derived in Appendix A. T

13.6.2.1 Poisson formula for the upper half-space The Poisson formula for the problem (13.75)–(13.77) has the form (see Appendix A) ∞ ∞ u(x, y, z) = K (x − x , y − y , z)g3 (x , y )dx dy , (13.78) −∞ −∞

where the kernel K is given by

and

⎛ ⎞ βz 2 (x − x )( r2 − 𝛾) ⎟ 1 ⎜ ⎜(y − y )( βz2 − 𝛾)⎟ , K ( x − x , y − y , z) = 2 ⎝ ⎠ 3 r 2πr βz 2 β z( r2 + 1 − 3 )

9 r = ( x − x )2 + ( y − y )2 + z 2 ,

β=

3α , α+1

𝛾=

1 , 1+α

α = 1 + λ/ μ .

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13.6 Response of an elastic 3D half-space to random excitations

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13.6.3 Stochastic boundary value problem: correlation tensor Let us suppose that the boundary displacement g3 (x, y) is a homogeneous Gaussian random field. With no loss in generality, assume that g = 0, so that u(x, y, z) is also a Gaussian random field with u = 0. This immediately follows from formula (13.78), since any linear transformation of a Gaussian random variable leads again to a Gaussian random variable. Hence the random field u(x, y, z) is uniquely defined by its correlation tensor. We include in our considerations also the case of white noise; the solutions being gen­ eralized random fields can be treated rigorously in relevant functional spaces (e.g. see [68; 69; 151; 83]). By the Poisson formula (13.78) for u, the correlation tensor B u (x1 ; x2 ) for the dis­ placements can be written as follows: (13.79) B u (x1 ; x2 ) = u(x1 , y1 , z1 ) ⊗ u(x2 , y2 , z2 )  = K (x1 − x 1 , y1 − y 1 , z1 ) ⊗ K (x2 − x 2 , y2 − y 2 , z2 ) B g (x 1 ; x 2 )dx 1 dx 2 . R4

We use here the notation ⊗ for the direct product of the vectors u(x1 , y1 , z1 ) and u(x2 , y2 , z2 ), and B g (x 1 ; x 2 ) for the correlation function of the random field g3 B g (x 1 ; x 2 ) = B g (x 1 , y 1 ; x 2 , y 2 ) =  g3 (x 1 , y 1 )g3 (x 2 , y 2 ) . Let us first consider the case when g3 is a Gaussian white noise. This implies that B g (x 1 ; x 2 ) = δ(x 1 − x 2 )δ(y 1 − y 2 ) .

(13.80)

Here δ(·) is the Dirac δ-function. In the next statement we give an explicit form of the correlation tensor of the so­ lution u. Theorem 13.11. The correlation tensor of the random field u(x, y, z) under the white noise boundary conditions has the following explicit form ⎛ ⎞ τ τ τ2 A − B Rx2 −B Rx 2 y 0⎟ ⎜ τ τ 𝛾2 ⎜ ⎟ τ2 ⎜ −B τ x τ2 y Bu = A − B Ry2 0⎟ 2 R ⎝ ⎠ 2πR τ τ τ 0 0 0 ⎛ ⎞ τ τ τ2 ( z −𝛾z ) τ z τ z ( r2x − 31 ) z xr2 y − βx + 1 𝛾r22 x ⎟ β𝛾 ⎜ τ 2y ⎜ τx τy τy ( z 1 −𝛾z 2 ) τ y z ⎟ 1 ⎜ ⎟ + z z ( − ) − + 2 2 2 r r 3 β 𝛾r ⎠ 2πr3 ⎝ τy ( z 2 −𝛾z 1 ) τ y z ( z 2 −𝛾z 1 ) τ x z τx z z2 1 1 + + ( − + ) 2 2 2 β 𝛾r β 𝛾r 𝛾 r 3 β ⎛ ⎞ τx τy τ 2x z2 τ x z (1 − 5 r 2 ) −5z r2 5 r2 − τ x ⎟ β 2 z1 z2 ⎜ ⎜ ⎟ τ 2y τx τy z2 τ ⎜ , (13.81) + −5z r2 z (1 − 5 r 2 ) 5 r 2 y − τ y ⎟ ⎠ 6πr5 ⎝ 2 5z τ y 5R 2τ 5z2 τ x − r 2 + τ x − r 2 + τ y z (2 − r 2 )

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300 | 13 Boundary value problems with random boundary conditions where A =1−

z , r

B=

 1  2 2 2z ( r − z ) + R ( 2r − 3z ) , τ r3

z = z1 + z2 ,

and 9 r = τ2x + τ2y + z 2 ,

τ x = x1 − x2 ,

9 R τ = τ2x + τ2y .

τ y = y1 − y2 ,

Proof. Substituting (13.80) in (13.79) yields ∞ ∞

B u (x1 ; x2 ) =

K (x1 − x 1 , y1 − y 1 , z1 ) ⊗ K (x2 − x 1 , y2 − y 1 , z2 )dx 1 dy 1 .

−∞ −∞

We use the Fourier transformation here to perform explicit evaluation of the integrals. Let us take a change of variables w x = x 1 − x2 and w y = y 1 − y2 , and introduce further new variables τ x = x1 − x2 , τ y = y1 − y2 . Then B u ( τ x , τ y ; z1 , z2 ) ∞ ∞ = K (τ x − w x , τ y − w y , z1 ) ⊗ K (−w x , −w y , z2 )dw x dw y . −∞ −∞

The last formula has a convolution form and can be written shortly as B u (τ x , τ y ; z1 , z2 ) = K (τ x , τ y , z1 ) ∗ K (−w x , −w y , z2 ) . The Fourier transform property for convolutions yields F −1 [B u ] = 2πF −1 [K (τ x , τ y , z1 )] ⊗ F −1 [K (−w x , −w y , z2 )] . So we have to find the inverse transforms F −1 [K ]. Using the Fourier transform formulae (13.102)–(13.104) from Appendix B we arrive at 2πF −1 [K (τ x , τ y , z1 )] = e−z1 where

Then F −1 [ B u ] = S u =

ξ x2 + ξ y2

G( ξ x , ξ y , z1 ) ,

β⎞ 2 − ıξ x z 3 ξ + ξ x y ⎟ ⎜ β⎟ ⎜ √ıξ y 𝛾 ⎜ ξ 2 +ξ 2 − ıξ y z 3 ⎟ ⎠ ⎝ x y9 β 1 + 3 z ξ x2 + ξ y2

⎛

G( ξ x , ξ y , z) =

√

√ıξ2x 𝛾

.

1 −(z1 +z2 )√ξ x2 +ξ y2 S ( ξ x , ξ y ; z1 , z2 ) e 2π

(13.82)

(13.83)

where S ( ξ x , ξ y ; z1 , z2 ) = G( ξ x , ξ y , z1 ) ⊗ G∗ ( ξ x , ξ y , z2 ) , and the star sign stands for the complex conjugate transpose.

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Note that we have taken the inverse Fourier transform of the correlation tensor with respect to the variables x, y. It means that we get a partial spectral tensor S u . Indeed, by definition S u (ξ x , ξ y , z1 , z2 ) = F −1 [B u (τ x , τ y , z1 , z2 )] ∞ ∞ 1 = e−i(ξ x τ x +ξ y τ y ) B u (τ x , τ y , z1 , z2 )dτ x dτ y , 2π −∞ −∞

hence S u ( ξ x , ξ y , z1 , z2 ) =

1 −√ξ x2 +ξ y2 (z1 +z2 ) G( ξ x , ξ y , z1 ) ⊗ G∗ ( ξ x , ξ y , z2 ) . e 2π

(13.84)

To find the correlation tensor B u it remains to invert the Fourier transform. Let us rewrite (13.83) explicitly as F −1 [ B u ] =

1 −(z1 +z2 )√ξ x2 +ξ y2 e 2π ⎛ 𝛾

ξ 2( −

βz1

𝛾

)( −

βz2

)

3 ρ 3 ⎜ x ρ ⎜ × ⎜ ξ x ξ y ( ρ𝛾 − βz3 1 )( ρ𝛾 − βz3 2 ) ⎝ 𝛾 −ıξ x ( ρ

−

βz2 3

)(1 +

βz1 ρ 3 )

𝛾

ξx ξy ( ρ − 𝛾

ξ y2 ( ρ − 𝛾 −ıξ y ( ρ

βz1 3

𝛾 βz1 3 )( ρ

−

𝛾

)( ρ − −

βz2 3 )

βz2 3

)

βz2 βz1 ρ 3 )(1 + 3 )

𝛾

ıξ x ( ρ − 𝛾

ıξ y ( ρ − (1 +

(13.85) ⎞

βz1 βz2 ρ 3 )(1 + 3 )⎟

βz1 3

)(1 +

βz2 ρ ⎟ ⎟ 3 )⎠

,

βz1 ρ βz2 ρ 3 )(1 + 3 )

9 where ρ = ξ x2 + ξ y2 . In the polar coordinates for the variables τ x , τ y as τ x = R τ cos ϕ, τ y = R τ sin ϕ, and ξ x = ρ cos ψ, ξ y = ρ sin ψ we obtain

B u (R τ , ϕ, z1 , z2 )  1 = e−ρ(z1 +z2) G(ρ, ψ, z1 ) ⊗ G∗ (ρ, ψ, z2 )e ıρR τ cos(ψ−ϕ) ρdρdψ . 4π2

(13.86)

R2

This integral is evaluated explicitly using of the formulae (13.111)–(13.117) (derived in Appendix C) which results in the final explicit formula (13.81).

13.6.4 Spectral representations for partially homogeneous random fields As seen from the previous section, we deal with the random field u(x, y, z) which is homogeneous with respect to the horizontal variables x, y, and is inhomogeneous with respect to the vertical coordinate z. Random fields with this property are called partially homogeneous random fields [160]; note that we deal with this kind of fields throughout this chapter. Having an explicit form of the correlation tensor and its spec­ tral tensor in the form (13.84), we can simulate the partially homogeneous random

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302 | 13 Boundary value problems with random boundary conditions field u(x) as described in [160; 193]. This leads to the simulation formula (where the case (k, m) = (0, 0) should be excluded): ∞ √  1 2 2 e−πz (k/R1) +(m/R2) u(x, y, z) ≈  2 R1 R2 k,m=−∞   kx my kx my × ζ k,m cos π ( + ) + η k,m sin π ( + ) R1 R2 R1 R2

(13.87)

where ζ k,m and η k,m are two families of Gaussian column vectors C km )T , ζ k,m = (η km A km , η km B km , ζ km

η k,m = (−ζ km A km , −ζ km B km , η km C km )T

where ζ km and η km are two independent families of standard independent Gaussian random variables whose lengths are defined by

k𝛾 βkz  − , 2 2 3R1 R1 ( k / R1 ) + ( m / R2 ) m𝛾 βmz =  − , 3R2 R 2 ( k / R 1 )2 + ( m / R 2 )2 9 βz =1+ ( k / R 1 )2 + ( m / R 2 )2 . 3

A k,m = B km C km

This model has a correlation tensor which is an approximation to the original cor­ relation tensor B u (similar to (13.87), the case (k, m) = (0, 0) is excluded in the sum­ mation): B u ( τ x , τ y , z1 , z2 ) ≈

∞ √  1 2 2 e−π(z1 +z2 ) (k/R1) +(m/R2) (13.88) 4R1 R2 k,m=−∞ . mτ y mτ y kτ x kτ x × S cos π ( + ) −  S sin π ( + ) , R1 R2 R1 R2

where  and  stand for the real and imaginary part, respectively, and we recall that the matrix S is S = G( ξ x , ξ y , z1 ) ⊗ G∗ ( ξ x , ξ y , z2 ) . The matrix S is explicitly given in (13.85), and the entries of S in (13.88) are taken for ξ x = πk /R1 , ξ y = πm/R2 . To check that the model (13.87) has the correlation tensor (13.88) we take the average u(x1 , y1 , z1 ) ⊗ u(x2 , y2 , z2 ) where u is substituted from (13.87). So it remains to prove that the correlation tensor approximated by (13.88) indeed tends to the exact correlation tensor B u . The idea of the proof is easy: just by defini­ tion, the correlation tensor is obtained as a 2D Fourier transform of the spectral ten­ sor which in our case is given by (13.85). Notice that the entries in this matrix which are real-valued (all entries except for S13 , S23 and S31 , S32 which are pure imaginary valued) are symmetric both in the ξ x and ξ y coordinates, while the pure imaginary

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entries, S13 , S23 , and S31 , S32 are antisymmetric in these coordinates. Taking this into account we obtain (13.88). In practical calculations, we have to use truncated versions of the series (13.87) and (13.88), which give rise to an error ε. This error cannot be easily estimated, but it is closely related to the smoothness of the covariance kernel and to the ratio between the length of process a and the correlation length L, (e.g. see [106; 180]). So for expo­ a nential correlations, the upper bound of ε is known to be scaled as ε ∼ nL where n is the number of retained terms.

13.6.5 Displacement correlations for the white noise excitations Let us present the correlations B ij (x) = (B u (x))ij , i, j = 1, 2, 3, as functions of the longitudinal coordinate x = x1 − x2 . In Figure 13.11, the correlation functions B ii (x), i = 1, 3, B31 (x) and B32 (x) are given for different values of the elasticity constant, α = 2 (left panel) and α = 20 (right panel), taken for z1 = 0.5, z2 = 1.5, and y1 − y2 = 0.5. It is seen that the correlation length of displacement fluctuations in the vertical direction is considerably smaller than those of the horizontal directions. In turn, the horizontal correlations B11 , B31 , as well as B32 behave differently. For large values of α (right panel), the negative correlations in B11 and B32 are larger. It should be noted that the form of the correlation functions is practically not changing with a further increase of α (we have made calculations till α = 200). This asymptotic limit corresponds to the Stokes equation (see [156]), so the correlations behave similar to the case of viscous flow correlations. 0.14

0.1 B 11 B31 B32 B33

0.08 0.06

B 11 B31 B32 B33

0.12 0.1 0.08 0.06

0.04

0.04 0.02

0.02

0 0 −0.02

−0.02 0

2

4 6 8 10 Longitudinal coordinate X

−0.04 12 0

2

4 6 8 10 Longitudinal coordinate X

12

Figure 13.11. Correlation functions B11 ( x), B31 ( x ), B32 ( x), and B33 ( x ), for the white noise boundary excitations, for different values of the elasticity constant α: α = 2 (left panel), α = 20 (right panel), for z1 = 0.5, z 2 = 1.5, y1 − y2 = 0.5.

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304 | 13 Boundary value problems with random boundary conditions 12

1.6 1.4

B 11, z=0.2

1.2

B31, z=0.2

B33, z=0.2 B33, z=0.15 B33, z=0.1

10

B 11, z=0.15

1

8

B31, z=0.15

0.8

B 11, z=0.1

0.6

6

B31, z=0.1

0.4 4

0.2 0

2

−0.2 0

−0.4 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Longitudinal coordinate X

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

Longitudinal coordinate X

Figure 13.12. Correlation functions B11 ( x ) and B31 ( x), for the white noise boundary excitations, for three values of height z = 0.1, z = 0.15, and z = 0.2 (left panel), and the correlation function B33 ( x ), for the same heights (right panel). In both cases, y1 = y2 .

An important question in the correlation analysis of the elastic displacements is how the correlations are changed with the increase of the distance to the bound­ ary where the dislocations are randomly distributed. In Figure 13.12, the correlation functions B11 (x) and B31 (x) (left panel) are presented for three different values of the height z: z = 0.1, 0.15, and z = 0.2, the elasticity constant α was taken as α = 2. First, it is clearly seen that with the height, the intensity of fluctuations is decreasing. Second conclusion concerns the correlation length, here the dependence is converse: the correlation length is increasing with the height. Both these conclusions are also true for the correlation function B33 (x) (right panel) which describes the correlations of the vertical displacement u 3 . It is interesting to compare the horizontal displacement correlation B11 with B31 . The function B31 (x) (left panel) at the height z = 0.1 has two characteristic correlation lengths: it increases rapidly in the range {x < 0.1}, then, it decays rapidly, reaching the zero level, and enters a negative region, coming back to zero around of x = 1, and finally tends to zero as x → ∞. Note that the correlation function B11 has no increas­ ing part for small values x: it starts with a rapid decrease, but also has a pronounced negative region. The function B33 (x) is rapidly decreasing with x and has only one characteristic correlation length. Note that the same behavior remains true for other heights, but the picture is less pronounced there (see the curves for z = 0.15 and z = 0.2 in Figure 13.12, right panel).

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Correlation function B 11(x,y)

13.6 Response of an elastic 3D half-space to random excitations

|

0.15

0.15 0.1

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0.15

0.1

0.1 B 11(x,y)

0.05

0.1

0.05

0.5

0

0.05

0.05 0

−0.05

0 y

0 −0.1 −0.5

0 x

0.5 0.5

0 y

−0.5

−0.05

0.5

0 −0.05

0 x

−0.5 −0.5

Figure 13.13. Correlation B11 ( x, y), as a function of x and y, at the height z = 0.2, for two different view angles.

It is interesting to note that the behavior of the correlation functions B31 and B11 is quite different from that described in [193] where the elastic half-space had different boundary conditions, namely, all the displacements were prescribed on the boundary. Thus the zero shearing forces are responsible for the appearance of two characteristic correlation lengths of B31 and a negative region of B11 . More details and features are seen in Figure 13.13 where we show the correlation function as a surface B11 (x, y), for two different view angles. Finally we mention that the dependence of B33 on the vertical coordinate z (not shown here) is similar to the dependence found in the case of the first boundary conditions presented in [193]; it is monotonically decreasing with the height, and has a singularity of z−2 near the boundary, as z → 0.

13.6.6 Homogeneous excitations Let us consider the boundary value problem (13.75)–(13.77) where g3 (x, y) is a homo­ geneous zero mean Gaussian random field with the correlation function B g ,



B g (x 1 − x 2 ) = exp−a| x1 −x2| cos [b (x 1 − x 2 )] .

(13.89)

Note that this correlation function does not depend on the transverse coordinate y which implies that the correlations B g at two arbitrary points along the lines x = const are all equal to 1. The constant a characterizes the global decorrelation decay along x, and b is the periodicity length characterizing the local correlations. We have used this type of correlation functions to describe the defect and dislocation position distribu­ tions in crystals [77], see also [25; 221]. The correlations of displacements caused by dislocations in crystals may have more complicated form, e.g. see [13]. Then, we can derive, following the scheme given in the proof of Theorem 13.12, a relevant general representation since it also reduces to a convolution with the “white noise” solution derived above in Theorem 13.11.

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306 | 13 Boundary value problems with random boundary conditions Theorem 13.12. The correlation tensor of the random field u(x, y, z) with homogeneous excitations on the boundary described by the correlation function (13.89) has the follow­ ing nonzero entries: {B u }11 (τ x , z1 , z2 )

z = π (1 + α )2

∞ $ −∞

2α 2 z1 z2 (z 2 − 3τ2x ) α (1 + α )(z2 − τ2x ) 1 + − 2 τ 2x + z (τ 2x + z 2 )3 (τ 2x + z 2 )2

× exp−a| τ x | cos bτ x dτ x ,

%

(13.90)

{B u }33 (τ x , z1 , z2 ) % ∞ $ 2α 2 z1 z2 (z 2 − 3τ2x ) α (z2 − τ2x ) z 1 + + = π τ 2x + z2 (1 + α )2 (τ 2x + z 2 )3 (1 + α )(τ 2x + z 2 )2 −∞

× exp−a| τ x | cos bτ x dτ x ,

(13.91)

where τ x = τ x − τ x , τ x = x1 − x2 , τ x = x 1 − x 2 . Proof. Since the correlation function B g (x 1 − x 2 ) depends on the variables x 1 , x 2 we integrate representation (13.79) over y 1 , y 2 , (here we use formulae (13.123), see Ap­ pendix D). Then,  B u (x1 ; x2 ) = K g (x1 − x 1 , z1 ) ⊗ K g (x2 − x 2 , z2 )B g (x1 − x 2 )dx 1 dx 2 , R2

(13.92) where the kernel K g is ⎞ −𝛾 ⎜ ⎟ 1 ⎜ ⎟. 0 K g ( x − x , z) = ⎝ ⎠ 2 2 π((x − x ) + z ) 2βz 2 β z( 3((x−x )2 +z2 ) + 1 − 3 ) ⎛

(x − x )



2βz 2 3(( x − x )2 + z 2 )

In the new variables τ x = x 1 − x 2 we get  B u (x1 ; x2 ) = K g (x1 − x 1 , z1 ) ⊗ K g (x2 − x 1 + τ x , z2 )B g (τ x )dx 1 dτ x , R2

and then, we tern to variables τ x = x1 − x2 and xˆ1 = x1 − x 1 . This yields ∞

B u ( τ x ; z1 , z2 ) =

∞

B g (τ x )dτ x −∞

hence

K g (xˆ1 , z1 ) ⊗ K g (xˆ1 − τ x + τ x , z2 )d xˆ1 , −∞

∞

B u ( τ x ; z1 , z2 ) =

I (τ x − τ x , z1 , z2 )B g (τ x )dτ x ,

(13.93)

−∞

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where ∞

I ( τ x − τ x , z1 , z2 ) =

K g (xˆ1 , z1 ) ⊗ K g (xˆ1 − τ, z2 )d xˆ1 ,

τ = τ x − τ x .

−∞

Thus I is represented in the convolution form I (τ, z1 , z2 ) = K g (xˆ1 , z1 ) ∗ K g (−τ, z2 ) . The Fourier transform property for convolutions yields F −1 [I ] = F −1 [K g (xˆ1 , z1 )] ⊗ F −1 [K g (−τ, z2 )] . So we have to find the inverse transforms F −1 [K g ]. Using the Fourier transform formulae (13.118)–(13.121) from Appendix D we arrive ⎛

at

⎜ F −1 [K g (xˆ1 , z1 )] = e−z1 | ξ x | ⎜ ⎝

ısignξ x 𝛾 − ıξ x 0 β| ξ | z 1 + 3x 1

⎞

z1 β 3 ⎟

⎟. ⎠

Hence ⎛

f11 ⎜ F −1 [I ] = e−(z1 +z2 )| ξ | ⎝ 0 f31

0 0 0

⎞ f13 ⎟ 0⎠ , f33

where |ξ |β𝛾 β2 ( z2 + z1 ) + ξ 2 z1 z2 , 3 9 β β2 = ı(𝛾signξ + ξ (𝛾z2 − z1 ) − ξ |ξ |z1 z2 ) , 3 9 β β2 2 = 1 + |ξ |(z1 + z2 ) + ξ z1 z2 . 3 9

f11 = 𝛾2 − f13 f33

f31 = −f13 ,

After taking the Fourier transform (see (13.118)–(13.121)) we get ⎛

I11 ⎜ I ( τ x − τ x , z1 , z2 ) = ⎝ 0 I31

0 0 0

⎞ I13 ⎟ 0⎠, I33

where I11 =

𝛾2 z β𝛾z(τ2x − z 2 ) 2β 2 zz2 z1 (z 2 − 3τ2x ) + , 2 + 2 π(τ x + z ) 3π(τ2x + z 2 )2 9π(τ2x + z2 )3

I13 = −

𝛾τ x z 2βz τ x (𝛾z2 − z1 ) 2β 2 z2 z1 (3z 2 − τ2x )τ x − + , 2 π(τ2x + z ) 3π(τ2x + z 2 )2 9π(τ2x + z 2 )3

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308 | 13 Boundary value problems with random boundary conditions 0.06 B33, a=1, α=2 B33, a=0.1, α=2 B33, a=0.5, α=2

0.15

B 11, a=1, α=20 B 11, a=0.5, α=20 B 11, a=0.1, α=20

0.04

0.1

0.02

0.05

0 −0.02

0

−0.04 −0.05 0

5

10

15

20

25

30

−0.06 0

Longitudinal coordinate X

5

10

15

20

25

30

Longitudinal coordinate X

Figure 13.14. The case of exponential–cosine boundary excitations: the longitudinal correlation function B33 ( x ) for different values of the decorrelation parameter a (left panel), and fixed value b = 1. The correlation function B11 ( x ) for the elasticity constant α = 20 and the same values of the decorrelation parameter a (right panel), for z1 = 1.5, z 2 = 2.5, y1 − y2 = 0.5.

I31 =

𝛾τ x z 2βz τ x (𝛾z1 − z2 ) 2β 2 z2 z1 (3z 2 − τ2x )τ x + + , π(τ2x + z2 ) 3π(τ2x + z 2 )2 9π(τ2x + z 2 )3

I33 =

z βz(z 2 − τ2x ) 2β 2 zz2 z1 (z2 − 3τ2x ) . 2 + 2 2 + 2 2 π(τ x + z ) 3π(τ x + z ) 9π(τ2x + z 2 )3

So substituting these explicit expressions and the correlation function (13.89) in (13.90) and (13.91) we can calculate the correlation tensor B u . Since the function B g is even in y, only the functions {B u }11 and {B u }33 are not equal to zero. In Figure 13.14 we present the correlations B33 (x) (left panel) and B11 (x) (right panel) as functions of the longitudinal coordinate x, for fixed values z1 = 1.5, z2 = 2.5, y1 − y2 = 0.5. All results are presented for the fixed value b = 1, and different values of a and elasticity constant α. It is clearly seen that for a = 1 (solid curve in the left panel), the correlations B33 (x) are positive, and decrease with x, while the correlations B11 (x) have a negative minimum at x ∼ 4 (right panel). For smaller values of the decorrela­ tion parameter a (a = 0.1 and a = 0.5), all correlations have an oscillating form. An increase in the elastic constant α from 2 to 20 does not affect much the oscillations of the correlation functions.

13.6.7 Conclusions and discussion Response of an elastic half-space to random excitations of displacements on the boundary is analyzed under the condition of no shearing forces. To make possible the explicit evaluation of the correlation function of the displacements we have derived the Poisson formula for the Lamé equation for the isotropically elastic half-space

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under the following boundary conditions: there are no shearing forces, while in the normal direction, the displacement is a prescribed function. This exact formula is of independent interest in the elasticity theory since it presents an explicit representa­ tion of the displacement vector for an elastic half-space with zero shearing forces and arbitrarily prescribed normal displacements. The results are derived for general random fluctuations of displacements pre­ scribed on the boundary, but we consider in detail also two important cases of ran­ dom boundary displacement excitations: (1) the white noise excitations which are of fundamental importance in the analysis of any random fluctuations, and are used in particular to simulate the heat fluctuations, and (2) the periodically correlated excitations, which are commonly used to model the partially ordered defects on the boundary, for instance, dislocations in crystals, whose positions are governed by an exponential–cosine-type correlation function. We present calculations of correlation functions which, first, validate the derived formulae, and second, show the character of the correlations transformation when moving away from the boundary inside the half-space: the intensity of fluctuations is decreasing while the characteristic cor­ relation length is rapidly increasing. The calculations also show the impact of the elasticity constants on the correlation functions. Correlations of the displacements in the case of exponential–cosine correlations of dislocations in crystals are calculated. The structure of these correlations is important both for theoretical and numeri­ cal studies of the X-ray diffraction peaks as suggested in our recent study [77]; see also [18] and [25]. Other applications in crystallography and geophysics are possible, see e.g. [214; 129].

13.6.8 Appendix A: the Poisson formula The Poisson formula is a special case of the Green-type integral representations (e.g. see [155]). It should be noted that for the half-space, the Poisson formula is known for the case when all the forces are prescribed on the boundary (e.g. see [5]), or all the displacements (e.g. see [96]). Here we deal with mixed boundary conditions when tangential forces and the normal displacement are prescribed on the boundary. We seek the solution to boundary value problem (13.75)–(13.77) in the form (see [96]) 1 u(x, y, z) = 2π

∞ ∞

K (x − x , y − y , z)g3 (x , y )dx dy ,

(13.94)

−∞ −∞

where kernel K has the form ⎛ β (x−x )z2 1

K ( x − x , y − y , z) =

5 ⎜ β1 (y−r y )z2 −⎜ ⎝ r5 β3 z3 r5

⎞

𝛾1 ( x − x ) r3 𝛾 (y−y ) ⎟ + 1 r3 ⎟ ⎠ 𝛾z + r33

+

(13.95)

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310 | 13 Boundary value problems with random boundary conditions and

9 r = ( x − x )2 + ( y − y )2 + z 2 .

The values of the coefficients 𝛾1 , 𝛾3 , β 1 , and β 3 can be found from equation (13.75) and the boundary conditions (13.76). To satisfy the equation Δ∗ u(x, y, z) = 0, we have the first relations between the constants to be found: 2β 1 β 1 = β 3 , 𝛾3 = 𝛾1 + (13.96) , α = 1 + λ/ μ . 3α Let us consider the third boundary condition u 3 (x ) = g3 (x )

x ∈ Γ = ∂D+ .

It is easy to see that on the plane z = 0, the third component of K (x − x , y − y , z) in the integral (13.94) equals to zero except for the point r = 0. Since the function u 3 (x ) is continuous till the boundary, and 1 2π 1 2π

∞ ∞ −∞ −∞

∞ ∞ −∞ −∞

z 1 dx dy = r3 2π z3 1 dx dy = r5 2π

∞ 2π 0 0

∞ 2π 0 0

zρ dρdφ = 1 , ( ρ 2 + z 2 ) 3 /2

(13.97)

z3 ρ 1 dρdφ = , (ρ 2 + z2 )5/2 3

(13.98)

where x − x = ρ cos φ, y − y = ρ sin φ, we get 𝛾3 + From the boundary condition



σ 13 |x=x = μ

β1 = 1. 3

(13.99)

 ∂u 1 ∂u 3   + =0 ∂z ∂x x=x

we can find the next relation for the unknown constants. Using the integral represen­ tations for the displacements (13.94) we get

∞ ∞ (x − x )z 1 10β 1 (x − x )z3 ( 2β − 3𝛾 − 3𝛾 ) − g3 (x , y )dx dy . σ 13 (x) = 1 3 1 2π r5 r7 −∞ −∞

From this, and taking into account that ∞ 0

ρ2 1 dρ = 2 , 2 2 5 / 2 (ρ + z ) 3z

∞ 0

(ρ2

ρ2 2 dρ = 2 7 / 2 +z ) 15z4

we obtain the last relation 2β 1 − 3𝛾3 − 3𝛾1 = 2β 1 .

(13.100)

It should be mentioned that the second boundary condition in (13.76) (written for σ 23 ) leads to the same relation, so resolving the system of equations (13.96), (13.99), and −1 (13.100) we find β 1 = α3α +1 and 𝛾1 = −(1 + α ) , which completes the derivation of the Poisson formula.

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13.6.9 Appendix B: some 2D Fourier transform formulae The direct and inverse Fourier transforms of a 2D function are related by 1 G(ξ x , ξ y , ·) = F [g(x, y, ·)] = 2π −1

∞ ∞

e−ı(xξ x+yξ y ) g(x, y, ·)dxdy ,

−∞ −∞

and 1 g(x, y, ·) = F [G(ξ x , ξ y , ·)] = 2π

∞ ∞

e ı(xξ x +yξ y ) G(ξ x , ξ y , ·)dξ x dξ y .

−∞ −∞

We use the simple property of the Fourier transformation α +𝛾

F −1 [D x,y g(x, y, z)] = (ıξ )α+𝛾F −1 [g(x, y, z)] , F

−1

[D αz g(x,

y, z)] =

D αz F −1 [g(x,

It is well known that (see [96]) - . √2 2 1 1 F −1 = 9 e−z ξ x +ξ y , r ξ x2 + ξ y2

(13.101)

y, z)] .

9 r = x2 + y2 + z2 ,

where x and y are the variables of the Fourier transform, and z is a free variable. Then, taking the derivatives we find - . . √2 2 z ∂ 1 = e−z ξ x +ξ y , (13.102) F −1 3 = − F −1 r ∂z r and F −1

-

x r3

.

= − F −1

-

. - . √2 2 ∂ 1 1 ıξ x = −ıξ x F −1 = −9 e−z ξ x +ξ y . ∂x r r ξ x2 + ξ y2

The same with respect to y: $ % - . - . √2 2 ıξ y y ∂ 1 −1 −1 −1 1 = − = − = −9 F F ıξ F e−z ξ x +ξ y . y 3 r ∂y r r ξ x2 + ξ y2

(13.103)

(13.104)

The following formulae are used in our considerations: $ % - . 2 √2 2 zξ x2 x z z − z ξ + ξ − 1 −1 x y = 3F − −9 e F , r5 r3 ξ x2 + ξ y2 $ % - . 2 √ zξ y2 z − z ξ x2 + ξ y2 −1 zy −1 9 −F − e = 3F , 5 3 2 2 r r ξx + ξy and F

−1

-

xy r5

.

1 = F −1 3

$

∂2 1 ∂x∂y r

%

√ 2 2 1 ξx ξy =− 9 e−z ξ x +ξ y . 2 2 3 ξx + ξy

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312 | 13 Boundary value problems with random boundary conditions By the property (13.101) we get - . - . √ 2 2 √ zx − z ξ x2 + ξ y2 −1 zy −ıξ x e−z ξ x +ξ y = 3F −1 , − ıξ e = 3F , y r5 r5 √ $ %  . 2 2  9 z3 z2 ∂ 1 e−z ξ x +ξ y  1 + z ξ x2 + ξ y2 , = F −1 5 = − F −1 3 r 3 ∂z r 3 hence z

9

ξ x2

+

ξ y2 e−z

√

$ ξ x2 + ξ y2

= 3F

−1

z3 r5

% − F −1

-

z r3

(13.105) (13.106)

.

.

We need also the next representations ⎡ ⎤ $ % 2 2 √2 2 √ 2 2 ξ ∂ 3z x 2 −z ξ x +ξ y −z ξ x +ξ y ⎦ −1 15zx ⎣ 9 =− e − 5 , ξx e = −F ∂z r7 r ξ x2 + ξ y2 ⎡ ⎤ $ % √2 2 √ 2 2 ξ y2 ∂ 15zy2 3z e − z ξ x + ξ y ⎦ = − F −1 − , ξ y2 e−z ξ x +ξ y = − ⎣ 9 ∂z r7 r5 ξ x2 + ξ y2 ⎛ ⎞ . √ √2 2 ∂ ⎝ ξx ξy zxy − z ξ x2 + ξ y2 − z ξ + ξ x y ⎠ = −15F −1 9 =− e , ξx ξy e ∂z r7 ξ x2 + ξ y2

(13.107)

(13.108)

(13.109)

and −ıξ x

9

ξ x2

+

ξ y2 e−z

√

ξ x2 + ξ y2

9 √2 2 −ıξ y ξ x2 + ξ y2 e−z ξ x +ξ y

$ %  √ ∂  15z2 x − z ξ x2 + ξ y2 −1 3x = ıξ x e − , = −F ∂z r5 r7 $ % √ 2 2 ∂  3y 15z2 y = ıξ y e−z ξ x +ξ y = −F −1 − . ∂z r5 r7

(13.110)

13.6.10 Appendix C: some 2D integrals Here we evaluate the integrals representing the entries (B u )ij in (13.86). The following formulae are derived by using some integrals given in [63]. In the polar coordinates ξ x = ρ cos φ, ξ y = ρ sin φ, τ x = R τ cos ϕ, τ y = R τ sin ϕ we have $

F

−1

e

−z

√

ξ x2 + ξ y2

ξ x2 2 ξ x + ξ y2

% =

1 2π

∞ 2π 0 0

e ıR τ ρ cos(φ−ϕ) e−ρz

ξ x2 dφdρ . ρ

(13.111)

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We change the variable by shift, φ = φ − ϕ; then, 1 2π

∞ 2π−ϕ



e ıR τ ρ cos φ e−ρz ρ (cos2 φ cos2 ϕ + sin2 φ sin2 ϕ)dφ dρ

0 −ϕ

(13.112)

%

J 1 (ρR τ ) J 1 (ρR τ ) 2 2 = e − J 2 (ρR τ ) cos ϕ + sin ϕ ρdρ ρR τ ρR τ 0 9 9 ⎛ ⎞ 2z2 ( z2 + R2τ − z) + R2τ (2 z2 + R2τ − 3z) 1 ⎝ z ⎠ − cos2 ϕ 9 = 2 1− 9 . Rτ z2 + R2τ R2τ ( z2 + R2τ )3/2 - √ . ξ2 2 2 Similar calculation with F −1 e−z ξ x +ξ y ξ 2 +y ξ 2 yields ∞

$

− ρz

x

1 2π

∞ 2π

e ıR τ ρ cos(φ−ϕ) e−ρz

0 0

1 = 2π

∞ 2π 0 0

ξ y2 ρ

  dφdρ 

y

ξ y = ρ sin φ,φ = φ − ϕ

(13.113)



e ıR τ ρ cos φ e−ρz ρ (cos2 φ sin2 ϕ + sin2 φ cos2 ϕ)dφ dρ

% J 1 (ρR τ ) J 1 (ρR τ ) − J 2 (ρR τ ) sin2 ϕ + cos2 ϕ ρdρ ρR τ ρR τ 0 9 9 ⎛ ⎞ 2 ( z2 + R2τ − z) + R2τ (2 z2 + R2τ − 3z) 2z 1 ⎝ z ⎠ − sin2 ϕ 9 = 2 1− 9 . Rτ z2 + R2τ R2τ ( z2 + R2τ )3/2 √2 2 ξξ . For F −1 e−z ξ x +ξ y ξ 2x+ξy 2 we have ∞

=

$

e−ρz

x

∞ 2π

1 2π

y

e ıR τ ρ cos(φ−ϕ) e−ρz

0 0

∞ 2π

1 = 2π

 ξx ξy  dφdρ  ξ x = ρ cos φ,ξ y= ρ sin φ ρ



e ıR τ ρ cos φ e−ρz ρ cos ϕ sin ϕdφ dρ

0 0

∞ =−

e−ρz J 2 (ρR τ ) cos ϕ sin ϕρdρ

0

9 9 2z2 ( z2 + R2τ − z) + R2τ (2 z2 + R2τ − 3z) 9 = − cos ϕ sin ϕ . R2τ ( z2 + R2τ )3/2

(13.114)

Finally, for the entries (B u )13 , (B u )23 we find 1 2π

∞ 2π

  e ıR τ ρ cos(φ−ϕ) e−ρz ξ x dφdρ 

ξ x = ρ cos φ

(13.115)

0 0

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314 | 13 Boundary value problems with random boundary conditions

1 = 2π

∞ 2π



e ıR τ ρ cos φ e−ρz ρ cos φ cos ϕdφ dρ

0 0

∞ =ı

e−ρz J 1 (ρR τ ) cos ϕρdρ = −

0

ıR τ cos ϕ ıτ x =− 2 , 2 2 3 / 2 (z + R τ ) (z + R2τ )3/2

and 1 2π

∞ 2π

  e ıR τ ρ cos(φ−ϕ) e−ρz ξ y dφdρ 

(13.116)

ξ y = ρ sin φ,φ = φ + ϕ

0 0

1 = 2π

∞ 2π



e ıR τ ρ cos φ e−ρz ρ cos φ sin ϕdφ dρ

0 0

∞ =ı

e−ρz J 1 (ρR τ ) sin ϕρdρ = −

0

1 2π

∞ 2π

e ıR τ ρ cos(φ−ϕ) e−ρz ρdφdρ =

0 0

ıτ y ıR τ sin ϕ =− 2 , 2 2 3 / 2 (z + R τ ) (z + R2τ )3/2

z . (z2 + R2τ )3/2

(13.117)

13.6.11 Appendix D: some further Fourier transform formulae In the notations h = F −1 [u (x, z)] =

∞

e−ıxξ u (x, z)dx ,

F [h](x, z) =

−∞

1 2π

∞

e ıxξ h(ξ, z)dξ , −∞

the following formulae are well known (see [63]) z ]= F [ π ( x2 + z2 ) −1

∞

e−ıxξ

−∞

z dx = e−| ξ | z . π ( x2 + z2 )

(13.118)

Taking derivatives we get ⎤ ⎡∞  ∞ ∂ ⎣ zdx zxdx − ıxξ ⎦ = −ı = −zsignξe−| ξ | z , e e−ıxξ ∂ξ π ( x2 + z2 ) π ( x2 + z2 ) −∞

−∞

i.e. x F [ ] π ( x2 + z2 ) −1

∞

e−ıxξ

−∞

xdx = −ısignξe−| ξ | z . π ( x2 + z2 )

(13.119)

We note also that by (13.118) F −1 [

π(x2

1 e−| ξ | z ]= , 2 +z ) z

(13.120)

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and taking the derivatives D z we get F −1 [

2z3 ] = (1 + z|ξ |)e−| ξ | z . π ( x 2 + z 2 )2

(13.121)

We need also the next Fourier transform formulae $ %  . x2 − z2 ∂ z −1 = F = −|ξ |e−| ξ | z , F −1 π ( x 2 + z 2 )2 ∂z π(x2 + z2 ) .  . ∂ z −2xz −1 = = ıξe−| ξ | z , F −1 F π ( x 2 + z 2 )2 ∂x π(x2 + z2 ) $ % $  % 2 2 ∂2 z −1 2z( z − 3x ) −1 F =F (13.122) = ξ 2 e−| ξ | z , π ( x 2 + z 2 )3 ∂z2 π(x2 + z2 ) $ % $ %   2x(3z2 − x2 ) ∂2 z −1 F −1 F = = −ıξ |ξ |e−| ξ | z . π ( x 2 + z 2 )3 ∂z∂x π(x2 + z2 ) Finally, in the proof of Theorem 2 we use the formulae ∞ −∞

dy 2 = 2, 2 2 3 / 2 (y + a ) a

∞ −∞

(y2

dy 4 = . 2 5 / 2 +a ) 3a4

(13.123)

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326 | Bibliography [209] S. Vempala. The Random projection method. DIMACS Series in Discrete Math. and Theoretical Comp. Science Vol. 65, American Math. Soc., Providence, R. I., 2004. [210] V. S. Vladimirov. Equations of mathematical physics. Nauka, Moscow, 1967 (in Russian). [211] Ju. V. Vorobiev. Stochastic iteration process. J. Comp. Math. and Math. Physics 4 (1964), 1088–1092 and 5: 787–795 (in Russian). [212] Ju. V. Vorobiev. Stochastic iteration process in the method of variable directions. J. Comp. Math. and Math. Physics., 8, number 3 (1968), 663–670 (in Russian). [213] A. J. Walker. New fast method for generating discrete random numbers with arbitrary friquency distributions. Electronic Letters, 10 (1974), 127–128. [214] K. Walker. Fourier integral representation of the Green function for an anisotropic elastic half-space. Proc. R. Soc. Lond. A, 443 (1993), 367–389. [215] W. R. Wasow and G. E. Forsythe. Finite-Difference Methods for Partial Differential Equations. Wiley, New York, 1960. [216] K. W. Wojciechowski and J. Narojczyk. Influence of disorder on the Poisson’s ratio of static solids in two dimensions. Rev. Adv. Mater. Sci., 12 (2006), 120–126. [217] K. Wong and L. Ling. Optimality of the method of fundamental solutions. Engineering Analysis with boundary elements 35 (2011), 42–46. [218] F. Woolfe, E. Liberty, V. Rokhlin and M. Tygert. A fast randomized algorithm for the approxima­ tion of matrices. Applied and Computational Harmonic Analysis 25 (2008), 335–366. [219] D. Xiu and J. Shen. An efficient spectral method for acoustic scattering from rough surfaces. Communications in computational physics, 2, Number 1 (2007), 54–72. [220] X. Frank Xu. A multiscale stochastic finite element method on elliptic problems involving un­ certainties. Computer Methods in Applied Mechanics and Engineering, 196, numbers 25–28 (2007), 2723–2736. [221] A. M. Yaglom. Correlation theory of stationary and related random functions I. Basic results. Springer-Verlag, New York – Heidelberg – Berlin, 1987. [222] J. Yang, D. Zhang and Z. Lu. Stochastic analysis of sturated-unsaturated flow in heterogeneous media by combining Karhunen-Loeve expansion and perturbation method. Journal of Hydrol­ ogy, 294 (2004), 18–38. [223] D. M. Young. Iterative solution of large linear systems. Academic Press, New York and London, 1971. [224] L. Zalcman. Mean values and differential equations. Israel Journal of Mathematics, 14, number 4 (1973), 339–352. [225] H.-Z. Zhou, A. Szabo, J. Douglas and J. Hubbard. A Brownian dynamics algorithm for calculating the hydrodynamics friction and the electrostatic capacitance of an arbitrary shaped object, J. Chem. Phys., 100, number 5 (1994), 3821–3826. [226] H.-X. Zhou. Boundary element solution of macromolecular electrostatics: interaction energy between two proteins, Biophys. J., 65 (1993), 955–963.

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Index A adjoint algorithm 248 adjoint estimator 248 alternative Schwarz procedure 144 B Balayage operator 32 balloid 21 Bessel function 7 Betty formula 62 biharmonic equation 34, 214, 282 body forces 244 branching process 139 C capacitance 220 confocal ellipsoids 20 consistent numbering 160 correlation tensor 299 coupled biharmonic-harmonic equation 58 covariance operator 277 covariance tensor 261 D Darboux equation 16 diffusion equation 15 Dirichlet problem 1 discrete Random Walk method 177 displacement correlations 303 double layer potential 206 Double Randomization 251 E ε-spherical process 122 eigenvalue problem 269 elastic disc 197 elastic half-plane 199 elastic half-space 201 ellipsoidal mean value relation 20 Euler integrals 77 exterior Dirichlet problem 221

Generalized spatial Poisson formula 86 Green formula 9 Green tensor 244 H harmonic function 1 heat equations 20 Heaviside step function 20 Helmholtz equation 15 High-order elliptic equations 32 homogeneous boundary excitations 273 Hooke law 110 hydrodynamics friction 220 I integral equation 12 isotropic elasticity 197 isotropic elastostatics 165 iterations of the spherical mean operator 121 K Karhunen–Loève expansion 263 Kupradze–Aleksidze method 210 L L-curve method 223 Lamé equation 60, 285 least squares solution 232 linear algebraic equations 157 M Markov chains 120 mathematical expectation 120 mean value relation 1 Mercer’s theorem 264, 271 metaharmonic equations 48 method of fundamental solutions 207 Monte Carlo method 120 N Neumann boundary conditions 274 Neumann series 13

F Fourier transform 262 Fredholm operator 156

O one-component diffusion system 55 overlapping discs 153

G Gauss theorem 110 Gaussian random field 260, 267

P parabolic equations 20 parabolic means 27

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328 | Index partially homogeneous random fields 263, 301 Pizetti formula 8 plane elasticity 74 Poisson formula 298 Poisson formula for displacements in polar coordinates 83 Poisson formula for the Lamé equation 74 Poisson integral formula 267 Poisson–Jensen formula 14 polyharmonic equations 48 pseudovibration elastic equation 66 R random boundary conditions 260 random field 261 random loads 241, 249 random vector 120 random walk on boundary algorithm 208 Random Walk on Fixed Spheres 148 random walk on spheres 120 randomized PCA 231 randomized SVD 222, 230 S separable matrix kernel 197 separable Poisson kernel 214 shearing loads 298 SOR method 173 spectral density function 267 spectral radius 163 spectral representations 261

spherical mean value relation 5 spherical mean value theorem 60 spherical means for the displacements 102 spheroid 21 stochastic boundary collocation 205 stochastic iterative procedure 168 stochastic projection method 203 stochastic spectral projection method 186 strain tensor 105 stress tensor 105 strong mean value relation 1 surface tractions 111 SVD decomposition 222 systems of elasticity theory 60 T tangential surface forces 298 thermo-elastic equation 73 transition densities 178 triangular systems of elliptic equations 55 two-component diffusion 56 U unit mass 33 upper half-plane 276 W Walker sampling method 178 wave equation 29 weak converse mean value relation 1 white noise excitations 267, 303

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