Parametric Estimates by the Monte Carlo Method [Reprint 2018 ed.] 9783110941951, 9783110460353

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Parametric Estimates by the Monte Carlo Method

PARAMETRIC ESTIMATES BY THE MONTE CARLO METHOD G.A.

MIKHAILOV

MYSPM Utrecht, The Netherlands, 1999

VSP BV P.O. Box 346 3700 AH Zeist The Netherlands

Tel: +31 30 692 5790 Fax: +31 30 693 2081 E-mail: [email protected] Home Page: http://www.vsppub.com

© V S P BV 1999 First published in 1999 ISBN 90-6764-297-5

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

Printed in The Netherlands

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bv,

Ridderkerk.

Contents 1. Introduction: weight unbiased Monte Carlo estimates 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9.

Integral equations, linear functionals Terminating Markov chains Standard weight estimates in the Monte Carlo method, biasedness . . . . Variances of the standard estimates The main approaches to variance reduction The use of recurrent representations Randomization Vector estimates related to the triangular system of integral equations . . Calculation of parametric derivatives and the main eigenvalues of integral operators 1.10. Test integral equations and problems 1.11. The extension of unbiasedness conditions 1.12. Approximate confidence intervals

2. Parametric estimates for solving problems of mathematical physics 2.1. 2.2. 2.3. 2.4. 2.5. 2.6.

Introductory information Solving the Helmholtz equation with a complex parameter Solution of boundary value problems of the second and third kinds . . . . Solution of the Dirichlet problem for the vector and nonlinear Helmholtz equations Estimating the main eigenvalue of the Laplace operator Global algorithms of the Monte Carlo method for solving «-dimensional difference equations

1

1 2 3 8 10 16 22 27 29 33 36 40

42 42 48 56 63 74 80

3. Parametric estimates for studying the radiation transfer in inhomogeneous media 95 3.1. 3.2. 3.3.

3.4.

Introductory information 95 Calculation of parametric derivatives and critical values of parameters . . 98 Use of the averaged estimates by the Monte Carlo method for the study of the effects of medium stochasticity 104 3.3.1. Modelling the homogeneous stochastic fields 105 3.3.2. Partially averaged weight estimates 106 3.3.3. Finiteness conditions for the variance of a partially averaged weight estimate 107 3.3.4. Asymptotic estimation of the passage probability 108 3.3.5. Test problem 112 3.3.6. Additional remarks 114 Critical parameters of the particle transport process with multiplication in a stochastic medium 115 3.4.1. Averaging the constants and the solution of the transfer equation . 116 3.4.2. Use of the diffusion approximation 119 3.4.3. Estimation by the Monte Carlo method 120

Contenta

3.5. 3.6.

3.4.4. Use of the simplest mathematical models 3.4.5. Use of the second order parametric derivatives New approach to path estimates in the Monte Carlo method Monte Carlo estimates for derivatives of polarized radiation

124 131 132 138

A. The improvement of random number generators by modulo 1 summation 148 A.l. Estimates of the nonuniformity of distributions of the congruent sums of random quantities 148 A.2. Congruent sums of grid random quantities 158 A.3. Improvement in the random number generators by congruent summation . 160 B. On modelling chemical reactions by the Monte Carlo method 166 B.l. Introduction 166 B.2. General scheme of chemical reaction modelling by the Monte Carlo method 167 B.3. Conditions of coexistence of steady states in chemical systems 172 B.4. Calculation of quasi-potentials of dynamic systems 176 B.5. Examples

178

C. One unsolved minimax problem

181

References

184

PREFACE This monograph is devoted to the further development of parametric weight Monte Carlo estimates for solving linear and nonlinear integral equations and boundary value problems for equations with partial derivatives. These estimates are related to special auxiliary weight factors which explicitly depend on parameters while simulated Markov chains do not depend on them. Using these estimates, it is possible to construct effective Monte Carlo methods for calculating parametric and coordinate multiple derivatives of solutions and linear functionals. On this basis it is possible to estimate the critical values of parameters which sometimes coincide with the main eigenvalues of the corresponding operators. It is well known that Monte Carlo methods are especially effective for solving problems with stochastic parameters. In a special chapter of the monograph, the new Monte Carlo methods for solving stochastic radiation transfer problems are presented, including the estimation of probabilistic moments of the corresponding critical parameters. In a sense, the monograph can be regarded as the sequel of the previous anthor's monograph "New Monte Carlo Methods with Estimating Derivatives" (VSP, 1995). Some material from this monograph is repeated here with necessary modifications, but mainly the material herein is new and contains many references to the previous monograph. The first chapter of the monograph is in fact a short manual on the theory of weight Monte Carlo methods including the improved grounds of the parametric differentiation and the additional randomization of weight estimates. Some proofs in an improved form are stated here which correspond to a new university lecture course. The second chapter contains new material on solving boundary value problems with complex parameters, mixed problems to parabolic equations, boundary value problems of the second and third kinds, and some improved techniques related to vector and nonlinear Helmholtz equations. There is a section on estimation of the main eigenvalue of the Laplace operator by calculating multiple parametric derivatives when solving the special Helmholtz equation. Special attention is paid to the foundation and optimization of the known global 'walk on grid' method for solving the Helmholtz difference equation. It seems that this method is appropriate for solving any elliptic difference equation and it is more effective than the 'walk on spheres' method for the global estimation of a solution. However, the walk-on-spheres method is preferable if it is necessary to estimate parametric or coordinate derivatives of a solution. In the third chapter, random media models related to homogeneous point fluxes are used. For these models new asymptotic estimates of the radiation transfer are obtained and the comparative Monte Carlo calculations are done. It turned out that the corresponding weight estimates can be partly averaged with respect to values of a random density. Using these new estimates the influence of a model dimension on the radiation transfer was estimated. In the special extended section 3.4, probabilistic moments of the time constant and the effective multiplication factor in a stochastic medium with scattering, absorption and fission are estimated. For this purpose, corresponding formulae for averaged constants of integro-differential transfer equations are constructed, and the diffusion approximation and simple probabilistic models, as well as a special iteration process realized with the Monte Carlo method, sure used. Also considered is the possibility of estimating critical values of parameters of averaged solutions. Computational results for inform and two-layer balls with stochastic density values are presented. Section 3.5 presents a new approach to constructing and substantiating weight 'path estimates' on the basis of introducing an artificial capture coefficient. Section 3.6 is devoted to the computation of parametric derivatives when solving problems of radiation transfer with polarization. In the first appendix, the theoretical aspects of the familiar method of improving random number generators by modulo 1 summation are developed. There is considered the vector version

of congruent summation which allows us to decrease the possible dependence of numbers by enhancing the property of uniformity in multidimensional distributions. In the second appendix, algorithms of statistical modelling of chemical reactions are considered. The parametric condition of coexistence of steady states in chemical systems is derived and the corresponding estimates are constructed. In the third appendix, one unsolved minimax problem related to optimizing parametric estimates is presented. The work was supported by the Russian Basic Research Foundation (Grant No. 97-010855) and by Grant No. 3759 of the Scientific Programme "Universities of Russia-fundamental researches".

Chapter 1

Introduction: weight unbiased Monte Carlo estimates 1.1. Integral equations, linear functionals Let us consider an integral equation of the second kind, 1, is equivalent to the inequality p(K) < 1, where the spectral radius p(K) is determined by the relation: p(K) = p(K') We recall that \(f,h)\

= lim \\Kn\\1/n

= inf HJril 1 /".

< ||/||L • \\h\\L., where ( f , h ) = f f(x)h(x)dx, x

\\K\\ =

\\K'\\,

1. Introduction: weight unbiased Monte Carlo estimates

2

(.Kf,

h)

=

(/,

K'h),

[K'h]{x)

=

x

By definition, L „ is a space conjugate to functions characterized by a norm INU« =

J k ( x , x')h(x')

Ax'.

i.e. the space of bounded (almost everywhere)

vrai

sup|/i(x)|,

x e X .

In this chapter we consider Monte Carlo algorithms for estimating functionals of the form h = { x, if k(x\x) / 0. That is, we must require K(X)

±

0 if f(x) ± 0, p(x', x) ± 0 if k{x', x) ^ 0.

(1.7)

The random variable £ is a commonly used and most convenient estimator of the functional I^ = ( xn leads to a break of the trajectory, n I 1 otherwise, so that this chain is formally of infinite length;