Inverse and Improperly Posed Problems in Differential Equations: Proceedings of the Conference on Mathematical and Numerical Methods held in Halle, Saale (GDR) from May 29 to June 2, 1979 [Reprint 2021 ed.] 9783112480281, 9783112480274

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MATHEMATICAL MATHEMATISCHE RESEARCH FORSCHUNG Inverse and I m p r o p e r l y Posed Problems in D i f f e r e n t i a l Equations edited by G.Anger

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Inverse and Improperly Posed Problems in Differential Equations

Mathematical Research

Mathematische Forschung

Wissenschaftliche Beiträge herausgegeben von der Akademie der Wissenschaften der DDR Zentralinstitut für Mathematik und Mechanik

Band 1

Inverse and Improperly Posed Problems in Differential Equations

Inverse and Improperly Posed Problems in Differential Equations Proceedings of the Conference on Mathematical and Numerical Methods held in Halle/Saale (GDR) from May 29 to June 2, 1979

edited by Prof. Dr. Gottfried Anger Martin-Luther-Universität Halle-Wittenberg

Akademie-Verlag • Berlin 1979

Erschienen im Akademie-Verlag, D D R - 1 0 8 Berlin, Leipziger Str. 3 — 4 Lektor: Dr. Reinhard Höppner © Akademie-Verlag Berlin 1979 Lizenznummer: 202 • 100/426/79 Umschlag: Rolf Kunze Gesamtherstellung: V E B Druckerei „Thomas Müntzer", 582 Bad Langensalza Bestellnummer: 762 7 4 4 1 (2182/1) • L S V 1065 Printed in G D R DDR 3 7 — M

P R E F A C E The Department of Mathematics of the Martin Luther University Halle- Wittenberg was the sponsor of a Conference on Mathematical and. Numerical Methods for Inverse and Improperly Posed Problems in Differential Equations, which was held in Halle (Saale) from May 29 to June 2, 1979» The aim of this conference was to bring together both pure and applied mathematicians from diverse areas in science and technology. The invited lectures and some short communications are published in these Proceedings. It is well known that in the past some famous mathematicians were of the opinion that improperly posed problems were not of interest in mathematics and its applications. During the last twenty years this situation has changed completely. Most problems in applications, for instance in physics, chemistry, biology, engineering and so on, are inverse problems. Previously these problems were studied as direct problems. Often the solution an inverse problem does not depend continuously on the measured data and is therefore improperly posed. Each physical system must be stable, i.e. it must depend continuously on the exterieur data accessible to Measurements. This is one reason why improperly posed problems have been the subject to such extensive study recently. The practical importance of inverse and improperly posed problems is such that they may be considered among the pressing problems of current mathematical research. Many problems remain unsolved in the study of inverse problems. Before attacking a specific inverse problem we should know the solution of the corresponding direct problem. Therefore new theories for the solution of inverse problems are a further development of the hitherto existing mathematical theories for direct problems. The organizer takes this opportunity to thank all mathematicians and physisists who took part in the work of the conference for their contributions. The organizer would like to express his warmest thanks to the Martin Luther University for its constant encouragement and for sponsoring this conference. He is indebted to many members of the University for freely offering their help and advice, especially to Prof. Dr. W. Walsch, Dr. H.-H. Buchsteiner, R. Aderhold, A. Juhl, E. Pfrepper and B. Thaler. The organizer also thanks the Akademie-Verlag Berlin, especially Mrs. R. HeXle and Dr. R. Hoppner, for its help in preparing these Proceedings.

Halle (Saale;, May 1979

Gottfried Anger

C O N T E N T S

6. Anger and B. Czerner Solution of an Inverse Problem for the Heat Equation by Methods of Modern Potential Theorie

9

V. Barcilon and G. Turchetti On an Inverse Eigenvalue Problem with Truncated Spectra Data

25

J. R. Cannon and R. E* Eving Quasilinear Parabolic Systems with Non-linear Boundary Conditions

35

G. Chavent About the Stability of the Optimal Control Solution of Inverse Problems

45

D. Colton The Approximation of Solutions to the Backwards Heat Equation by Solutions of Pseudoparabolic Equations

59

L. Eld&n

73

Regularization of the Backward Solution of Parabolic Problems

V. Friedrich Zur stochastisehen Kegularisierung nichtkorrekter Gleichungen in Hilberträumen

83

I. Galligani Quasi-Linearization and the Identification of Parameters in Partial Differential Equations of Parabolic Type

89

K. Glashoff Über die Behandlung Inversor Probleme bei streng zeichenfesten Kernen

99

M. Goebel

Optimal Control of Coefficients in Linear Elliptic Equations II. Necessary Optimality Conditions

105

A. GBpfert Hadamard's and Optimization-theoretical Instability

115

S. Gorenflo Numerical Treatment of Abel Integral Equations

125

V. M. Isakov On Uniqueness of Solutions for Inverse Problems of Potential Theory

135

R. Kluge

14-1

On Some Inverse Problems for Variational and Quasi-variational Inequalities

8

R. J. Knops and L. E. Payne Uniqueness and Continuous Dependence of the Null Solution in the Cauchy Problem for a Nonlinear Elliptic System R. Kiihnau

Bemerkung zu einer klassischen Arbeit von J. Blumenfeld und V. Mayer

151

161

M. M. Lavrentjev Some Problems of Analytic Continuation

165

B. Michel

171

J. Nedoma

About Inverse Problems in Continuum Mechanics and Solid State Physics SH-Wave Propagation in Quasianisotropic Socks: Problem of Partial Differential Equation with Rapidly Oscillating Coefficients

181

R. Nürnberg On the Determination of Functional Parameters in Non-linear Evolution Equations of the Navier-Stokes Type

189

J. Peil

197

Inverse Problems in Growth Dynamics

V. S. Romanov Inverse Problems and Energy Inequalities

215

P. C. Sabatier On Extremal Solutions of Sturm Liouville Inverse Problems

223

T. I. Seidman Ill-Posed Problems Arising in Boundary Control and Observation for Diffusion Equations

233

G. Stoyan

249

M. Tasche

Identification of a Spatially Varying Coefficient in a Parabolic Equation. A Report on Numerical Experiments Some Constructions of Right Inverses and Generalized Inverses

259

D. Zidarov New Approach to the Solution of the Integral Equation of First Kind

271

V. A. Morozov Regular Method for Solving Nonlinear Problems

289

SOLUTION OF AN INVERSE PROBLEM FOR THE HEAT EQUATION BY METHODS OF MODERN POTENTIAL THEORY G. ANGER and R. CZERNER

We study an inverse problem for the heat equation. The right-hand side (sources) is to be determined from the values u on the boundary 3 Q . of a domain

XL

C

where x

= t is the time coordinate. The sources are positive measures on ; n the functions considered are potentials of a positive measure. Because this problem cannot be solved uniquely, we study the set of all positive measures on -O-

which create outside X L the same potential. This set is convex and weakly compact. Therefore we can apply the theorem of Choquet - Krein - Milman. In this theorem the extremal points are important. We characterize classes of extremal points (measures) in the case R

= {x )f(z)

dz

-

The same r e l a t i o n s hold f o r the s o l u t i o n o f t h e i n i t i a l value problem o f t h e heat equation r e l a t i v e t o t h e hyperplane x n = "C • The s o l u t i o n i s given f o r x q > f u ( x ) = j u ( z ) S w ( x , z ) dz = : J u ( y )

4.

by

y) .

We s h a l l now study some extremal measures o f the convex s e t

(20)

2(^0)

:= ( v e C j U j , )

where x ° = 0 and u ( x j = 1 on S

: V = 0,

J u d£ x o = j u dv , u £

.

S ^ o ( f ) = f ( 0 ) . There e x i s t s a s o l u t i o n u £ D ( I 2 T )

. Therefore

j d\3 = 1 f o r a l l V e i § ( 5 * o ) .

satisfying

[3] i t was proved

I n

that e ex & & x o ) f o r a l l ~ s a t i s f y i n g 0 < T k T and

S^o fe ex ¡^iS^o). We now c o n s i d e r the f o l l o w i n g measures

(2D

v = vK +

e

£(^

0

)

,

where supp \>K C K C -Ap. K compact and supp f

. We want t o determine t h e

s t r u c t u r e o f ¡ f . Let u fc D(£1 T ) and h = r e s t ^ ^ u . We g e t , because the r e l a t i o n j u d ^ o = j f d ^ 0 = J u dvfc + J u djr= j h d U * ^ ) + J h

d^

holds, (22)

j ' h d ^ = j u d£ x o - j u dVK = J h d ^ 0 - j h d(A*v K ) .

The measure j f i s uniquely determined, because the s e t o f a l l h = r e s t ^ ^ u eD(HT), (23)

i s equal t o 0 ^ ( 3 ^ ) .

g ( z ) := G w ( z , 0 ) - ' J g w ( z , x ) d v K ( x ) ,

where z = ( z ^ , . . . (24)

Let

,zn_1tT).

Applying ( 1 9 ) t o ( 2 2 ) we g e t f o r the measure ( 2 1 )

v(f) =VK(f) + ^K = 0 and jf i 0 . From ¡¡-i

A consequence o f the l a s t i n e q u a l i t y i s t h a t (25)

u , T

0 it

f o l l o w s t h a t g=0.

Vg = 0 has t o s a t i s f y t h e i n e q u a l i t y

G w ( z , 0 ) = ] g w ( z , x ) dV^(x) f o r z„ = T and 1 - j dVR = 0 .

2 At f i r s t we c o n s i d e r the heat equation i n R . Let

14

XIT = ( x

&R2,

Cx2

0

=

&R2, x2 =

[x

.

Tj

We s t u d y extremal measures o f t h e form (26)

:= a % + ^

&

£( 0, 2 =

(b,t),

and (27)

\j , = a ' ^ a y

+ a'^2 y

The measure (26) i s

+ T 0

&

£(&>).

an extremal element o f

y1 = ( - b . t ) ,

^(£¿0)

if

y2 = (b/M,

Sa0(z°) =

b,a> >

0.

where

2 (28)

z ° = bT/tr

, a Q = (1 - f / T ) 1 / 2

exp(-

) .

Then 1 (29)

sa a

°

C*-,) = 1

z2 exp(- - ¿ ) 4T

2^tF

(1 -

( z t - b T ) 3

exp(

4 t

I n t h e c a s e 0 < a < a Q we o b t a i n t h e important (30)

= Sao^-P

Inequality

(30) i s

b < (2(1

+

b

" ~ T —- 7t

ex

2 2 , g P) is positive if (40)

>

£

(r2

. Stftiil, .

Now we have to study the solutions of (38), i.e. of g'(y) = 0. The number ^ =

0

is a solution of this equation. The function g"(£) attains its minimum if (41)

r2
) = 0, g"(g) > 0, has to satisfy (38)

and (44)

o


1-1 G w (z,x)/G w (z,0) = C(T,c) exp(- C

n1 + ^ C ' j X j

"

T

n-i C ^j)4(T_T)T

it follows that the function h tends to zero for |z | — > 0 0 . For a = 1 we obtain G^(z) = g(z) = GW(Z,0)(1

-

h(z))

and from g > 0 we obtain the inequality 1 > h(z) > 0. There exists at least one point z°

= C^i&Q^,) we can construct two sequences of functions

dz1

um

0

»

um

6

eHdj^) having the same properties. From v^ s= u^ + u^ and (55)

v B ( x ) = j G w ( z , x ) v m ( z ) da1 = G w ( z ° , x ) j * v m ( z ) dz 1 + o

x

)

• f -

J(Z1"

z

X

( z )

dz1

* J ( z 1" z 1 ) 2 V ' "

, X > T

-

( E )

dZ1

i t follows that

—

d1 G w ( z

+

d2

> 1 Remark 2: In the case of k roots of g - 0 in z , quence of functions (56)

vm(x)

*

'

in

k , z

we can construct a se-

vm = um1 + . . . + u o k + um1 + . . . + umlc

such that

d ^ Gw(z1,x) + . . . + d1k Gw(zk,x) +

There are 2k independent parameters d,^, . . . ,

, d21,

d2k

in formula

( 5 6 ) . Let K be a set of 2k points. Under certain conditions on K i t i s possible to approximate each value ^f dv R by functionsjv^ dVR.

7«

We now want to determine the structure of the extremal measures V= "V^ + £

£

e

in R 2 . In this case g = 0 o n ^ - ^ . Using T a y l o r ' s formula r e l a t i v e

we obtain r o r -as ( z ° , x ) u ( x ) = j G w ( z , x ) u ( z ) dz^ = G w ( z ° , x ) j u ( z ) dz,, + ^ j i z . , - z ° ) u ( z ) dz,,

+|(Zl

- z®) 2 R G w ( z , x ) d Z l ,

where u e H(_a,p) and R Gw (2,X) = \ r2,Z G w ( i , x ) / -3z 2

.

Let g ( z ° ) = 0, g ( Z l ) > 0 f o r z 1 4 z ° . Then we get, I 1 := [z^

e R, z°-6=

z1 = z °

I 2 :=

if RM^

21

(57)

-ÔG ( z ° , x ) , u ( x ) = | G w ( z , * ) u ( z ) d Z l + G w ( z ° , x ) j " u ( z ) dz,, + — j ( z 1 - z ° ) u ( z ) dz 1 0 f o r z^ ^ z ° . Then the support of V R

c o n s i s t s at most of two

22 points. In the case of k roots the supportV^ consists at most of 2k points. Proof: This Theorem 6 was proved for the Laplace equation in [5] . Let g(z°) = 0, / z° and Kj = {x1} , i = 1, 2, 3. Then

g( Zl ) > 0 for d

i1 • c i G w ( z

>'

d

i2 = C i

'

The determinant of the system (60), i = 1,2 , is equal to

A*=

e*p(8JT^T-t1)(T-t2)

Q(z,x1,x2) .

exp(^(T-t^)

4(T-tg)

where 2

-X(2)

T - t2

Z

-X(1)

T - t1

£. 1 p From (33) and (34) it follows that Q(z,x ,x ) / 0, because 0 = a^, a g < © o .

Therefore ^u m (z) dz^ and J(z1 - z°)um(z) iz^ can be calculated from the system (60) for i = 1,2 . Let Then

ilf-umidr

= J l f -Uml«dZ1 —

||f - uB||i/l(.^)

0

°

Prom (58) and (59) it follows that the sequence (c*) , i = 1, 2 , converges. Also j\iB converges to ^ f dv^. Therefore (61)

^

and

- 2 °> u m < z )

dz

1

converge. Let K^ = -jx^ . The value jf dv^ can be arbitrary, because V £ ex ¡£(oxo) From (60) it follows that this is not possible. Remark 3: The whole proof of Theorem 6 can be found in flO], Many problems remain unsolved, for instance in S^, n = 3. In future much work must be done to study completely our inverse problem for the heat equation.

23 R E F E R E N C E S ANGER, G.: Funktionalanalytische Betrachtungen bei Differentialgleichungen unter Verwendung von Methoden der Potentialtheorie. Berlin 196? ANGER, G.: Direct and inverse problems in potential theory. In Colloquium on Non-Linear Evolution Equationj and Potential Theory. Prajue 1975» pp. 11 - A4 ANGER, G.s Ein inverses Problem für die Wärmeleitungsgleichung, I. In Proceedings of the Third Finnish-Romanien Seminar on Complex Analysis, Bucuresti 1976 ANGER, G.: Convex sets in inverse problems. In Trudy Vsesojuznoj Konferencii po Uravnenijam s Castymi Proizvodnymi, Moscow 1976 (Russian) ANGER, G.: Uniquely determined mass distributions in inverse problems. In Veröffentlichungen des Zentralinstitutes fiir Physik der Erde, Potsdam, 52, 2 (1977), PP. 633 - 656 ANGER, G.: Some remarks on inverse problems. In Colloquium on Elliptic Differential Equations, Rostock 1977, to be published in the Rostocker Mathematisches Kolloquium ANGER, G.: Lectures on potential theory and inverse problems. Summer School held at the Sektion Geowissenschaften der Bergakademie Freiberg (Sachsen), June 6 - 1 6 , 1978; to be published in the series Nationalkomitee für Geodäsie und Geophysik bei der Akademie der Wissenschaften der DDR, Geodätische und Geophysikalische Reihe, Berlin 1979 ANGER, G. and CZERNER, R.: The extreme measures in the inverse problem of the heat equation. The same Proceedings as [6] CHOQUET, G.: Lectures on Analysis I,. II, III. New York - Amsterdam 1969 CZERNER, R.: Charakterisierung von konvexen Mengen beim inversen Problem der Wärmeleitungsgleichung. Dissertation, Martin-Luther-Universität Halle - Wittenberg, Halle (Saale) 1979 DOUGLAS, R. G.: On the extremal measures and subspace density. Michi . math. J. 11 (1964), pp. 243 - 246 JUHL, A.: An inverse problem for the wave equation. The same Proceeding as |6'] MICHEL, B.s About inverse problems in continuum mechanics and solid state physics. These Proceedings on Inverse and Improperly Posed Problems in Differential Equations KARR, A. F. and PITTENGER, A. 0.: An inverse balayage problem for Brownian motion. Department of Mathematical Sciences, The John Hopkins University, Technical Report No. 277, Baltimore, October 1977 PFREPPER, E.: Das inverse Problem für die Wärmeleitungsgleichung. Diplomarbeit, Sektion Mathematik der Martin-Luther-Universität Halle Wittenberg, Halle (Saale) 1978 SCHULZE, B.-W und WILDENHAIN, G.: Methoden der Potentialtheorie für elliptische Differentialgleichungen beliebiger Ordnung. Akademie-Verlag Berlin and Birkhäuser-Verlag Basel-Stuttgart 1977

ON AN INVERSE EIGENVALUE PROBLEM WITH TRUNCATED SPECTRAL DATA VICTOR BARCILON1 AND G. TURCHETTI2

Summary. The inverse problem for a vibrating string whose first N eigenvalues are given is considered.

It is shown that among all the strings which share this

truncated spectrum, the density p(x) of that string which minimizes a linear functional F[p] cannot be strictly positive.

For the case N=l, we can actually

prove that p(x) is in fact a delta function. 1.

Introduction. For most inverse problems arising in Geophysics, the available data set

is much smaller than that required for the construction of a unique solution. Typically, this data set consists only of a finite number of measurements. In view of the nature of the data set, the inverse problem becomes one of characterization of a class of compatible solutions rather than one of reconstruction of a single solution.

Such a characterization can be accomplished

by means of extremal solutions which provide global bounds for all the solutions in that class.

This approach was used by Parker (1974, 1975) in his investigation

of the inverse problem for gravity anomalies which led him to the development of the theory of ideal bodies. Th? same approach can be adopted for inverse eigenvalue problems.

Using

the vibrating string as a prototype, the typical problem can be stated as follows: —

N

u

Given the first N natural frequencies of vibration i n J n _j » characterize the class of strings (of unit length, taut by a unit tension) which have this truncated spectrum in common.

For future reference, the normal mode problem for

these strings which are labelled by their densities p(x), is: u" + 0 n

(6a)

and 0 < x, < x, < . . . < x < 1 . — i j n —

(6b)

In other words, we conjecture that the ideal string is made up of N point N . For the case N=l, we shall be able to prove that this n=l conjecture is true.

masses {m } n

2.

Structure of the ideal string. Let us denote by R the class of strings whose f i r s t N eigenvalues

— N coincide with {u } , i.e. « n=l R - ip(x) :

p(x) > 0 , ion[p] = û n , n=l, 2

N)

(7)

The ideal string p(x) is obviously a member of that class.

We shall consider

nearly ideal strings with densities o(x;e) = (yJJT:0 • en(x))2 ,

(8)

where e is a small parameter and n(x) is a yet unspecified function. given n(x), the

For a

nth eigenvalue associated with one such nearly ideal string is: n„(e) = *>n[a(x;E)].

(9)

It is easy to show that dil, dc~

f1

_ = e=0

.2

(10)

u

" n \\P( x ) T1 ( x ) u n ( x ) dx Jn

where " n (*) is the nth normalized eigenfunction of the ideal string. nearly ideal strings to be in R, the above derivative must vanish:

For these

we shall

impose these conditions and require that n(x) be such that j 1 n(x)>/S00 u^(x)dx = 0,

n=l, 2, ...N.

(11)

The function n(x) is therefore an arbitrary function lying in the subspace orthogonal to that spanned by Kip Jp u 2J} n n= n=l On the other hand, since F[a(x;e)] is a minimum for e=0, we must also have dF JCl de

e=0

= 2 I

nixi^/SÖOfixidx == 0. n(x\l5(x)f(x)dx

(12)

Jo

Consequently,

does not form a complete set.

There-

fore, the density of the ideal string must vanish somewhere in the interval. In retrospect, this result is easy to understand.

Obviously, in order

to minimize F[p], we must construct a string with a density as small as possible

28 over as large a fraction of the interval [0,1] as possible, without violating the eigenvalue and inequality constraints.

Hiis suggests that the structure of

the ideal string might be the following: N P(x) = I m 6 ( x - x j . n n=l n 3.

(14)

The case N=1 For the case of a single eigenvalue constraint, we shall be able to

prove that £(x) = m ^ C x - x ^ . The proof is in two parts.

(15)

First, we partition the interval (0,1) in

I equal intervals and consider the problem (2)-(4) for the subset of strings with piecewise constant density over these intervals.

We shall show that

the desired string has a non-zero density over a single interval.

Next, we

consider the limit I •*• •», and show that the mass of the string tends to a constant.

This approach is closely related to that used by Sabatier (1977a,b)

in his analysis of linear inverse problems with positivity constraints. We shall represent a given string either by a piecewise constant function p(x), i.e. p(x) = p.

< x < i ,

(16)

X or by a point in R

with coordinates (pj^, p 2 , ..., Pj).

The subset of R

corresponding to these piecewise constant strings is represented geometrically by the first sheet of that portion of the hypersurface in R 1 with the equation oijlp] - ^ for which Pj^ > 0, i = 1, 2, ..., I.

= 0 ,

(17)

Finally, the optimization criterion can

be written thus: F = f • p"

(18)

where the components of the vector f are given by i f. = I

f(x)dx .

J i-1 Li I

(19)

29 Consequently, our first task is to solve the following nonlinear programming problem:

Find the vector p which minimizes *

F = f • p

(20a)

subject to the constraints UJIp] - UJ = 0 ,

(20b)

and f >0 .

(20c)

The fact that (20) is a nonlinear programming problem complicates matters a great deal.

However, for this single eigenvalue constraint case,

we shall be able to show that the surface Wj[p] - Uj • 0 is concave.

With

this property, the solution can be found in a standard manner. In order to study the curvature of the surface oij[p] - Wj = 0, we -j-fO") consider two nearby points p

v

•»•

' and p on this surface.

The difference between

these two vectors can be written in terms of the local tangent and normal vectors, namely

•*• "Vol 2"*" p - p l u J = et + e n

(21)

t • Vuj = 0

(22)

where

and n = C Vu

;

(23)

C is a constant we shall determine and e a small ordering parameter.

Since

u^[p] is an analytic function of p, we can write

32(I),

2 +

\

tVk

rarR+ J

2n

i

+

0(£3)

j

'

where the summation convention has been used. Since p and p lie on the -V surface u^ - to^ = 0 and since t is tangent to this surface, we deduce that

1

30

where

aV * • V k

•

•

u = o , u ( 0 ) (0) cosa - u (0) '(0) sina = 0 , u ( 0 ) (l)

COSY

+ u (0) '(l) siny = 0 ,

where the subscript 1 has been omitted. I. v

—2

(0)

+ (a p

l

(27) y

On the other hand, v and w satisfy: -2

•'v + u

(0) tu

(28)

s 0

and w" + ^ p ( 0 ) w + u^tv + i ^ n u ^ = 0 .

(29)

The boundary conditions for v(x) and w(x) are identical to those for u ^ ( x ) . Multiplying (28) by u ^

and integrating, yields ,.(0)2 (x)dx = 0 . f 1 t(x) u Jo

(30)

Reverting to geometric pictures in R*, we can write (30) thus: t •U = 0

(30')

where U is a vector with components . i

jJ"

(0)2

(x)dx .

(31)

31

Since t is an arbitrary tangent vector it follows that U is aligned with Vtij. This implies that Vuij > 0 . Similarly, multiplying (29) by u ^

(32)

and integrating, we obtain

•1

2

f 1 t(x)u( ' (x)v(x)dx + I n(x)u (0) (x)dx = 0 . Jo Jo

(33)

A more useful form of this equation can be obtained by deriving an expression for the first integral in terms of v solely.

Omitting some straightforward

calculations, we rewrite (33) as follows:

(0)

1

p(x)v':(x)dx . J

(34)

J

0.

0

Classical results associated with Rayleigh's variational characterization of the first eigenvalue imply that the right hand side of (34) is positive. 9pT3p7 > 1 ]

V j

Therefore

0

and consequently the surface

"

0 and

=j

v

1=1

Xi

I I Xi - 1 1

i=l

p^

(36)

32 Since the surface

ID^[P]

=

Û^

lies above this plane, we can represent any point

on this plane thus: p = p p + vh

(37)

where p^ is a point on the plane, v the unit normal to the plane pointing toward the surface and h thé distance between the plane and the surface. Note that h is therefore a function of Xj. • ••, XjMaking use of this representation, the linear form to be minimized becomes F =

1 X, (f • p / ) • (C • ?)h 1 i=l 1

(38)

If -*• f

* ' »j

min = i=l, ....

*

it' is an easy matter to prove that the minimum of F is reached by setting x

i " 6 ij '

and that this minimum is equal to f • -p. since h(0, ..., 1, ...0) = 0 . Thus, for the restricted subclass of piecewise constant strings, the minimum of F is achieved for a string with a zero density everywhere except on a single interval.

As the partition is made finer and finer, the subset of

piecewise constant string becomes larger and larger and in the limit coincides with R.

Hius, provided that we can show that p.

*

/I -»• m as I -»• », we would have

proved that the ideal string has a delta function structure. *

To that effect, let us consider a generic string p^ . The eigenfunction u

j*(x) corresponding to this string can be written as follows:

33 r^CA^sinr^ + Bi cos i^) (x + tana), x
n

+

lleCU>-gCU)ll2,sT + l|FCU)-F(U)

+ ||B(U)-B(U)||2>Qt + ||c(U)-coolly] .

Proof. The proof is similar to that presented in [1].

References J.R. Cannon, W.T. Ford and A.V. Lair, Quaslllnear parabolic systems. J. of Differential Equations, Vol. 20(1976), 441-472. A. Friedman, Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, N.J., 1964. O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Uralceva, Linear and Quasi-linear Equations of Parabolic Type, A.M.S. Transl. Math. Mono., 23, Providence, R.I., 1968. J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, I, Springer-Verlag, N.Y.-Heidelberg-Berlin, 1972. H.L. Royden, Real Analysis. 2nd Ed., Macmillan Company, Toronto, Ontario, Canada, 1968.

ABOUT T H E STABILITY O F T H E O P T I M A L C O N T R O L SOLUTION O F INVSERSE PROBLEMS

Guy C H A V E N T *

SUMMARY : A definition of parameter identiflability (OLSI) well fitted to the commonly used output least square m e t h o d is g i v e n . This definition requires the injectivity of the mapping parameter

observation (often taken as identifiability

tion) but takes also in account model a n d m e a s u r e m e n t errors. A sufficient

defini-

condition

of identifiability is given and applied to a n actual inverse p r o b l e m in seismics.

*) IRIA-LABORIA - BP 105 - 78150 LE CHESNAY - FRANCE Université PARIS-IX - 75116 PARIS C e d e * 16 - F R A N C E

46 I - INTRODUCTION

M o t i v a t e d by important and diverse possible applications, the study of the inverse p r o b l e m in partial differential equations has been rather intensively

studied

during the last years. Then were a-priori two possible points of view for attacking

the

problem :

- the "theoretical" point of view, w h i c h could be defined by saying that the concerned people are interested in gaining the more possible insight into the inverse p r o b l e m using m a i n l y a pencil, paper and their b r a i n . This point of view is predominent in the area of theoretical physics [l], [2], [3]. A central point of a general of parameter estimations in P.D.E. w o u l d be a good theory of identifiability

theory

; due to

the difficulty of the problem, only very lacuneous answer have b e e n given : some people have hinted on the importance of state observability to get identifiability, as for example in [4], [5], [6], [7], other people define the identifiability as the unicity of the parameter achieving a g i v e n observation (in the no model-error case), as for example in [6], [8], [9], [10]-

- the "numerical" point of view, which is the point of v i e w of the people beeing concerned m a i n l y by the p r o b l e m of getting on a computer satisfying

estimation

of an unknown parameter in an approximate model and from noisy data . Among those people, one of the most widely used m e t h o d is the so-called "output least square m e t h o d " , as for instance in [lO] to [23]. This m e t h o d is, however, rather critically considered by the "theoretical" people, because it is very difficult to prove any theoretical result on it, except the existence of at least one solution of the m i n i m i z a t i o n p r o b l e m and the d e r i vability of the criterion ([22], [23], [24]). M o r e o v e r it is possible to prove that in v e r y simple cases the m e t h o d w o r k s very poorly

[2l].

So the task of this paper is b o t h ambitious and modest :

- ambitious because we shall try to construct a (tiny) bridge between the two above m e n t i o n n e d point of view : w e shall dare to give an identifiability

definition,

well fitted to the output least square m e t h o d and usable in presence of b o t h model measurement

and

errors.

- modest because w e shall be able to use this definition in only one (but at least one...) actual identification p r o b l e m [l9], [20], and because w e are conscious that the sufficient given in § III are too restrictive for many practical

applications.

However we think that this k i n d of definition could be used, if a less restrictive

suffi-

cient condition w e r e found, in cases where some "numerical identifiability" has b e e n o b s e r v e d without any theoretical result beeing yet available

[22].

47 II - T H E OUTPUT LEAST SQUARE IDENTIFIABILITY

(OLSI)

W e recall first on figure 1 the "output least square m e t h o d " noise

current

estimate

Figure 1 : The output least square m e t h o d

Where

is the set of admissible parameters, y(a) the state computed using the state

equations w i t h parameter a, Cr the observation operator, z the recorded observation a n d J(a) the output least square criterion to be m i n i m i z e d over G ^

•

So the "output least square estimate" a of â is given by :

(2.1)

Find à Ç d

. such that J(â) < ad

J(a)

Va €

Vz € IT

= {z e * T | d ( z , Z )
1 :

(3.9)

V =

. . . , a N + 1 .

In order to get the same number of unknown a^ than of observation, w e

suppose

(which is the case in marine seismics) :

(4.4)

ij > 0 is k n o w n

and we take, as p a r a m e t e r s p a c e d and set

of admissible impedances :

a=R (4.5) = (a = ( a 2 , . . . , a N + 1 ) £ R N | 0 < a £ a ^

5

i = 2,...,N+l,

j u . 1 + 1 - a.|< V) where a, a are given lower and upper bound of the impedance, and v is given upper b o u n d of the total v a r i a t i o n of the seeken impedance

distribution.

So let be given :

(4.6)

z £ R

= given observation

(seismogram)

then the inverse seismic problem may be formulaterias

(4.7)

_ find a € d

, which minimizes ad

2 ||C-y(a) - z||„ over O . N ad

for which one we prove :

Proposition 3 : With the hypothesis and notation (4.1) to (4.7 y and if :

(4.8)

g(0) = 0, g(At) ¡É 0

then the Impedance distribution a is OLSI and & ^ for the observation operator

We refer to reference [17] for a complete proof of that proposition and a detailed theoretical and numerical study of the seismic inverse problem. We would like however to illustrate, to close this paragraph, how the numerical results for the above problem confirm the theoretically expected continuity of the mapping z • à.

We have e s t i m a t e d ^ a medium made of N - 400 layers of travel-time thickness Ax = 4 ms from seismic recording z of duration 3 s. Eighty actual., field recordings z are shown the figure 3. They nave been recorded at regular time intervals on a ship moving with uniform spe0, t = 0 ( l ) .

2

K

l.

( ^ O l l 1 / ,2

0+ '•> ' 2 JZsi

By deforming the c i r c l e

contour p i c t u r e d below in Figure 1

r ( n + 1 /2- )

|u - — | - 6

(2.9) i n ( 2 . 5 ) onto the

c.

Figure 1 we can r e w r i t e

K(R,t)

in the form

K(R,t) = — —

vt/t

Jc

exp(- — z + z 2 ] g (z)dz

Jt

e

(2.10)

65

where

•E4

g e ( z ) - exp|

.

(2.11)

By using ad hoc methods, i t i s new possible to obtain a complete asymptotic expansion for

K(R,t)

as

e •+• 0

in the form

2 exp(- fj-5 [l+d 1 (|)+d 2 (|) 2 + . . . + d n (|) n + 0 ( ( f ) O f l ) ]

K(R.t) -

(2.12)

Tft

where the c o e f f i c i e n t s ([5]).

dj

are expressible in terms of Hermite polyncmials

In particular

d, - — i j H iAz> + 2 8 2*4 2^ 4

H6(-M 2^t

(2.13)

with similar expressions holding for the higher order c o e f f i c i e n t s . (2.. 13) H n (z)

In

denotes Hermite's polynomial.

The expansions in a l l of the above cases may be differentiated termwise. Ill.

Approximation of Solutions to the "Backwards" Pseudo-Heat Equation. We now discuss the problem of approximating solutions of i n i t i a l -

boundary value problems for the "backwards" pseudo-heat equation defined by (1.2a) - ( 1 . 2 c ) .

One approach proceeds as follows.

By replacing

t

by

t Q - t , and using the Fourier transform or Borel transform to construct a solution to the pure i n i t i a l value problem, we can reduce problem (1.2a) - ( 1 . 2 c ) to an Initial-boundary value problem of the form E A 3 u t - ufc - ¿ 3 u - 0 in D x ( 0 , t o )

(3.1a)

u - f ( j f , t ) on 3D x ( 0 , t o )

(3.1b)

u($,0) - 0 in D.

(3.1c)

Note that (3.1a) - (3.1c) d i f f e r s from (2.1a) - (2.1c) only in a sign change in the d i f f e r e n t i a l equation.

Although this has no e f f e c t on the well-posedness

of the problem, as we s h a l l see i t unfortunately has serious Implications on

the problem of constructing approximate solutions for small values of the (positive) parameter e. case as

e + 0

equation.

This Is, of course, not surprising for the limiting

is now an Improperly posed problem, i.e. the backwards heat

If we follow the analysis of the previous section, it is seen that

we can represent the solution of (3.1a) - (3.1c) In the form «(¡E,t) - ^ where

P(£,T)

fo

j 3 D P t f r O ^ i - r(R,T-t)dsdx

(3.2)

is determined as the solution of an integral equation of

Fredholm-Volterra type and the notation is the same as in (2.3).

He observe

that the only difference between (3.2) and (3.3) is that the argument of the fundamental solution is

T-t

instead of

t-T.

tunately complicates the evaluation of

r(R,t)

in the asymptotic behaviour of

for

r(R,t)

This change, however, unforsince we are now Interested

t < 0

instead of

t > 0.

Except

for the expansion (2.8) the analysis of Section II breaks down and we are forced into adopting an ad hoc approach yielding expansions of

r(R,t)

in

terms of a series of generalized hypergeonetric functions or a BesselLaguerre series (c.f.[A}). tion of

r(R,t)

for

Such expansions are of limited value in the evalua-

t " 0(1)

and hence we are led to look for other approxi-

mation procedures. To present such an alternate approach we return to the formulation (1.2a) - (1.2c).

We first note that if 11m j-«.

A

Xj

Is a sequence

2 > 0 j

then in a sufficiently small complex neighbourhood of [0,tQ] we can approximate any analytic function defined in this neighbourhood by a finite linear com-X 2 t " binatlon of functions taken from the set

{e

j

This result follows

from a theorem of Levin in the theory of entire functions and the reader is referred to [12], p. 219, for details.

From the results of [2] we can now

conclude that any continuous function defined in D can be approximated

67

In D by a linear combination of the functions .

O'k)

< y >

P m (z) an associated Legendre n

where J (z) denotes Bessel's function and n function. He observe that 1. u

(

,,:)

nmj 0

— t ] 1 + eA2

***

{u^^}

(3

'5)

is a complete set of

the order of the equation is increased

and hence except for exceptional circumstances we would expect that extra functions would have to be introduced to supplement the above set. exceptional case occurs for example when D is a sphere and eigenvalues of the Laplacian in D.)

(An

{X^} are the

In particular from the Runge approxima-

tion property for pseudoparabolic equations ([17]) and the results of Section II (approximating the pseudo-heat potential defined on the cylinder S x [O.t^] 3 D x [0,tQ] by numerical quadrature) we have the following 2 Theorem, where completeness is with respect to the L norm over D x [0,t ]. o Theorem 3: Let

points on a sphere

0

where

v -

Hi

"

r

of

S Z> D. Then the functions *

W * '

be a dense set

{X^} be as in (3.3) and let

2

W

2

n

2 3»

P

n(cos

0)

[im

+ " ^ ¡ 2 »1

/ir | form a complete set of solutions to (1.2a) defined in Is, I

D x [0,tQ]. Now let Theorem 3.

{¥ , denote the complete set of functions defined in n n»i

In order to find an approximate solution to (1.2a) - (1.2c)

we minimize the functional

N

N

+ IIn1- 1 V nn-n, n ^ ' Vo - •0, t-O(l)

can now be obtained for

(4.3) R>0, t-O(e)

from

from (2.12) since these expansions can be inte-

grated terrorise with respect to R over the range (1,«°).

In particular from

(2.12) we have r(R,t) -

- —

•wt '

2 exp(- |r)

JZrt.

8t

(s 2 - l ) - 1 / 2 ds + 0(f)

exp(-

r

c

2 i exp(- f - ^ ) (p 2 - 1)" '2 dp + 0 ( f )

Jl

'

8t

Ko(!f>

C4.4)

+ 0 (

t)-

/4irt The coefficients of the higher order terms can be expressed in terms of Whittaker functions.

Finally, the analysis of Section I I I proceeds in the

same manner except, that the functions _ . +ine are now J (X.r)ern J Acknowledgement:

r

'2 ] n +.1,/2 (A.r) Pm (cos 0)e"*Jn

Uber allen linearen Vorschriften = Rz + q, wobei $ und ^ als stochastisch unabhängige Vektoren vorausgesetzt werden, E für die Erwar= tungswertbildung steht, E^j EjEj genutzt und Ef = 0 angenommen wird und die Bezeichnungen x = Ej, B = cöv^, C = cov^ verwendet werden. Sind,/ und £ nor^ialverteilte Vektoren, so stellt (3) bezüglich des Kriteriums (4) die beste Schätzvorschrift Uberhaupt dar, und das Verhältnis von trace BA*(ABA*+C)~1AB zur a-priori-Kenntnis -x|2) = tr B charakterisiert den mittleren erreichbaren Informationsgewinn durch Auswertung der Meßdaten unter Ausnutzung von (1) und der a-priori-Daten x und B. In jedem konkreten Anwendungsfall ist die Gegenüberstellung dieser Größen zweckmäßig, um die Wirksamkeit einer mathematischen Bearbeitung der Meßdaten beurteilen zu können. Betrachtet man £ und £ als zufällige Elemente in Hilberträumen X und Y, so ist eine formale Anwendung von (3) nicht möglich, da Kovarianzoperatoren zufälliger Elemente mit der Eigenschaft < 00 nicht stetig invertierbar sind und folglich auch ABA*+C nicht stetig invertiert werden kann. Für Gaußsche zufällige Elemente J[ und £ besitzt das Definitionsgebiet des Operators (A3A*+C)~1 nur das Maß 0 (bezüglich der Verteilung von£) [9j . In [5] wurde für Gaußsche Elemente $ , % eine Erweiterung R des Operators R = BA (ABA +G) angegeben, die für fast alle z (bezüglich des Maßes vonjj) anwendbar ist und die bestmögliche Schätzung von x bezüglich (4) darstellt. Zur Konstruktion von Sf, welche i.a. nicht mit einer Abschließung von R zusammenfällt, wurden KarhunenLoeve-Zerlegungen nach den Eigensystemen der nuklearen Operatoren B und ABA*+C benutzt. Da R i.a. ein unbeschränkter Operator ist, entsteht die Frage nach solchen beschränkten Operatoren R d i e für 0 die gleiche mittlere Genauigkeit wie R garantieren, und nach dem durch eine zusätzliche Regularisierung eintretenden mittleren Genauigkeitsverlust. Die gleichen Fragen entstehen auch bei der näherungsweissn Bestimmung

85 linearer Punktionale (f,x) Uber dem unbekannten Zustand x. In den oben beschriebenen Fällen ist

s(,-D5

f (BÌP-DyCM-D) . C ( p - D y ( u - 1 ) and 1 X,

and s(P-D

(t x) =

ul^-Cv^w^j.

i=1

The initial guess y ^ C t x) o£ the vector y(t x) is supposed to be given everywhere. Under weak assumptions, the functional (5) may be represented as a quadratic functional on a suitable (convex and closed) Hilbert space X. Thus, the functional (5) has an unique minimum u^ 1 ^ = ( v ^ w ^ ) , which P P satisfies the characterization equation: (9)

for all u (u) e

( u ( p ) , \7J(u^w)))w= 0

x

where :W(Ox;uW) • P

P

-ft

(10) v J O i ^ ) = 3 V' P

(y + wp l-- g 6

is the gradient of J(u^-') in u1-*1-1 ((•,')JC denotes inner product in X) . P 3zCp)(tx;u^J) 9 z (u) The k-th component (k=l,2,...,n) of is 3v 3v m I

IK

3 — dX^

cos(v,x.) and zCnJ ^ =E z V M J ( t x ) i -Cu) rt v,.CuJ 1 p

solution of the adjoint system to (6)-(8):

is the

92 (11) -z{") = z

I i=l

1

x

i

i

• I ((B^"1))1^) 1 x i=l i

+ y ( p ) (t x;u£ u ) ) - y(t x)

. ( C ^ " 1 ) ) 1 z(">

inftx[0,t*)

(12) z ( p ) (t*x) = 0

in n

(13) z ( y : ) ( t ^ = 0

on rx[0,t*]

(B?"" 1 )) 1 and ( C ^ - 1 ^ 1 are the tranposes of the matrices b(w~1-) and C ^ ,

respectively.

It is assumed that z ^

is sufficiently differentiable.

Now, at each outer iteration p, the vector u^-* is determined as the limit point of a weakly convergent sequence generated by the following conjugate gradients method (A =0,1...): u(y

x+l)_ u(y

x)

+

-^ q (p

x)

(14) q

( ^+ D =

- V J ( u ^ X + 1 >) • o.qi"

X)

where u (y0)= u (v-l) > q(y 0) _ and P X is chosen so that J(u (M

, 5 x = JvJ(u(w x+1) )| 2 /JvJ(u (:i1 X)+

P x q C,J

X)

)= inf J(U(Y P

X)

+P X Q T Y

X)

X)

)|2

) .

X

It is possible to obtain an explicit expression for p^. Here the superscript X stands for inner iteration number. This iterative process realizes the infimum

of J(u^y-') when |vJ(u^ X + ^ ) | ^eps for

a given eps > 0.

2. A PARAMETER ESTIMATION PROBLEM OF GROUNDWATER HYDROLOGY. The above quasi-linearization method has been adapted to the problem of identifying the transmissivity parameter in an underground water reservoir. We shall consider the case of a two dimensional inhomogeneous isotropic aquifer extending over a region £1 e R

with boundary r = r + r^. The equa

tion of flow valid for this aquifer is: (15)

lEi!L*l = div(cr(x)grad p (t x)J + S(tx)

where a(x) denotes the unknown transmissivity parameter, 9(tx) is a known function related to the production rate of the wells drilled into the

93 aquifer and p(t is the piezometric head. The initial-boundary conditions associated with (15) are: (16)

p(Ox)=h(x)

in ft

(17)

p(t x> = g(tx)

on r * [ 0, t ]

Let be given measured values p(t x) , K(x) and g(t x) of the functions p(t x), h(x) and g(tx) , respectively; then it is required to determine the functions a(x), h(x) and g(t x) which, under the constrains (15)-(17), will "best" fit the measured values. In order to solve this parameter identification problem, it is convenient to think the parameter o(x) as a dependent variable and to put y x (tx)

p(tx)

y 2 (t x3

o(x)

y(t x) 5

Thus the equation (15) may be written in the form: 3y,(t tf — ^ = div(y 2 (t xjgrad y ^ t x ) ) +0(tx) (18)

in n x (0,t ]

3y 2 (tx) = e A. y 2 (tx) at y x ( 0 x ) = h(x)

(19)

(20)

in Si

y 2 (0 x) = o (x)

on r x[0,t ]

y x (t 3i = g(t x)

where e > 0 is allowed to go to zero. By using the quasi-linearization method, we have finally to find a sequence of vectors { u ^ } , u = 1.2... such that each u ^ extremizes P P the functional: J(u(y)) = f t + ( { | y ^ - ^ | 2 + | y ^ - y i ^ | 2 } d t d x o S3 vM

2

+

+

| v(w).v(p-D I ^ J ^ j / l w ^ - l l ^ l w W - w ^ " 1 ) | 2 }dtds o r

94 along a trajectory y ^ = y ^ ( t x ) o£ a linear parabolic system. In order to evaluate the computational performance of this method some numerical studies have been carried out. The domain ft is composed by a rectangle R with eventually some lateral rectangular subregions removed: the sides of R are parallel to the axes Xj and x^. In the removed subregions "small" values was used) of a(x) and a(t x) are assigned. The parameter e is taken e = 10~3. On R a rectangular network of 2 1 x 1 9 mesh-points is overlapped. Two studies concern the estimation of a(x) in a problem with p(t. x) , 9(tx) ,h(x) ,g(tx) and a(x) given exactly: p(tx) and a(x) are slowly varying functions with values included into (l.,2.) and (0.5,1.) respectively. In the first study p(t x) -'p(t x) ,h(x)=h(x) ,g(t x)=g(t x) and y ^ ( t x ) is obtained by adding a white noise to the truth values of P (t x) and a (x) . In the second study the previous values of y ^ ( t $ are used but the measured values of p(t x) are changed by subtracting a white noise from the exact values. In both studies cr(x) was determined with three significant digits after 3 and 5 outer iterations (and 7 (=4+2+1) and 13=(6+3+2+l+l) inner iterations) in 280 and 620 sec. of IBM 370/168 computer time, respectively. A further study is related to the numerical simulation of a'three-year (15-days time steps) pumping test conducted on a realistic aquifer with 48 wells distributed in a domain composed by a square of 12 Km width with some lateral rectangular subregions removed. We suppose that the history of the production rate of the aquifer is known and, at a fixed time t = 0, the piezometric and transmissivity maps are available: it means that the functions ft(tx),h(x) and > A ^ . In this paper we will limit the analysis of the system (6)-(8) to the extreme cases B^ = 0 and B^ >> A^ with m = 2 and n composed by a rectangle with sides parallel to the axes x, and x-.

96 a) Case B^ = 0, i = 1,2 Consider Q Ax' AX set of points {t }, t

and T fit along with n,r and [0,t ]. Let TA4. be the At - t = At : t = 0,1...,t Q = 0.

x

Let us construct an alternating direction difference scheme for system (6)-(8) of the form: , T , T + l/2 , t . At , T . „T % U - At — (D1+H1))'P = (I + -2"( D 2 + H 2 ^ ' > n

(21)

n

T . At _T

T

ft AtfriT +. Htj))ip T-v » T+l = _ (I At, n T + .HjJ fT UT-V , T+l/2 . At (I - -2"CD + . -j-CDj Jq> + ~Y s T 2 2

Here n>° and s T are restrictions of v(x) and s(t x), respectively, on a rectilinear network overlapping a with mesh-spacings A x ^ a n d A x 2 . The p th order matrix D T = D^ + D^ consists of the elements which represent the spatial differencing in (6) at time t = t are

: therefore these elements

inversely proportional to the square of the mesh-spacings Ax^ and

Ax 2 - The p th order matrix H T = H ^ + H 2 consists of the elements which are independent upon the mesh-spacings. We assume that D? + (DT) T (i=l,2) is negative definite: this condition is satisfied by many conventional difference schemes. For a fixed discretization of the spatial variables of (6) at time t = t T , the transition

matrix

mJnJmJn^, where mT=(I - ^ ( d T + h T ) ) - 1 ,

nT « I + 4 £ ( d ? + h 7 ) ) , (i=l,2) has a matrix power series development about At = 0

< i , i-l,2) that agrees through at least linear

terms with the expansion of e x p ( A t ( D T + H T ) ) . In order to have stability for this scheme, it is sufficient that the matrices "r~f(M*Ni'M?,N*) b e uniformly bounded for all At < At and T « A t < £ , ^ *=0 L 1 1 L where t is a fixed time-station. When the matrices A^ and A 2 are independent of time-, by using a theorem on the uniform boundedness of sparse matrices proved in [ 5], it 2

2

can be shown that, if At/Ax^ and At/Ax 2 are arbitrary positive constants, the difference scheme (21) is stable. Thus, if we continually refine the time step size At, and 2hence also the spatial mesh-spacings Ax^ and Ax 2 (in such way that At/Ax^ is an arbitra ry positive constant, i=l,2), and require more and more steps to reach a fixed time-station i, the computed solution is always bounded. b) Case B. >> A ^

i-1,2

In this case, on each interval [

»

2 . . . ( t Q = 0 ) , we

split the system (6)-(8) into two systems (y(t Q x) = v(xj):

9?

2 (22)

- M , •t " 1. =1

B

"' Ct T-l X)

=

iJ. * VA 1• y(t

=

in 8 * ( t

0

t-l'S + l

in n = n + r

T-lXj

2

in ft x (.t in fi

(23) i(tx) = w(t x)

on r * [ t

y ( t T + 1 x ) = V 1£R

|gx ( x , l ) | * a 2 ( x ) + p 2 | l | J

A

2

«R

, a.e.

1

such that

x «12.

( 1 1 ) ( a ) h ( . , u ) i s measurable onXXfor every u € R 1 . ( b ) h ( x , . ) i s continuously d i f f e r e n t i a b l e in R1 f o r almost every x c X l ; h u ( x , . ) denotes the derivation. ( c ) For every 7l > 0 e x i s t s an element a^€L 1 (Cl) such that |h(x,u)l , j h u ( x , u ) | ^OJJ(x) l^u«R 1 with f u| ) *

oonst*

i s v a l i d . This means K(v)€ ^(¿"(£1) ,H - i (CI.)) ) . Using the continuous properties of d ' ( v ) mentioned above i t can be shown that the map v — m ( v ) i s a continuous one from J ^ - C L (CI) into ^ ( L " ( n ) , « j r ( H ^ ( Q . ) , H ~ 1 ( Q ) ) ) . Hence the proof w i l l be f i n i s h e d , i f the r e l a t i o n K ( v ) = /A'(v) i s proved. Let 6 be a real positive number, such that v + u e j ' - f o r every u4L°"(fl) with //uJ»at 0 and J 2 ( x ) ~ ? i ( ) * const. > 0 a.e. x t ¿1 (cf. (1)), then there exist q+1 real numbers yMQ,-..yK^ with 0 and |/«0| +.».+ |/«q/>0 such that L

u,u-S(jr'V/V-"/V

Vxx&V

-

These assertions immediately follow by results decribed in / 1 , V (It must be noted that the adjoint states in /1 / and / V , here given by the solutions of (20), differ in their sign.) The optimality conditions in (ii) are integral inequalities. Applying a lemma by KRARNOSELSKI from these we can imply local conditions, too. Namely, if we introduce the notation L

v,u(y'z; / V - ' / V

=

JHvf3r'z5 /,o'"'^q:?(x)Ll(x)

dx

(cf. (19)), then under the above assumptions the integral conditions are equivalent to Hfi f y , z 0 ; / « 0 , . . . ^ q J (x)(u - u(x)) *0 V U « [ ? / * ) > F 2 (*)J

a.e.

X t O.

with the corresponding parameters Hereby and by the maximum principle for solutions of elliptic equations one can conclude bang-bang properties for the optimal control in various particular cases. We refer to /1/. In some ways the above theorem gives final results. But, in the case q > 0 a result of preliminary character is of special interest. It reads as follows:

113

Supposing that the assumptions of the theorem are fulfilled, that the functions h°(.), j = 1(1)q> given by the defining relations Lg^Cf.Zj)- (1972), 442.456.

[17]

WEISS, R.: Product Integration for the Generalized Abel Equation. Math, of Comp. 26 (1972), 177-190.

Incorrectly Posed Problems.

Austra-

ISNM 31 (1976), 57-63.

Integralgleichungen. De Gruyter, Berlin und LeipBIT $

ON UNIQUENESS OP SOLUTIONS FOR INVERSE PROBLEMS OF POTENTIAL THEORY by V. M. ISAKOV Institute of Mathematics, Sibirian Branch USSR Academy of Sciences, Novosibirsk 630090

In this talk we will outline some recent progress we have made in the investigation of uniqueness of solutions of the inverse problem for the heat equation and the wave equation In general form the inverse problem of potential theory is formulated in the following manner: The value of the gravitational field is given outside a given domain. It is required to find the domain and the density which create the field. It is connected with important problems in geophysics and celestical mechanics. This problem has been extensively studied by P.S. Novlkov, A.N. Tikhonov, V.K. Ivanov, M.M. Lavrentiev, L.N. Sretensky^ A.I. Prylepko, etc. [4], Theorems on uniqueness, stability and existence of solutions have been obtained; the conditions of the latter ones being rather strict due to incorrectness of the problem. In the present paper we outline several results which are natural analogies of the inverse problem in potential theory for classical non-stationary equations of mathematical physics. Only questions on uniqeness which play an important role in studying the problems according to A.N. Tikhonov [5'j are considered. Let E.^ be an n-dimensional Euclidean space with elements x = (x^, ... , * n ), |x| the distance between 0 and x. Let further E q + 1 be the space of points X = (t,x), t 6 ¡ ^ and D a cylinder (0, T) x G in E n + 1 » where G C E(0,T) the set {x : 0 < t < T } , by .

*

D

3

is a domain. Denote by

the side boundary "3D \"3E(0,T) of a 3

set D. C E(0,T) and by Xd.! t h e characteristic function of D.. Fix the density •J J J ^C where

( o , H

( B ) , ju > 0 in D ,

(1)

Consider the bounded subsets D

j

^

C

t 0,T l

X G)

'

5 = 1,2

'

which satisfy the following conditions: D^ f\ {t = 0 ] , (D \D..)f| [t = 9 ^ are domains in E n for 6 €r(0,T);

(2)

136

the boundary U

D. is in a neighbourhood of each of its points equal to the x 0

graph of the function x t = f(t,x 1 t ... , * i _ 1 , x . + 1 , ... , x n ) , f fc C ( 1 » X ) ( E n ) .

(3)

In the following the non-empty cylinder (0,T) x G o in D \ (IT U'lJg) is of interest, where G is a ball in E . o n Theorem 1: If the sets ( D - s ^ U S g ^ O I* =

6

}

are

connected

W

at 6 € (0,T)

and the u^ are (generalized) solutions of the Cauchy problem -

J

satisfying u

1

= u

2

° n D, u"5 = 0 on {o} x 3, j = 1,2

on (0,T) x G q ,

,

(5)

(6)

then D^ = I>2. Here we will outline the proof of Theorem 1. The following lemma holds. Lemma 1:

U" f| {t = 0 J

=

T>2 0 {t = 0 J .

Denote by D~ the cylinder (-T,0) x G^, where G 1 is the interior of the crosss e c t i o n ^ A { t = Oj.Using properties of thermal potentials, condition (1), condition (2) and Lemma 1 it follows that u = u2 - u1 £ where

=

C(1,2)(K.), (7) 3 = DgX^,, are compact subsets of E(G,T). Using (5), (6) and

(7) we obtain the following statemant Lemma 2: There exists a t 0. Due to (5) we have ut - A u

^

- X-jj ) on D, u = 0 on ¿o} x G 1

(9)

137 From equality (6) due to conditions (2), (4) we obtain that u = 0 on D \ 0 3 1 U T 2 )

(10)

We multiply both parts of equation (9) by u and integrate over the domain E(0,t): * "fc jdTj(u_-Au)udx

°

=

jdt

Gc

o

Gr

The left-side integral due to (9), (10) is greater or equal to \

( u2(t,x)dx . Q

From Lemma 2 it follows that u = 0 on E(0,t0), which contradicts assumption (8). Therefore D 1 = Dg. Conditions (2), (4) are o? topological character and hold, in particular, for domains D^ which'are starlike (in a certain sense). The complete proof of Theoreml is given in Jl}

in a more general form. Uniqueness theorems for solutions of a

variational inequality of parabolic type have been essentially used [3j •

We now consider the wave operator

11

"Vt2

We suppose that the density^. satisfies the condition / ¿ e C 2 ^ ) , 0 < £ ^ < £ " 1 o n E4 .

(11)

In Theorem 2 we consider the following domains Dj = ( X s | x | < d ^ t , j x ~ 1 | x ) } where d.. € c3(E1 X"3B(0;1)) ,

|(dj)t|
0, cec(b.v), vfeD .

(19)

We denote the solution operator of (19) by S.Besides of (pQ.V.I.) we consider the parametric variational inequality (pV.I.)s

148

(pV.I.)

Find [b.d.v^tBXZ

such that bfcC(d,v)

and

(A(b,d,v)-b*,c-b) ^ 0, c€:C(d,v) , vt-U.

(20)

We denote the solution operator of (20) by S Q . Lemma 15. If G(SQ) is with yC =max ^ , .

(j , ,'S")-closed,then

G(S) is

(oc/C)-closed

Proof. The proof of this Lemma is the same as for parametric quasi-minimum problems in C^J. Lemma 16. Let U be 'c.-closed, K € (U>—^2 ) satisfies the condition er-Hm Then Z=G(K)

K(v n )C K(v)

for v n ^ > v

in Ü.

(21)

is ( S'.'C)-closed.

Proof, See Lemma 2 of section 2.3 in On the base of the Lemmata 15 and 16 we can prove the closedness condition's of G(SQ) by the results of section 3.2 and solve the inverse problem for the parametric quasi-variational inequality (19) by Theorem 11 in an obvious.way (see for inverse problems in parametric quasi-minimum-problems References [l] Browder,P.E..Nonlinear operators and nonlinear equations of evolution ih Banach space.Proc.Symp.Pure Math.Vol. XVIII.Part 2.Providence 1976. Browder,F.E.,Hess,P.,Nonlineare mappings of monotone type in Banach spaces.J.Funct.Anal* 11 (1972) ,251-294. Kluge,R.,Optimal control with minimum problems and variational [3] inequalities.Lecture Notes in Computer sciences 27 (1975) »377382. £4] Kluge,5., Über eine Klasse von Minimumproblemen mit Neberibedingungen.Math.Nachr. 78 (1977),37-64. [5] Kluge,R.,On some parameter determination problems and quasi-variational inequalities.InsTheory of nonlinear operators.Abhandl. der AdW der DDR.N6.Akademie-Verlag,Berlin 1978.Ed.R.Kluge.S.129-139. [¡s] Kluge, R^Nichtlineare Variationsungleichungen und Extremalauf gaben. VEB Deutscher Verlag der Wissenschaften.Berlin 1979. jj] Kluge,R.,On some parameter determination prpblems in variational and quasi-variational inequalities.I.Math.Nachr.

14-9 [8] Kluge,R.,Langmach,H.,On some problem of determination of functional parameter in partial differential equations.In:Distributed Parameter Systems.Modelling and Identification.IFIP Working Conf. Lect. Notes in Control and Information Sciences 1,Springer-Verlag, Berlin-Heide lberg-Bew York 1978,p.298-309. [9] Kluge,R.,Langmach,H.,On the determination of some rheologic properties of mechanical media.In:Theory of nonlinear operators.Abhandl. der AdW der DDR. N6.Akademie-Verlag,Berlin 1978,Ed. R.Kluge.S. 141-158. [10] Kuratowski.K.,Topology.I.Academic Press,Hew York 1966. fill k a n g m a c h , H . » t l i e determination of functional parameters in some parabolic differential equation.In:Theory of nonlinear operators. Abhandlungen der AdW der DDR.N6.Akademie-Verlag .Berlin 1978.Ed. R.Kluge.S.175-184. [12J Rockafellar,R.T.,On the maxi mal monotonicity of subdifferentials mappings.Pacif.J.Math. 33 (1970),209-216. [13] Rockafellar,R.T.,0n the maximaIity of sums of nonlinear monotone operators.Trans.Amer.Math.Soc. 149 (1970),75-88.

Uniqueness and Continuous Dependence of the Null Solution in the Cauchy Problem for a Nonlinear Elliptic System

by

Introduction.

R.J. Knops

Heriot-Watt University, Edinburgh

L.E. Payne

Cornell University, Ithaca, N.Y.

In this paper we investigate the Cauchy problem for a class of

nonlinear elliptic systems using a method which was employed by Payne [U], [5] to study questions of uniqueness and continuous dependence in the Cauchy problem, first for the Laplace equation, and later for a more general second order elliptic equation. This method relies on logarithmic convexity arguments which have been employed by several authors to analyze a variety of properly and improperly posed problems. interested reader is referred to a recent survey on the subject by Pavne [6],

The The

major aim of this paper is to establish a fundamental differential inequality which will permit the application of logarithmic convexity techniques.

The derivation of

uniqueness and continuous dependence of the null solution (as well as other results) from the convexity property is by now standard and is therefore only sketched. The nonlinear system which we consider is .not covered by the results of Trytten [10] and Schaefer [8], [9], since they did not consider coupling in the highest order derivatives as occurs for instance in the system of equations of nonlinear elasticity and in other equations of continuum physics.

It should be emphasized, however, that

we establish uniqueness and continuous dependence results only for the null solution. Thus our results must be regarded as a first step in what will hopefully be a more general theory for fully nonlinear systems. In Section 2 we set out the notation and basic equations and indicate the relation to nonlinear elasticity.

Section 3 is devoted entirely to the proof of the fundamental

inequality, while the last section discusses uniqueness and continuous dependence results.

152 2.

Notation and Basic Formulation. N Let

D Z

portion that

Z

be a domain in

R

whose boundary

3D

is Liapunov and contains a

of finite area over which Cauchy data is specified.

It is assumed

is a (^-surface.

Let

p: R •* R

be a nonlinear continuously di fferentiable function which satisfies

the inequalities p(s) > 0, s > 0,

(2.1)

¿ S p(t)dt < sp(s),

(2.2)

N and let

u: D ->• R

u^, i = 1

N.

be twice continuously differentiable with components

We adopt the summation and comma conventions and consider the

nonlinear system of equations (a rk p(a)u i)r ), k = 0, x f D

(2.3)

in which 5

and the

a

" arkui,rui,k'

a

a^j

are positive-definite in

ijSi5j i V i E i '

for some positive constant n^

'U)

are symmetric functions with uniformly bounded first derivatives.

We suppose also that the

Let

(2

x f

D;

D

Z

'5)

a^.

be the components of the unit outward normal on

conormal derivative on

(2

Z , and define the

by

3u. 17=p(a)arkVi,r

2

(3

-lU)

We may then use (3.12) and (3.1k) together with the Cauchy-Schwarz inequality, easily derive the inequality

>*'aa-a>2i

CT

. 1

a ,
.

r

iaIwr(ip(t)dt)as-W,kui,4arkui,rp(a)dx

" 2

,

+ ?2 I B (£ p(t)dt)dx + 2± ' p(a)gp man p 1u.m EaP.P 0 V ' '

1u. ,n

dx

(3.19)

>. - c^ j. ap(o)dS - c"2 ^ op(a)dx,

(3.20)

where we have used the positive-definiteness of derivatives of

a^

and of

a^j, the boundedness of the

g^, and assumption (2.2).

positive and are determined from bounds for

f, a.,

The constants

c

i >°2

are

and their derivatives.

Use

of (2.2) again in the left of inequality (3.20) finally leads to a lower bound for the expression in square brackets in (3.15).

**'a 1 werden aus e i n e i komplexen Eigenfunktion w(z) mit Eigenwert - q < - 1 Funktionen F , G usw. mit l e i c h t zu modifizierendem E e h r a e u t i g k e i t s v e r h a l t e n g e b i l d e t . Wieder f o l g t h = C (= r e e l l e Konstante). In ( 3 ) i s t ^ durch 1/j,^ zu e r s e t z e n , entsprechend i n ( 4 ) , i n ( 5 ) z u s ä t z l i c h noch 2N durch 2H - 1 , ganz. Zur Bestimmung der I n t e g r a t i o n s k o n s t a n t e n i n ( 4 ) nutzen wir =0 aus entsprechend der Tatsache, daß G i n z eine ungerade Funktion i s t . Dem wird durch Y0-1 -

1

§ S* = A

2

-

1

§

+ ( | + nTT) i , n ganz,

Rechnung getragen. So erhalten wir dann (S)

q

Nach entsprechender Prozedur wie oben haben wir b e i (9)

qN ' f y j & f ^ * S p f ^ j .

» " 1. 2. 3

d i e Eigenwerte - q = - q ^ der aus einem Zuge bestehenden Cassiniseilen Kurve f ü r diesen F a l l 0 < 1. V. Im Gegensatz zu ( 9 ) werden f ü r diesen F a l l 0 < angegeben, die s i c h zu (10)

-

j ^ f f l ^ L j

, 5 - 1 ,

i n [ l ] Werte

2,

umschreiben l a s s e n . (Der F a l l ^ > 1 wird i n [ 1 ] n i c h t b e h a n d e l t ) . Und zwar e r g i b t s i c h ( 1 0 ) aus [ 1 ] ( S e i t e n 2 0 4 3 / 4 4 ) , weil dort die Sunue über d i e ungeraden m = Differenz der Summe über a l l e m und der Sua^e

164, ü b e r d i e g e r a d e n m. E r s t e r e l ä ß t s i c h n a c h [ 3 ] » 1 6 . 2 3 . 3 auf S e i t e 5 7 5 ( d o r t u = 0 g e s e t z t ) , zusammenfassen, l e t z t e r e ebenso nach v o r h e r i g e r Landenscher Transformation des Ausdruckes K'/K ( v g l . [10], S e i t e n 237/ 8 ) . Daß ( 1 0 ) t a t s ä c h l i c h n i c h t d i e E i g e n w e r t e d a r s t e l l e n k a n n , e r k e n n t man ü b r i g e n s a u c h s c h o n a u s f ü r y^ 1 (bei festem N). Dies g e h t n ä m l i c h n i c h t a n , da l e n d s e i n muß.

d a b e i n a c h [ o ] , S e i t e 1 2 0 9 , monoton

fal-

Schrifttum [ 1 ] BLUMüHFELD, J . u n a «¡AYEE, W.s Über P o i n c a r f e ' s e h e F u n d a m e n t a l f u n ü t i o n e n S i t z u n g s b e r . d . K a i s e r l . Akad. d . W i e s . , M a t h . Nat. K l . , J 2 2 , Abt. I I a (1914), 2011-2047. [ 2 ] GAIER, D.s K o n s t r u k t i v e M e t h o d e n d o r K o n f o r m e n A b b i l d u n g . G ö t t i n g e n - R e i d e l b e r g 1964.

Berlin-

[ 3 ] Handbook o f m a t h e m a t i c a l f u n c t i o n s w i t h f o r m u l a s , g r a p h s , a n d m a t h e m a t i c a l t a b l e s . E d i t e d b y M. ABEALIQ'jTIiZ and I . A. SxEGUiT New York 1965« [ 4 ] KÜHNAU, Ew: E i n e f u n k c i o n e n t h e o r e t i s c h e R a n d w e r t a u f g a b e i n d e r Theorie der quasikonformen Abbildungen. Indiana Univ. Math. J . 2 1 (.1971), 1 - 1 0 . [5]

i

[6]

,

[7]

,

[9]

>

Zur a n a l y t i s c h e n D a r s t e l l u n g g e w i s s e r E x t r e m a l f u n k t i o n e n d e r q u a s i k o n f o r m e n A b b i l d u n g , M a t h . N a c h r . 60 (1974), 53-62. Eine V e r s c h ä r f u n g des Koebeschen V i e r t e l s a t z e s f ü r q u a s i k o n f o r m f o r t s e t z b & r e A b b i l d u n g e n . Ann. A c a d . S e i . F e n n . , A. I . M a t h . , l ( 1 9 7 5 ) , 7 7 - Ö 3 -

Eine Integralgleichung in der Theorie der quasikonformen A b b i l d u n g e n . M a t h . N a c h r . £ 6 ( 1 9 7 7 ) , 1 3 9 - 1 5 2 . [ ö ] SCHIFFES, M . : The F r e d h o l m e i g e n v a l u e s o f p l a n e d o m a i n s . P a c . J . Math. 2 (1957), 1167-1225. t Fredholm e i g e n v a l u e s of m u l t i p l y - c o n n e c t e d domains. P a c . J . Math. £ (1959), 211-269. [ 1 0 ] TRICOKI, F . s E l l i p t i s c h e F u n k t i o n e n , ü b e r s e t z t und b e a r b e i t e t v o n M. KRAFFT, L e i p z i g 1 9 4 6 .

SOME PROBLEMS OP ANALYTIC CONTINUATION M.M.LAVRENTJEV 1. A large class of an analytic continuation problems for the functions of one variable can be formulated in the following way. Let f(Z) be analytic function of complex variable in the domain of complex plane. Some additional information on function ^(Z) can be given. For example, j(J) is continuous in JjJ (closure of ) and bounded in $ with given constant:

IfCDl Let

A

t

* e ,

3

be some subsets of

$

,

A c 3 c ¡0 and values of j?(2) on the set A are known. The problem is to find the values of f(2) on the set ft . 2. Analytic continuation problem are classical problems in analytic functions theory and in the theory of complex variable functions; some uniqueness theorems for this problems were proved in the last century and are presented in most of monographs'. If we consider the analytic functions in »0 with no additional restrictions, then analytic continuation problems are linear. Thus the uniqueness of solution in the analytic continuation problem is equivalent to the statement: Let fCZ) be analytic function in the domain ,

X 6 Pi Then

{(Z)*0,

26

_

&

166

Note, that in uniqueness theorem for analytic continuation ^problems set

B

includes the domain of analyticity B

D

iO

:

SO

3. We present now the classical results on uniqueness of a solution in analytic continuation problems (see 2, 3, 5)« f(2)

Theorem 1. Let and

be analytic function in the domain

is a subset of

,

.'Then, if

fC2)=-0,

we have

f ( Z ) ' C ,

Theorem 2. Let

ZC-. 0

1

e. A

' { ( 2 )

/"*

if

-

0,

.

{ ( 2 )

Z & X )

-

be analytic function in the domain ¿0 f

, and

f e z )

w e h a v e

.

_ '

Theorem 5. Let f(2) boundary

, and set with

. Then, if

-

0

=

0 ,

C- f

i

with

be some straightened curve. Then,

i c - r ,

1

Z € ) 0

.

A notion of riemannian manifold is connected v;ith analytic continuation. He give here some corresponding definitions (see 5)• it" the totality Definition 1. .Ve call "an analytic element" of a domain in

#

/I

in complex plane and analytic function and values / (Z) are given in the set /) 1) To determine values f(2) in some set Ji .7» j\ .

168

2) To determine some characteristics (say, topological) of the-Riemannian surface of the function £ (X) . 5) One needs to determine the singular points of the function fCZ) and behaviour of the function in the neighbourhoods of this points, in particular, types of the singular point®, and behaviour of the function in the neighbourhoods of the singular points of the Hiemannien surface. The following theorem is valid (see 3, 5). ¡Theorem 4. Let the set A includes limiting point inside the Riemannian surface of analytic function ) . Then Riemannian surface of function j!( Z) is uniquely determined by values of ^(Z) on the set A 6. We present now the formulation of analytic continuation problem' for the function of several variables, an analogue of formulation in one-dimensional case. Let I z y ' f 2n) be analytic function of /I complex variables in the domain ¡0 of /7 -dimensional complex Bpace, A and B be some subsets of ¡0 *

A

c

B c

J5

The values of a function ^fft,^*,--,^)are known in the set i n tlie set B> * fo determine values • • •, Problems of analytic continuation theory for the functions of several variables were treated from the begining of our centupy. Analytic continuation playes an important role in the monographs on the theory of analytic functions of several variables (see 1, 4, 5). The space of (l complex variables ..^Jwill be denoted by C" • pefinition 7. A domain & C , strictly including iO , is called holomorphic expansion of ¡0 , if any in iQ analytic function , 2 n ) has an analytic continuation onto

G

•

It is known, that the domains in complex plane have no holomorphic expansion, but in the spaces of two and more variables Such domains do exist. Next theorem establishes the existence of 4>ne class of such domains: __ 7 (pheorcm_5. (Hartoga). Let £1 ha a domain in. £ be a

domain i n t h e con®lex p l a n e , M

be a boundary of

»

be a s e t M * [ x > x r , ] u [ f t , . . . . , * : )

»here

¿0^

(zf,

)

*

€ JQ

Then every f u n c t i o n a n a l y t i c i n some neighbourhood of s e t M ( i n t h e space Q ) can be a n a l y t i c a l l y c o n t i n u a t e d t o t h e domain: £

-

$

*

SO 4

7 . The s o l u t i o n s of some a n a l y t i c c o n t i n u a t i o n problems cont i n u o u s l y depend on d a t a . I n case of one v a r i a b l e i t i s t h e case, when s e t A i s equal t o /"* , boundary of t h e domain ¿Q . For s e v e r a l v a r i a b l e s i t i s , f o r example, an a n a l y t i c c o n t i n u a t i o n i n t o holomorphic expansion, which was mentioned i n the theorem 5» For t h e f u n c t i o n of one v a r i a b l e t h e a n a l y t i c c o n t i n u a t i o n problems from t h e i n n e r s e t of t h e r e g u l a r i t y domain and from t h e p a r t of the boundary of t h e domain a r e u n c o r r e c t i n a c l a s s i c a l s e n s e , a n d c h a r a c t e r of t h e i n s t a b i l i t y i s the same, as i n s t a b i l i t y i n t h e u n c o r r e c t Cauchy problems f o r Laplace and h e a t conduction equations. 8 . As known, t h e s o l u t i o n s of some d i f f e r e n t i a l e q u a t i o n s are t h e a n a l y t i c f u n c t i o n s i n t h e r e g u l a r i t y domains. I n p a r t i c u l a r , t h e s o l u t i o n s of e l l i p t i c e q u a t i o n s with a n a l y t i c c o e f f i c i e n t s have t h i s p r o p e r t y . I n t h i s connection t h e f o l l o w i n g problems a r i s e ( a l s o c a l l them a n a l y t i c c o n t i n u a t i o n p r o b l e m s ) . Let be a d i f f e r e n t i a l o p e r a t o r i n t h e space (Xj, with the following property: livery s o l u t i o n of the equation u> (1) .ZU = C i s an a n a l y t i c f u n c t i o n of v a l i a b l e s •, i n t h e regul a r i t y domain. I t i s known, t h a t equation (1) has r e g u l a r s o l u t i o n i n t h e domain SO . U(Xt,...,X*J

170

Soma additional information on the function

U (Xf)

Xn)

can be given, for example,

lui*a

,

(**,

, *») & £

let A , B be some subsets of £ . The values , and we need to of the solution (1) are known in the set A determine the values of the solution in the set Q .

R e f e r e n c e s 1. Vladimirov V.S. Methods in the theory of the functions in several complex variables. Moscow, Nauka, 1964- (Russian). 2. Lavrentjev U.A., Shabat B.V. Method in tbie theory of the fun-f ctions of complex variable. Moscow, Fizmatgiz, 1958 (Russian). 3« Markushevich A.I. Theory of analytic functions. Moscow-Lenin-f grad, Gostehizdat, 1950 (Russian). 4. Fux B.A. Introduction to the theory of analytic functions in several complex variable. Moscow, Fizmatgiz, 1962 (Russian). 5. Shabat B.V. Introduction to the complex analysis. Moscow, Nauka, 1969 (Russian).

Prof. Dr. M. M. Lavrentjev, USSR Academy of Sciences, Sibirian Branch, Computing Centre, Novosibirsk 630090

ABOUT INVERSE PROBLEMS IN CONTINUUM MECHANICS AND SOLID STATE PHYSICS

BERND

M I C H E L

Summary: Applying linear continuum mechanics of solids containing structural defects, the author deals with different physical problems leading to inverse and improperly posed problems of potential theory. Higherorder elliptic equations, such as biharmonic and modified biharmonic equations, are derived for special cases of crystal symmetry. It is shown that essential advances in the determination both of the sources of field quantities and related material parameters can be achieved, if the solutions of inverse problems as presented in this paper are available. 1.Introduction The present paper is not concerned with the derivation of solutions for special cases of inverse problems, but an attempt is made to formulate certain inverse problems corresponding to elliptic differential equations. For simplification the author restricts his investigations within the field of linear continuum mechanics of solids. Continuum mechanics of solids has become more and more important in recent years, since it is necessary for the progress of many branches of solid state physics and modern material sciences, in general,e.g. fracture mechanics /21/ and soil mechanics /4/,/22/. Practically all of modern physics deals with fields: potential fields, probability fields, electromagnetic fields, tensor fields,and spinor fields. Classical theory of elasticity as a typical field theory is strongly related to potential theory. The famous works of PAPKOVITCH /19/ and NEUBER /20/ are concerned with this subject. Clas-

Br. B. Michel, Sektion Fertigungstechnik Technische Hochschule, DDR 901 Karl-Marx-Stadt

172 sical theory of elasticity of solid bodies deals with their behaviour under the action excerted by external loads. Most modern branches of linear continuum mechanics of the solid state, on the other hand, are concerned with a medium containing structural defects(dislocation theory etc. /8/). The properties of these defects are essential for the physical (not only the mechanical!)behaviour both on a microscopic and on a macroscopic scale. Prom the geometrical point of view one can divide the defects into several classes; 1. zero-dimensional or point defects (impurity atoms, such as interstitials,vacancies etc.), 2. one-dimensional or line defects (e.g. dislocations), * 3. two-dimensional defects or plane imperfections (stacking faults . and grain boundaries), and 4. three-dimensional defects or space imperfections (volume defects su'ch as inclusions, precipitations, interface layers, regions of decomposition etc.). The theoretical study of the deformation and stress fields in the defected region presents extremely difficult problems, since both the exact distribution function of the defects and the local material parameters usually are not available. Continuum mechanics describes the deformed solid by suitable field functions such as stress and strain tensors /10/. The sources of the field are represented by socalled "source functions" (usually density functions). Th? latter are strongly related both to the defect properties and.to the material parameters in the defected region /13/-/15/» For the physisist the following two basic problems result: 1. To get information about the source functions (without being able to obtain experimental values immediately within the defected region) . 2. To get information about the material properties in the defected region(e.g. this problem is related to questions about the exact values of elastic constants,so-called local elastic constants).

173

It is impossible, in general, to separate the two problems from each other, because the "strength" of the source function, as a rule, is not independent of the real values of the material parameters immediately at the defects. That is why it is 6n approximation only to apply, for instance, YOUNG'S modulus, obtained by a compact tension test, for a workpiece after machining in order to calculate the deformations remaining in the strongly distorted inhomogeneous surface region of the same solid. An exact calculation should,however, take into account the changed material constants in the sub-surface layer of the solid /17/. Local material parameters usually are not available, but their values may be very different frome those, which were

obtained by con-

ventional experimental methods for the whole solid. This problem has not yet been, solved up to now. So the values of elastic parameters outside the defects are different from those inside them. This kind of questions has become typical for many branches of modern physics and material sciences /15/.atomic physics /25/ and soil mechanics as well /4/. It always leads to inverse and improperly posed problems. 2. Inverse problems for solids with "quasiplastic" deformation due to structural defects 2.1. Distortions in an anisotropic medium In the framework of linear continuum mechanics it is possible to carry out separate investigations for stress and strain fields connected with structural defects on the one hand and for stress and strain fields due to external loads on the other. This fact is expressed by the theorem of COLONETTI, which states that the elastic energy of static interaction between a source of internal stress and a field of external stress vanishes. The structural defects, described above, are the sources of additional stress and strain fields /8/,/11/-/13/,/16/.

The

elastic medium (matrix), with possible exceptions of some singular points, lines, or planes, can be described by a regular displacement field u, the tensor of total distortion (* being given by the relation

174

fi'uv where

fl

t

is composed of two terms (i m

fl4l

t

•'

( 2 )

It is assumed that the elastic part flei is related to the stress tensor ff by generalized HOOKE•s law

The quantity U^(E) is given by the expression

where k is the BOLTZMAOT constant. The corresponding inverse problem consists in the determination of the drift potential E, especially in the immediate vicinity of drifting defects. In recent works of the author a way is shown to find an approach to understand drift diffusion processes near solid surfaces and interfaces /11/,/18/. From the drift potential one can deduce new assertions about the governing elementary processes on microscopic scale and about interaction phenomena between the defects /11/. The newly developed methods and results known from the inverse quantum mechanical scattering problem should be applied to solve this kind of inverse problems too /25/. 2.5. Elastic coupling between''temperature field and deformations The well known basic equations for the displacement field u* of a

178 thermoelastic solid i f / r Jj M r f l f - —

4-2V

Z"u *

'

G

f t *

( 18 )

6

is coupled with a generalized heat conduction equation of the kind

k4 AT + a4 A-y-g* * bj K denotes mass forces, whereas k ^ a ^ b ^

s

and c 1

Cf-r

. ( 19 )

are material para-

meters. We have to mention the importance of the term

a^div u

in

(19), which is responsible for the coupling between heat production and elastic deformation. It represents the change of elastic dilatation ¿ j = div T? with time. Usually this term is not taken into consideration. (18) and (19) can be uncoupled /3/. This leads to a temperature equation /3/»/4/. For the case of a quickly changing periodic temperature field (of constant frequancy V

) one can de-

rive equations of the following type /14/,/l5/ z where J\

*

f

,< 20 )

is the polyharmonic operator of the order i. f is known,

outside the sources. It is a single -valued function of the heat sources and of severa.1 thermomechanical parameters. The inverse problem is the determination of f and T'inside the source region, T and f being given outside the sources only. The progress in this field is connected with real advances in several branches of material sciences, which are of technological importance, since the sources of heat production play an important role in mechanical . engineering sciences such as metal cutting and metal forming

/'7/,

3. Concluding remarks The kind of inverse problems, which have been outlined within this paper, has become typical for coupled problems in many branches of solid state physics and continuum mechanics as well. For this reason one likes to speak about the continuum thermodynamics of solids today. For simplification we were forced to restrict our investigations to linear models here. The more general conception of both nonlinear constitutive laws (general constitutive theory, which

179

also includes plastic and viscous flow) and geometrically non-linear relations(such as large deformations) provide the basis for new and more complicated types of equations and corresponding inverse problems. Including higher-order terms of the stresses,distortions and further field quantities (e.g. magnetoelastic or electro-mechanical ones) is possible, but sensible only after achieving real advances in the study of the analysis of inverse problems for the linear case. Acknoledemente The author is grateful to Professor G.Anger and Professor P.Loges of the Martin Luther University Halle-Wittenberg and to Professor H.Weber of the Technische Hochschule Karl-Marx-Stadt for useful discussions. Bibliography /1/ Anger,G.,Die Rolle der modernen Potentialtheorie in der inversen Aufgabenstellung,Gerlands Beitr.z.Geophys. 85(1976),1-20. /2/ Cahn,J.W.,0n spinodal decomposition,Acta Metallurg. 9(1961),795. /3/ Carlson,D.E.,Thermoelasticity,in: Handbuch d.Physik,Bd.VIa/2, Springer-Verlag,Berlin,Heidelberg,New York, 1972,328. /4/ Chamecky,S.,Soil structure interactions in the analysis of raft formulations,2o.PolishSolid Mech. Conf.,Porabka-Kozubnik,1978. /5/ Deuretzbacher,G.,Das Eigenspannungsfeld eines Gleitbandes in einem kubischen Kristall,Diss«Vniv.Halle-Wittenberg,1973/6/ Hieke,M.,Eine indirekte Bestimmung der Airyschen Fläche bei unstetigen WärmeSpannungen,ZAUM 35,8(1955),285-294. /7/ Hieke,M.,Die Anwendung der inhomogenen Differentialgleichung 4. Ordnung und die Festkörperphysik,Beitr.z.Analysis,4(1972),61-67. /8/ Koväcs,I.,Zsoldos,L.,Dislocations and plastic deformation, Budapest ,Akademiai Kiado,1973. /9/ Maschke,H.,Frischbier,R., Zur Integration der Grundgleichungen für Eigenspannungsquellen in kubischen Kristallen, Kristall u. Technik 12,1o(1977),1001-1oo6. /1o/ Michel,B.,Loges,F.,Probleme der Kontinuumsmechanik inhomogener Werkstoffe,Wiss.Z. Univ.Halle 25M,5(1976),5-15. /11/ Michel,B.,Influence of solid surfaces on the energy of point defects and inclusions,2o.Polish Solid Mech.Conf.,Porabka-Kozubnik, 1978. /12/ Michel,B.,Beitrag zur Ermittlung der elastischen Energie von Punktdefekten, Einschlüssen und Entmischungsbezirken, Acta Phys. Hung. 44,2(1978),147-156. /13/ Michel,B.,Diss.z.Prom.B., Univ.Halle-Wittenberg,1979.

180

/14/ Michel,B.,Spannungen und Verformungen in Pestkörpern und inverse Aufgaben der Potentialtheorie,Sommerschule über inverse Probleme der Geophysik, Freiberg,June 1978, u. Veröff. d. Nationalkom. f. Geodäsie u, Geophys., Potsdam (in print). /15/ Michel,B.,Über das inverse Problem in der Kontinuumsmechanik fester Körper, Vortrag,Math.Koll.Sektion Math. Univ.Halle-Wittenberg, 12.10.1978. /16/ Michel,B., A note on the interface interaction of point defects and volume defects, phys.stat.sol.(b) 89(1978),K179-K181. /17/ Michel,B., A study on calculations of self-stresses in machined surfaces,Research Report, Techn. Hochschule Karl-Marx-Stadt, Mart 1977. /18/ Michel,B., phys.stat.sol.(a) 1979(in print). /19/ Papkovitch,P.P.,On the general solution of thermal stress (in Russian),Prikl.Math.Mech.1,1(1937). /2o/ Neuber,H.,ZÄMM 14(1934),203. /21/ Rice,J.R.,The elastic-plastic mechanics of crack growth,2o.Polish Solid Mech. Conf.,Porabka-Kozubnik,1978. /22/ Sawczuk,A.,Stutz,P.,0n the formulation of stress-strain relationships for soils at failure,ZAMP 19(1968),770-778. /23/ Seeger,A.,A separability of the diffusion equation with drift term, phys.stat.sol.(b) 41(197o),509. /24/ Yoo,H., Butler ,Yi.H., Steady-state diffusion of point defects in the interaction force field,phys.stat.sol.(b) 77(1976),181. /25/ Chadan,K.,Sabatier,R.C., Inverse problems in quantum scattering theory,Texts and Monographs in Physics,Springer,Berlin,Heidelberg, New York, 1977.

SH-WAVE PROPAGATION IN QUASIANISOTROPIC ROCKS: PROBLEM OP PARTIAL DIFFERENTIAL EQUATION WITH RAPIDLY OSCILLATING COEFFICIENTS +

JIRI NEDOMA

)

Summary: In this paper we propose a solution of partial differential equations with rapidly oscillating periodic coefficients and its applications to the study of problems in SH-wave propagation in a quasianisotropic rock. 1. Introduction In studies of the propagation of elastic waveB through the earth it is usually assumed that the medium composing the earth is isotropic. But any solid material which forms the core of the earth and the UDt>er mantle is usually equiped by a crystallic structure. Its crystallic structure is characterized by a regular form and unequal properties at the other directions - by anisotropy. On the strength of prevailing theories regarding the composition and the mechanical behaviour of the upper mantle these results suggest an upper mantle which is transversely isotropic to seismic wave propagation. Backus [1] shows that in the long wavelength limit a sequence of isotropic layers acts as a transversely isotropic medium (quasianisotropic medium) which is described by five independent elastic constants. Elastic constants are related to the velocities of elastic waves in a medium through the equations of small motion, i. e.

where u^ is the particle displacement vector in Cartesian coordinates x^, o J , (1)

3G =

jj q

, r 0 s30s = { ( x ^ x j ) ! * , 2

R, = {(x1fx3)| I x ^ R , x 3 = oj, = |(x v x 3 )|

Ix^iR,

X3=HJ,

+ x 3 2 = r2, r> o ] ,

r2 = ^(x1fx3)| x 1= R, 0 0, M T

T

12

= M

12 =

21 c

=

M £

66

22 we

12

=

0

°44

N

N

c

12 = 66 ¿12

£12 =

then according to Tg-j = c ^ ^23'

have

«12

- i - ^ T j ^

-^£,2 ,

C 1 2 = ( Y H ^ J»i )"1 T 1 2 * 0 ,

, i = 1.....H , and

i « "1 f

11

= £

22

=

£

33

= f

23

= f

=

31

Analogically we have T

23

=

°44 £ 23

= K

22 £ 23

Then

N

(

h i »•1

} 1 £

n foo = Mil 23 " 22 ur23 "=

So that T23 "J 1 ! £23^ » 1

=

£

11

= £

22

= £

33

23 '

N

i 4 — • "ii¡L

v

, «4

>^23 •

C

h

i 4 3

}

*

Jl, and then , • A

1,...,H ,

L T-(l) ^ „~23 (2) " _. . . 23 "

and

As

=

= t

32

L

(B) "_ t 2 3 23

= £

12 *

0 , i = 1,...,N then M 1 1 > 0, M 2 2 > 0 , and then M ^ i S 2 2 ,

i.e. Cgg> c ^ . It is true as M ^ = Cgg is an arithmetic mean and M

22

=

°44

is a

geometric (harmonic) mean o f j ^ , i = 1,...,N.

The assumption that the wavelength is sufficiently large compared to the thickness of the system of layers is essential as only in these cases we can take these materials as homogeneous quasianisotropic media. As in our investigations of quasianisotropic media h — » 0 , the

187

assumption discussed above is fulfilled. Now we shall study the limit of velocity of SH-waves for A—»0 if waves propagate perpendicular to the layer. The velocity in the direction perpendicular to the system of layers is like the marginal value the average velocity c S H defined „ N as Cgjj1 = ( y h^ c ^ )"1 , where c ^ = ( . We compare this value with the marginal value of velocity for wavelength), i.e. C g ^ = g = f

we have

. As

gh(x1,x3) G

d* = g 1 + i g 2

0

= f

i

(

+

-jr2

>»

1 , 1 + . 1 ^2 1 , ^ 1 , ^2 , 2 s - -2 5H = A (^ "TT" )• c.c -SH •T T ( ^ TT" 7T" e; )' =- -? Jl, + -rr JX2 + TTTT" 12 Prom there cSH " 2 - cSK " Then in the direction perpendicular to the system of layers we have lim

Cot, ^ lim

c?n .

This relation also holds in the case if the SH-waves propagate parallel to the system of layers. This result follows from the fact that c ^ and egg as the mean values are smaller than the greatest of the jij , i = 1,...,N. References [l] Backus, G. E.s long-wave Elastic Anisotropy Produced by Horizontal Layering. J. Geophys. Res. 67(1962), 4427-4441.

188

\2] Babuska, I.: Solution of Interface Problems by Homogenization. I. Tech. Hote BN-782, January 1974, II. Tech. Note BN-787, March 1974, SIAM J. Math. Anal. 7(1976), No. 5, 603-645. [3] Lions, J. I.s Asyptotic Behaviour of Solution of Variational Inequalities with Highly Oscillating Coefficients. In Applications of Methods of Functional Analysis to Problems in Mechanics, lecture Notes in Mathematics, No. 503, Berlin-Heidelberg-New York 1975. [4] Babuska, I.: Error Bounds for Finite Element Method. Numer. Math. 16(1971), 322-333. [5] Marek, I. and Nedoma, J.: Using Finite Element Procedure in the SH-wave Propagation (manuscript). [öj Handlovii, J., Nedoma, J. and Smid, J.s Issledovanie razprostranenija uprugich SH-voln v sloistoj srede s lokalnoj neodnorodnost'ju metodami vycislitelnovo arializa. Numerische Methoden in der Geophysik, Geoph. Inst. Czech. Acad. Sei., Prague 1973» [7] Nedoma, J.: On the Application of Homogenization to Study of the Ultrabasic Rocks. In IV. Conference on Basic Problems of Numerical Analysis. Plzen, September 4-8, 1978. [8] Nedoma, J.s Investigation of Linear Harmonic Field of SH-waves in a Stratified Inhomogeneous Medium Using the Finite Difference Method. Zeitschrift f. Geophys. 38(1972), 529-542.

+

) J. Nedoma, Geofysikälni Ostav CSAV, Bocni II, 14131 Praha - Sporilov (CSSR)

ON TEE DETERMINATION OF FUNCTIONAL PARAMETERS IN NONLINEAR EVOLUTION EQUATIONS OF THE NAVIER-STOKES TYPE Reiner N'drnberg

*)

In this paper vie deal with the problem of determination of viscosity properties of an incompressible fluid, by means of optimality criteria using measurement data of a corresponding flow. The viscous behavior of a non-newtonian fluid are characterized by a nonlinear function which gives a relation between the stresses and the velocity of deformations. The basic equations describing the flow of a viscous fluid are the Navier-Stokes equations. The corresponding boundary and initial value problems of these partial differential equations lead to evolution equations in function spaces. In a first part we give an answer to the question of existence of an "optimal" functional parameter for evolution equations in a general case. To be able to take Bingham effects into considerations we regard evolution equations with multivalued mappings. In a second part we apply these results to Navier-Stokes equations and to the problem of determination of the viscosity of a fluid.

1. Let X Q be a real reflexive Banach space imbedded in a Hilbert space H 0 , where the injection of X Q into H Q is continuous and X Q is dense in H 0 . Identifying H 0 with H ^ i t h e dual of h*0) and H 0 * with a linear subspace of X ^ C t h e dual of X 0 ) we have X

o

C I I

o

C X

o * '

where the injections are continuous and each space is dense in the following one. V;e denote by (.,.)Q the scalar product in H Q as well as between X„ and X o o For 0 < T < +«o and 1 p ^ q < +oo let X be a reflexive Banach space with

Lc'(0fT;Xo) O X

C. LP(0,T;X o ) ,

o which is imbedded in the Kilbert space K = L c (C,T;H 0 ). *) ZIMK der AdW , DDR 108 Berlin , KohrenstraBe 39

190

Identifying H with H* and H* with a linear subspace of X* we obtain I C

H C

X1 ,

where the injections are continuous and dense. By (.,.) •

I

(.,.) dt 0 0 _ we denote the scalar product in H as well as between X and Ir, and we have Lp'(0,TsXo*) C Z & G

^'(O.TjXJJ")

with 1 + 1, = 1 and 1 + 1, = 1 . p q q' p More precisely we make the following assumption to the space X. let F o i (i=1,...,m) be reflexive Banach spaces and » L Pi (0,T}F ol ) with p & P i « q. Then let S Q =[ F = with ( H be continuous linear mappings with (V.)(t) «V 0 (.(t)) f.a.a. te(0,I), where Tf0»i0 = I 0 s H Q — > H Q andV-i = I s H — > H are the identities. Next we introduce the reflexive Banach space Y = [ x « X : ff«X*J, where

means the derivative in the sense of distributions.

Remark 1. Por x, y e X with ^ J( jf , y )0dt + }( x , g

,

£ X* w.e have x, y £ C(0,T;Ho) and

)0dt = (x(T), y(T))0 - (x(0), y(0))o .

Finally let U be a nonempty bounded closed convex subset of a reflexive Banach space V. Then we consider the following minimum problem: Find a pair [x, uj C G(S) such that

f (x, u) = min^$(y, v) s [y, v]cG(S)J ,

(1)

where G(S) is the graph of the solution operator Y

S j U—f2

of the parametric evolution equation

gf + A(x, u) » f , x(0) = x Q ,

1

J

here $ s Y * U — > R1 is a functional, A : X * U — i s mapping, f C X^ and x Q C H Q .

(2)

a multivalued

191

Now we introduce some notions. X*

Definition 1. The mapping A ( . , u) s X—• 2 i s oalled monotone i f for a l l [ x , f ] , [ y , g ] e G(A(., u)) the relation ( f - g , x - y )

»0

holds; maximal monotone i f i t i s monotone and for each [ x , f j e XxX* the inequality ( f - g , x - y ) * 0 implies

for a l l [ y , g ] C'G(A(., u))

[ x , f ] C G(A(., u ) ) .

Definition 2. The mapping A(x, . ) : U—>2 i s called increasing lower semicontinuous i f for each sequence u^c U with u^—*u in V and each f A(x, u) theie exists a sequence ^ C A i x , u ^ with f ^ — > f in X*. X* Definition 3. The mapping A s X * U —f 2 i s called pseudomonotone i f (1) the set A(x, u) i s closed and convex for a l l [ x , u] e D(A), and (2) for each sequence [ x i f u^ f j « G(A), [ x ^ u i t f j J — » [ x , u, f ] in X«CV*X* and Tim ( f ^ x ± ) £ ( f , x ) imply £x, u, f j C G(A) and lim ( f i , x^) = ( f , x ) . Lemma 1» The mapping A s X X U —>2** s a t i s f i e s the following conditions: —( — ! ) —A— ( . , u) s X — » 2 J* _ is maximal monotone for a l l ucU, and x (2) A(x, . ) s U—>2 i s increasing lower semicontinuous for a l l x ( Z . Then the mapping A i s pseudomonotone. The proof i s given in K1UGE [ 5 ] , NtiRNBERG [ 9 ] . We are going to study the properties of the solution operator S. Theorem 1. The mapping A has the properties: (1) A : X*U —>2 X * i s bounded with D(A) = X*U and there exist a constant c 0 and a function c : H+—> B+ with c(r)—> +00 for r —> +•• such that min } ( g , x l d s * - c*|x|| 0 0 t«[0,T] 0

and

II x II c(0,T;H ) + ( s ' X ) > o i l * ! ) ' ! » ! f o r a 1 1 [ * • s j C G . Y* (2) A : YxU —>2 is pseudomonotone. Y Then the solution operator S s U —> 2 is defined on the whole set U and S i s a weakly closed and bounded mapping. Proof. A complete proof is given in NtJRNBERG [ 9 ] , here we give a scotch only.

192

Firstly we define a monotone and closed linear mapping L from X x H Q into ^ X H 0 by L(x, x Q ) = [ | f , x j for each z t ! with jjjf e X* and x(O) - x 0 , where D(L) is dense in X x H Q . If we take L and A canonically as mappings from X * D *H C into X*x H Q the parametric evolution equation (2) is equivalent to ( 1 + A )[x, u, x(0)J » [f, x 0 ] . The monotone linear mapping L* from X x H 0 into X * * H 0 defined by l*(y, y 0 ) = [ - If , y 0 - y(0)] for each [y, y 0 ] e X K H 0 with C X* and y(T) « 0, where D(L*) is dense in i * H 0 , is the adjoint mapping to I>. Therefore the mapping L is maximal monotone (see LIOIIS [sj ). H So we obtain the fact that the sum L + A » Y * U * H „ —• 2 o is 0 r i pseudomonotone (see also BROWDER-HESS [2J ). That means the graph G(S) is weakly closed in Y XV. Using the coercivity as s unit ion to A it is &asy to show that the mapping (L + A)"1 s X*x H 0 — » 2 Y * U * H o is bounded. Finally we prove the existence of a solution xcSu for a fixed u c U with the help of the method of elliptic régularisation (see also HONS [8] )• Theorem 2. Let the functional Y X U — * B 1 be a weakly lower semicontinuous one and assume the conditions of Theorem 1 to be fulfilled. Then the minimum problem (1) has a solution. The proof is based on the Weierstrass' theorem in its extension which was investigated in VAINBERG [llj . 2. Let A be a bounded domain of R n (n é 3) with a sufficiently regular boundary T, i.e. let P be locally Lipschitzian and H be locally located on one side of P. Further let P^ and P 2 be two parts of T withT« P-joPg and P^n P 2 By y we denote the exterior unit vector normal to the boundary. For ja finite interval [o.T] "e seek a vector field x : [o.TjxQ,—» R n , a scalar field p s [o,T]xil —» R1 and a tensor field f( s [0,T]*il — > R n * n representing the velocities, the pressure and the stresses, respectively, which satisfy the following equations and conditions:

193

yj- + (x«grad).x = div« + 7 - grad p in Q=(0,T)*.tt, where |€f| 4 u(lgrad x|) and « = a(Igrad if grad x 4 0 , div x = 0 in Q , «f-v + ^|xl2.V - p. V+ g ' onL2=(0,T)xr2 , x =0 on £•,*((>,T)*^ , x(0,.) - xQ in i l , where u : R+—» R+ is a material function which satisfies 0 < ci•(1 + r p ~ 2 ) * u'(r) 4 c 2 *(1 + r p ~ 2 ) and 0 6 u(O) * cQ for positive constants c 0 , c 1 t c 2 and p * ^ ^ , moreover let 7cL 2 (Q) n , g«I 2 (Z 2 ) n and xQc L 2 (il) n be given functions. The variational formulation of this problem leads to a generalized problem in function spaces ( see also LIONS [ 8 j , TEMAM [ l o j ). Therefore we introduce the following spaces: X0 = { x e (W1»p(jQ)i>IqCa))n t div x = 0 in ft, x - 0 onT,] , q = , 2 n HQ = [ x € (L (A) ) : div x = 0 in il, x-V = 0 on P, J , X ={xe(L p (0,T;W 1 ' p (a))AL q (0,T;I q (iD)) n : div x = 0 in Q, x • 0 on JL, } , H = | x t (L2(0,T;L2(il) )) n t div x = 0 in Q, x* v = 0 on£1 j . Furthermore we define mappings A.,(., T u) : X—>2 and A, » X—> X* by A,(x, u) =fgcX* : (g, y) = JJu(lgrad x|)«5-grad y dsdt, f.a. ycX, I oa where J c L"*(Q)n'n with 6 1 and «-igSHiif , T (A2 x, y) = - JjKx»

(udgrad x |) - u(|grad y|))(|, rad xj - J grad y|) » 0 if grad x 4 0 and grad y 4 0 , (udgrad x D . j g f j - f , - u(O)• 1 ) grad x (u(|grad x |) - u(0))|grad xj a 0 where % e I ^ i Q ) " ' " with

£

if grad x 4 0 and grad y » 0 ,

|5|61.

Taking into account the definition of the mapping A^ we obtain from the last inequalities by integration ( f - g , x - y ) ^ 0 for all f eA.,(x, u) and geA.,(y, u), that means,

u) is monotone.

3. The mapping A.j(., u) is demiclosed. Let x^^c X be a sequence with x^, — » x in X and let f^ci-jte^, u) be a sequence with f ^ — * f in X*. By definition of the mapping A 1 there exists a sequence T

1

« L"° (Q) n " n such that

( f i , y) - i£u(|grad x j where H J

6 1 and

^

=

jg|

grad y dsdt d

- | i j i f grad

for all y < X , 4 0.

Using the relation |u(|£;rad x i |) - u(|grad x|)j is c 2 .(1 + |grad Xi| p " 2 +|grad x | p " 2 ) •(|grad Xi| - | grad x J)

195

we obtain u(lgrad x ^ ) — » u(|grad x|) in L P '(Q). On the other hand there exists a subsequence — * 5 L"°(Q) n ° n . We have hence the unit ball in the space L ® ( Q ) n " n is weakly-«-closed. Further, for almost all (t,s)cQ with grad x(t,s) 4 0 there exists a subsequence grad x l t (t,s) 4 0 such that grad x ^ C t . s ) — > grad x(t,s). So we obtain ?,,(*..) = j g f H j i f t i f t

*

= ,gSHfHj| •

Finally it follows that T (f. y) = S S u(|grad xD'j.grad y dsdt for all y 6 X. OXL that means f&A^Cx, u). Obviously the mapping u) is bounded. This completes the proof. X* Lemma 3. The mapping A.,(x, .) $ U — » 2 is increasing lower semicontinuous for all x c X. Proof. let u^fc U be a sequence with u ^ — » u in V. Then we have u^c C(R + ) and u ^ C r ) — > u ( r ) for all r « R + . Hence we obtain u^lgrad x | ) — > u ( J g r a d x|) a.e. in Q for an arbitrary x«.X. Because of the growth condition lu^lgrad xj)| & (c0+c2)(1-^grad x | p ~ 1 ) a.e. in Q we can apply the theorem of Lebesgue. So it follows u^lgrad x|)—-*u(Igrad x|) in L p (Q). Nov« let u), i.e. there is a tensor function e L 0 0 (Q)11"n with If I k 1 and f = jp§g-§|if x ^ 0, which satisfies the equation T y) = J J udgrad x|)«!|'grad y dsdt for all y e X . OA. Then we define »P a sequence f - C Ai1 ( x , i u,) i by the equations (f-s. y) = J i (I grad xl).|«grad y dsdt for all yfiX. 1 Oft 1 , Using the convergence of the sequence u^(|-grad x|) in the space L p (Q) we obtain f ^ — » f in X 1 , and the lemma is proved. Lemma 4. The mapping A 2

s Y

—***

is

pseudomonotone.

Proof. The choice of the space X guaranties that the mapping A 2 is bounded and continuous from X into X*. Using the compact imbedding of the space Y into H we confirm that the mapping A , is weakly 1 n continuous from Y into L (Q) , hence Ag is weakly continuous from X into X*, too (see also LIONS [ 8 ] , TEHAM £l0] ). Finally we remark that A 0 has the property (A- x, x) = 0 for all x e X . From this it follows directly that the mapping A 2 Y—»Y is pseudomonotone. Corollary 1. The mapping A : Y x U — > 2 ^ is pseudomonotone. i-roof. It follows from Lemma 1, 2 and 3 that the mapping A1 : X x U — f 2 is pseudomonotone. Furthermore the sum of tw.o bounded

196

pseudomonotone mappings i s pseudomonotone, t o o ( s e e BROiYDER-HEStf £ 2 j ) . Remark 2. Using t h e continuous imbedding I p ( C , T ; W 1 » P ( a ) ) A C ( 0 , T ; L 2 ( f t ) ) C L q (0,T;L q Cft)) i t i s easy t o shovt t h a t the mapping A f u l f i l s t h e c o e r c i v i t y assumption of Theorem 1. Based on t h e p r o p e r t i e s of A g i v e n h e r e , we a r e a b l e t o apply t h e r e s u l t s of the p a r t 1 on the problem of t h e e x i s t e n c e of an optimal parameter f o r m u l a t e d i n t h i s p a r t . Remark 3 . Under t h e assumption above i t can be shown t h a t t h e s o l u t i o n of the e v o l u t i o n e q u a t i o n (2) i s unique f o r each u c U . References [ I ] Browder, F. E . , Problèmes non l i n é a i r e s . Les P r e s s e s d e l ' U n i v e r s i t é de Montréal 1966. 121 Browder, P. E . , Hess, P . , Nonlinear Mappings of Monotone Type i ù Banach Spaces. J . Funct. Anal. 12 (1972), 251-294.. [ 3 ] Kluge, R . , Optimal-Control with Minimum Problems and V a r i a t i o n a l I n e q u a l i t i e s . L e c t u r e Notes i n Computer Science 27, S p r i n g e r - V e r l a g , B e r l i n 1975, 377-382. [4I Kluge, R . , Über eine Klasse von Minimumproblemen mit Nebenbedingungen. Math. Nachr. 78 (1977), 37-64. Kluge, R . , On some Parameter D e t e r m i n a t i o n Problems and Quasiv a r i a t i o n a l I n e q u a l i t i e s . Abhandlungen Akad. '.Viss. DDR N6, Akademie-Verlag, B e r l i n 1978» 129-139. [6] Kluge, R . , Langmach, H., On some Probleme of Determination of F u n c t i o n a l Parameters i n P a r t i a l D i f f e r e n t i a l E q u a t i o n s . L e c t u r e Notes i n C o n t r o l and I n f o r m a t i o n S c i e n c e s 1, S p r i n g e r - V e r l a g , B e r l i n 1978, 298-309. [7] Langmach, H., On the D e t e r m i n a t i o n of F u n c t i o n a l Parameters i n some P a r a b o l i c D i f f e r e n t i a l E q u a t i o n . Abhandlungen Akad. Wiss. DDR N6, Akademie-Verlag, B e r l i n 1978, 175-184. Î8~l Lions, J . L . , Quelques méthodes de r é s o l u t i o n des problèmes aux l i m i t e s non l i n é a i r e s . Dunod, G a u t h i e r - V i l l a r s , P a r i s 1969. [9] Nürnberg, R . , E v o l u t i o n s u n g l e i c h u n g e n und ParameterbestimmungsL Probleme. D i s s . A, B e r l i n 1979. ri ] Temam, R . , N a v i e r - S t o k e s E q u a t i o n s , Theory and Numerical A n a l y s i s . L 0J S t u d i e s i n Mathematics and i t s A p p l i c a t i o n s 2, North-Holland 1977. [II] Vainberg, I.I. M., V a r i a t i o n a l Methods i n t h e Theory of Nonlinear Operators ( R u s s . ) . Moscow 1956.

INVERSE PROBLEMS IN GROWTH DYNAMICS 1 5 Jürgen PEIL 2 )

Summary The meaning of the tern "inverse problem" in a general biocybernetical context will be explained. The mathematical tasks of general inverse problems as well as special Inverse problems are demonstrated at the analysis of growth dynamics. To this end at first the main features of a phenomenologlc-mathematical kind of modelling of body length growth processes of man are sketched to give the concrete meaning of general Inverse problem within this framework, and to show the embedding of the special inverse problem on different levels of the general inverse problem. Then the special inverse problem will be specified as the task of parameter estimation for growth differential equations and for the corresponding functional expressions on the basis of measured growth courses. The numerical results of solving these special inverse problems are presented. 1. The meaning of Inverse Problem in Blocybernetlcs In Blocybernetlcs as Inverse Problem the task is named to find a-mathematical expression for a connection between variables which is experimentally given by a series of measured values. Such a collection of data may be a time-series as the result of examinations on the dynamical behaviour of the system in question, or it is the picture of a functional relationship between variables of the system. In every case the measured values-are the starting-point in the search for a theoretical description of the observed phenomenon. In respect to what end the mathematical expression is used one can meet two extreme situations. On the one hand the expression is intended only for a quantitative description of the observed process. It transforms the information which is diffused in the measured values to a comprehended form, and in this way it is suitable to serve as an objective 1 This work was supported by the Martin-Luther-Universitfit Halle on the basis of the research theme "Biomathematische Modellierung" 2 Dr.rer.nat.habil. JUrgen Peil, Anatomisches Instltut des Bereichs Medizin der MLU Halle, DDR-U02 Halle, GroBe Steinstr. 52

198 basis for comparisons. In most cases there Is to settle a class of empirical functions and the numerical task remains of determining that expression out of this class which gives the best fitted quantitative description of the measured course. This special inverse problem is called "parameter estimation problem". On the other hand, the general Inverse problem means the construction of mathematical models, 1. e. the deriving or defining an appropriate mathematical expression from the data. Such a mathematical model, has not only to give a quantitative description of a measured connection between variables, but It should express the main dynamical features of the system, its structural or functional properties, and moreover, the underlying general principles by which the system is governed. But this cannot be done by mathematics alone. As a rule, there is no unequivocal way from experimental data to the mathematical expression, and in the multi-step.process of model building and model proving one has to consider some empirical facts and hints from the background of the field from which the problem arises. It should be noticed that in this sense a mathematical model carries also prospective and designing features. Sometimes general inverse problems also are called as system Identification or system specification. Among the mathematical tasks of solving a general inverse problem parameter estimations (as special Inverse problems) play an important role. But before an estimation can be performed the general inverse problem must be specified: One has to decide what kind of mathematical description is appropriate for reflecting the structural, dynamical and so on features of the system, and one has to settle the class of mathematical expressions out of which the "best" expression is to determine by parameter estimation. In the following we use differential equations and analytical functions for a quantitative description or for modelling of growth processes. (This settlement Is qualified because we are not intended to consider aspects of time-delays which, from a control-theoretical point of view, may be important in certain growth processes of populations, see e. g. CUSHING /1977/. Our kind of modelling, see 2., has also a control-theoretical background, indeed, but the details of the control mechanisms are not yet well understood. So we don't take into consideration any time-delays). As differential equations several growth models come into question. The solutions of these growth differential equations constitute the mentioned classes of analytical functions.

199 2. General Inverse problem of growth dynamics Fig. 1 shows measured courses of the body length growth of man. The course at the top In fig. 1 Is a mean value course reaching from conception until the time of adolescens (the values were taken from anthropometric literature). There are a lot of single measurements (cross-sectional data) which enter the respective mean value at every time point.

» tody

length [em]

wo 160

H0 120 tooto to40 20 i 2 3 * B 6 7 t i

y-'faniryanic) body St

l i r n a a » a 16 17 is t••age [ y tars] y-My

length [an]

length

170 m 1SOw

Fig. 1: Examples of measured growth courses (body length growth of man)

I 23

+ 5 6 7 * 9 W t [month»]

»

a

V*

*

18 20 t> age [years]

At the left on the bottom In fig. 1 the embryonic part of body length increase Is drawn. These values are also mean values of cross-sectional data. The right part shows a longitudinal data series of the body length of an individual (one of twin-brothers), see also fig. 2. There may be several occasions, besides the purpose of modelling the growth dynamics, to give for a measured course a quantitative description, e. g. to provide an objective basis for comparisons of growth

200

processes of different populations in examining acceleration phenomena. The ways of choosing appropriate classes of functions and solving the special inverse problems of parameter estimation are discussed in detail in PEIL /1979/. For modelling growth processes at first differential equations are to prefer as mathematical expressions because for constructing a differential equation one has to know the underlying physical, physico-chemical, or biochemical etc. principles and processes which generate the observed phenomenon. (Attempts to construct such differential equations were undertaken by SCHARP /197k/ on the basis of laws of mechanics). The differential equations which will be used In 3. rest on simple and obvious assumptions. An analytical expression as solution of a differential equation of course carries the same aspects in modelling as the differential equation does it. The general Inverse problem of growth dynamics consists In seeking for a mathematical expression which is able to reflect the quantitative and qualitative features of growth dynamics represented *>y measured courses like those shown in fig. 1. It would be without chance of success to seek for an expression in form of a differential equation which is valid for the whole time interval of body length growth of man (fig. 1 at the top). Our attempts to get a quantitative description for an as large as possible time interval of growth (PEIL /1979/) showed the generalized logistic function to be proper for meeting this task to a certain extent. But its generating differential equation contains the time explicltely (see PEIL /1978/). Starting-point for our kind of modelling of the growth process is the well known empirical fact that growth of human beings takes place in form of more or less clearly marked growth spurts succeeding one another. Taking this into account we try to separate the whole growth into growth spurts mathematically. Following the principle of simplicity we assume spurts to have a symmetric-sigmoidal shape (in its time course). Furthermore it seems to be reasonable that the type of functions used for describing the single spurt must have such asymptotical properties which don't limit the validity of the corresponding mathematical term only to that time interval in which this spurt happens. (Otherwise one would be obliged to cut off the improper parts). The superposition of the single spurts has to ensure the smooth course of the growth character y as it Is in reality (see fig. 1). Fig. 2 shows an example of analyzing two longitudinal data series of twin-brothers according to the just mentioned manner. The data cover

201

the time Interval of the last part of the praepuberal development as well as the time Interval In which the puberal and the postpuberal growth spurts take place. After setting up a 3-parametrlc hyperbolic tangent - the solution (4a) of the differential equation (2) can be written In this form - for describing a single spurt the solution of the parameter estimation problem is also given In fig. 2. fUkp^Umtlail

glfi- 2= Example of longitudinal data series of body length development of twins (to the left) and phenomenologic-mathematical separation of growth spurts (to the right)" with their quantitative description by means of terms of a 3-parametric hyperbolic tangent function (from HELWIN et al. /1979/). ( 8: coefficient of determination, S R : residual standard deviation). As the graphs of the estimated expressions written down in the right-top corner of the figure the 2 curves in the left part are best fitted to the measured courses. The mathematical separation of growth spurts is only one step within the process of model construction. The biological relevance may be proved by considering the social and the biological circumstances in which the growth process happens (see HELWIN and PEIL /1978/, HELWIN et al. /1979/). Details of the suggested phenomenologlc-mathematical model are discussed in former papers (HELWIN and PEIL 1977, PEIL and HELWIN 1979). There are quoted also examples of performed separations. For the present theme it was the Intention to show that special as well as general Inverse problems may occur on different hierarchical levels within the process of model building and that they are mutually connected.

202 3. Mathematical models of growth spurts 3-1. Differential equations In the phenomenologic-mathematical modelling of growth processes as mentioned before an essential step is to set up the class of functions for a quantitative description of the single growth spurts. Because such a functional relation between time t and body length y should also have a meaning as a mathematical model we go back to the traditional growth differential equations. The general structure of a growth differential equation (of first order) is y = V

f

l

( y )

"

a

d -f 2 (y)

Oa w ,a d > 0)

(1)

The growth velocity y is a function only of the growth variable y itself in form of the balance between the growth term fj(y) (with Intensity a w ) and the damping term f 2 (y) (with intensity a d ). The functions f 1 and f 0 must be chosen in such a way that the solution of (1) can re1) fleet the slgmoldal shape of real growth processes . Besides this more formal argument the choice of the functions f 1 and f 2 is an essential point in the model,construction. They reflect the ideas about the (dynamical) properties of anabolism and catabolism the balance of which determines the growth of the organism. The most simple setting up of fj and f 2 already was done by VERHULST In 1838. The equation y = awy - a d y 2 (2) is the so-called logistic law of growth. (For a summarizing and detailed discussion of the logistic law of growth and its generalizations in respect to the differential equation forms as well as to their solutions see PEIL /1978/). The meaning of (2) is a quadratically damped 2)

exponential growth '. To get more general expressions there are 3 possibilities to parameterize VERHULST's law of growth. Parameterization of the exponent of the growth term alone yields $ = awyn - ady2

(0 < n < 2)

(3a)

An other version of this is the partly parameterized form of BERTALANFFY's growth equation y = a w y n - ady

(2/3 s n < 1)

(3b)

1 For instance, the choice fj(y)=y and f 2 (y)=ylny leads to the well known law of growth which was suggested by GOMPERTZ. 2 The growth term a w y alone would describe unlimited exponential growth.

203 (As to the setting up of the interval in which n may vary and concerning the linear form of the damping term see BERTALANFFY /1960/). Parameterization of the damping term gives y = awy - a d y m

(1 < m < -)

(3c)

(m,n > 0; m > n)

(3d)

The differential equation y = awyn - adym

is the completely parameterized version of the former logistic law of 1) 2) growth ' being able to describe sigmoidal growth courses '. At this stage one of the general inverse problems of growth dynamics was to fix the structure of the growth differential equation as a first order differential equation being composed of growth term and damping term, and the choice of the power form for fj and f2Now it is of interest to estimate the parameters in (3), the intensities of the growth term and the damping term as well as the exponential parameters n resp. m proceeding from measured values of y. To a certain extent this special Inverse problem - related to growth differential equations - appears on the level of growth functions as a general inverse problem. The differential equation (3d) has, if at all, different classes of functions as solutions dependent on the interval in which the values of the exponential parameters vary (e. g. see SCHARF /1975/, SAGER /1978/). 3.2. Growth functions Here we consider only those analytical expressions as growth functions which are integrals of the above mentioned differential equations. As solution of (2) we have the expression y = a1/[l+exp(a2-a3t)] with ajsa^a^,,

a2=ln[aw/(a(1-y0)-l],

a 3 =a w

(ta)3) and y o = y ) t = 0 as the ini-

tial value of the growth variable. The solution of (3c) is, written with the parameters of the dlfferen1 This version sometimes is called BERTALANFFY's (general) growth equation. For growth processes of the mass he sets the catabolism to be proportional to weight, that means m=l in (3d), see (3b). 2 Growth velocity becomes zero for y=0 and for y=(a(j/aw)1^m~n^ according to formulae (3). 3 This functional expression of logistic growth is the starting-point for another generalization which has a polynom of t as exponent in the e-function (for a discussion also of this generalization see PEIL /1978/).

204

t l a l equation: y=(aw/ad)1/(m-1)[l+(awy(1-m)/ad-l)exp(-(m-l)awt)]1/(1-m)

(lb)

o r , without the structure of the parameters: y=a1[l+exp(a2-a3t)]"ai'

( a 3 , a „ > 0)

(lb»)

As solution of (3b) one can get y=[(aw/ad)-{(aw/ad)-yj1-n)}exp(-(l-n)awt)]1/(1"n)

(He)

also written In the simpler form (1. e . neglecting the parameter structures Influenced by the parameters and c o e f f i c i e n t s of the d i f f e r e n t i a l equation): y=b1[l-exp(b2-b3t)]bil

(b 3 > 0, b|| i 3)

(He')

4. Nonlinear regression f o r solving the Inverse problems For solving the special Inverse problems of growth dynamics we estimate the values of the parameters in the d i f f e r e n t i a l equations (3b,c,d) as well as separately those in the analytical expressions d b ' . c ' ) and Cla). The appropriate principle f o r the estimation procedure is the GAUSS' principle of least squares because in the measured data series always a random component is contained. As measured data series we choose the values of the body length of the embryonic development of man. The mean values are shown to the l e f t at the bottom of f i g . i , each of them represents a large amount of single measurements. The reason for this particular choice is the idea that the embryonic body length growth seems to be a more "uniform" growth period than the following periods because i t may be determined, f i r s t at a l l , genetically. Therefore we are allowed to expect that our main concern now - the examination of numerical aspects in solving the spec i a l Inverse problems - w i l l be disturbed not too much by a possible superposition of growth spurts as one can see i t in the other parts of f i g . 1. The inverse problem related to the d i f f e r e n t i a l equations w i l l be solved by means of Internal regression which i s in the present cases a nonlinear 1 ^ approximation procedure because of the exponential parame1 Usually, internal regressions w i l l be performed to get i n i t i a l v a l ues f o r parameters which are nonlinearly contained in an analytical expression and which can be calculated from linear parameters (coe f f i c i e n t s ) in the d i f f e r e n t i a l equation the solution of which i s the analytical expression in question ( e . g. see HARTLEY /1948/).

ters. To this end the divided central differences DDy^ will be calculated from the measured values y(t1_1), y(t 1 + 1 ) (1=2,...,N-1, N: number of measured values). The DDy4 are plotted as dots in fig. 3. These first order divided differences serve as approximations for the values of the differential quotient y at the corresponding time-points tj. Then the course of DDy in its dependence of y will be least-squares approximated according to the respective growth differential equation. As to the estimation problem of the parameters of the growth functions two remarks must be stated: First, according to the parameters a w > a d and m or n in (3b), (3c) there are in the expressions Ctb'), Ctc1) only 3 independent parameters. The fourth parameter (a2 resp. b 2 ) is formed by the two linear parameters of the differential equation and by the initial value y Q which usually is to settle. But in (lb'), (1c') this parameter is treated also as an adjustable one. (For the numerical integration of (3b) this would mean to optimize the initial value y Q , too). Secondly, in completion thé approximation tasks posed by Clb'), (He') we furnish these expressions with a time-constant term consisting of a parameter a Q or b Q respectively. (This would allow to describe also growth processes which don't start from the zero-level. In the present connection of the embryonic growth period such a parameter may be of advantage if there are small "inconsistencies" between the value y. 1) and the course of measured values for t > 0 '.) Such a "level-parameter were to be taken into account also in the differential equations by replacing y by (y-aQ) allowing the growth velocity to become zero for values of y unequal zero. The nonlinear least-squares estimation of the parameter values exactly is a constrained optimization because the parameters have to vary in the above specified intervals. In the integrals of the differential equations (3b), (3c) these constraints are partly reflected by the structure of the analytical expression. The common structure of (kb) and (4c) could be written ysAjU+Ajexpt-Ajt)]*1»

(5)

Because of A2=(y0/A1)1/Ai,-1 and y 0 < Aj (Aj is the asymptotically attained largest value of the growth variable) for A^ < 0 follows A 2 > 0 (and the differential equation (3c)), for A^ > 0 follows A 2 < 0 (and the differential equation (3b)). 1 The time t=0 Is a singular point in respect to the embryonic growth dynamics and our considerations concerning the time-variations of the growth variable should be restricted to the time Interval t > 0.

206 The parameter estimations were performed by means of a computer program system for nonlinear approximation (PEIL /1975/). In this program the GAUSS-NEWTON iterative algorithm is completed by several damped versions of this algorithm which show different properties in relation to reliability and velocity of convergence. Since the values of linear parameters can be determined without any difficulties by a linear regression step an expression like Ctc') is preferable to Ctc). Two other computer programs for nonlinear regression were used to verify the numerical results because certain numerical effects (see 5.) have been happened. It should be excluded that these phenomena are artefacts caused by the used special procedure. In one of these two other programs (KNAUFF and HENNIG /1978/) there was the possibility to regard the constraints in a direct manner in the optimization algorithm, and parameter estimations were performed with constraints as well as without constraints1^. In the third program (NAGEL and WOLFF /197t/) a GAUSS-NEWTON step can be followed by a "dog-leg" search if the first one did not reduce the residual sum of squares (SQ). For numerical integration of the differential equation (3d) after parameter estimation the BURLISCH-STOER procedure was used (ORZ MLU /1972/). 5. Numerical results and discussion Starting values for the nonlinear parameters m, n in (3d) were chosen according to BERTALANFFY's approach: m ( o ) =l, n ( o ) = 0.67. The iterative approximation converged rapidly from 13.8 down to S Q ^ = 1.68 already in the fourth step. Only after the first and after the second GAUSS-NEWTON step dog-leg-searches were in need of. Without such intermediate searches the GAUSS-NEWTON Iteration did not converge starting with the above specified values for m and n. The estimated parameter values written in fig. 3 are the result obtained In the 8th step of the iteration. The curve I in fig. 4 was calculated by numerical integration of the differential equation I of fig. 3 assuming y|t =o =0 -l a s initial value. As measure of goodness of approximation we take the sum of squared differences between measured and calculated values (SQ), and the sum of (linear) differences (SD) may possibly indicate systematical deviations between measured and calculated courses. It should be mentioned that 1 I am indebted to H. KNAUFF, ORZ MLU Halle, for performing these estimations 2 For these verifying calculations I am grateful to Dr. R. SCHMIDT, VEB Chemische Werke Buna

207

»

5

10

IS

20

25

30

35

40

MS

SO if

Fig. 3: Course of divided central differences (dots) calculated from the measured values y(t,) of (embryonic) body length. Differential equations fitted by internal^ least squares regression. Parameter values of case fl are calculated from those of the growth function Ctb*) after its approximation to the measured course of (see case II in fig. 6). in some cases there may be a flat minimum of the SQ-surface in the parameter space 1 ^. For discriminating between various sets of parameter values one has to take into account the amount of the "natural variabi2)

lity" and any inconsistencies ' (relative to the applied model) which could be contained in the measured course. From this point of view hardly we can decide whether the curve I or II in fig. 4 better reflects the measured course. (This statement will not be influenced by the fact that the initial value - y ^ s o 2 0 " 1 f o r numerical integration in case I, y | f i o = ^ ® f o r t h e b a c l c w a r d numerical integration in case II - is preassigned and therefore not optimized in the former sense). Notice that in case II of fig. 4 the parameter values are quite different to those in case I which yield the attainable minimum value of SQ, see also fig. 3. They were obtained by a damped GAUSS-NEWTON procedure starting from the above mentioned parameter values. This set of parameter values is of interest as far as it lies in the "range" of the partly parameterized model (3c), the least squares fit and the estimated parameter values of which is shown by case II in fig. 3« Thus the goodness of fit of curve II, fig. 4, is nearly the same (SQ=3.37) as that 1 see for Instance fig. 5 2 Such an inconsistency in respect of the here applied growth differential equations and growth functions could revealed by the measured values of body length in the neighbourhood of t=8 months(corresponding to y=45 In fig. 3). In a former paper we showed by separation of growth spurts that at this time a growth spurt begins which is responsible for a smooth course of the body length increase also over the period of birth.

208

of the 3-parametric d i f f e r e n t i a l equation ( 3 c ) in f i g . 3- We can state that the f i t t i n g of the d i f f e r e n t i a l equation (3d) to the course of d i vided differences shows clearly the favourable status of the least-squares-estimated set of parameter values. This could not be demonstrated by means of the integral curves (of f i g . 4 ) .

sa

f= Wj length [cm] SO

a

30

—I' 20 "

2

• 1-766 . 000169 n-.Stt m'2.912 [y„-.!] (SO'•*.»(, SO-5.65) ym„.50.12 I - a j « . 369 n-1.025 m• 1.377 [y„'50] (SD'-.395, SO - 3.23) ymaJt. 52M # /> y C J 2 . , from

if fa RC\i)

j

(6)

21?

where Z^ is given operator, p £ Q ; R,(*ji)t 4 (Xji) are given functions, is JV" by J f matrix, V = Indeed, if [Q. ,U J and \ U J are the two solutions of a problem 0 ) - ( 4 ) , corresponding to the same initial data, but different data (4), then, denoting ^

V'U-U

,

= f j ] + 4 with the norm in C , which is less then given constant r| , and fixed c o n s t a n t ^ > 0 in (2). Now assume, that for ^ e Q the family of characteristic conoids is regular in the sence, that bicharacteristics, arising in one point, nowhere intersect each other. _ Denote by the set of functions (/>(*) with support in J I , which - t h derivative is bounded by given constant K • Assume that is continuous and bounded together with its partial derivatives with respect to X up to the order in the domain G 3) X £ 0, 7 J .Let

Hfll^)

=

jl(pO»Nx , f/RMIIcfft\*

Here

at>

HI Ulcm) - \ ^ I I™*

I^/OIMS', .

is an area element of the surface ^ , and are the components of matrix • The two following stability theorems are valid

218

Theorem 1 . L e t J

>

J -

0

Then there e x i s t such under conditions

s a t i s f y the e s t i mate (id uniformly with respect to s e t Q Theorem 2 . There e x i s t s such S > 0

(¡Lain

. • that under the conditions

< §

Q) < U T /

'

, uniformly the estimate ( 1 1 ) i s valid f o r the function (p £ with respect the s e t Q . Proofs of both theorems are s i m i l a r . Assuming f o r s i m p l i c i ty t h a t C(x) £~ 0 , we introduce an energy i n t e g r a l

j(i)* 2

[ C ^ i + Z, S

f

W , . - crx) V J J

Ksl J v) The i n t e g r a t i o n i s performed along the i n t e r s e c t i o n of the domain with the plane £ •= const. Using standard method of obtaining the energy r e l a t i o n s , we find;

- Ufa) -

l

fa

J

i(VK)T

By the condition ( 7 ) » taking into account t h a t the d e r i v a t i v e s ^ in yVZ are bounded, we obtain

219

where and

C*

0

and are independent of C^ - /!

IIc(e)

€ £0t T j

,

.

From the inequality (13) we have

or

~ / l ^ i i ^ ] , CD if C , 7 V > ' < i . This i n e q u a l i t y automatically holds i f the condition (10) i s s a t i s f i e d , or i f T (iD theorem 2) i s s u f f i c i e n t l y small and aligned with $ , so t h a t ( 1 2 ) i s true« To estimate !J(T) we represent the solution of the equation ( 6 ) with Cauchy data from ( 7 ) i n the form V = \'i + , where Vi i s a solution of nonhomogemeous equation ( 6 ) with zero Cauchy data, and v/j, i s a solution of homogeneous equat i o n ( 6 ) with Cauchy data from ( 7 ) . Let 1 (V and i m be th$ i n t e g r a l s , corresponding to V with V = Vx and V = ^a, . Naturally, J ( T ) < l [ ^ ( T ) + \ (T)] By energy estimates we have eCl7 X(T) * c^T /I tpilljA) (i5) At the same time si v r I t i s known, that a l l s i n g u l a r i t i e s of the function i) are situated on the l a t e r a l part of dependency conoid with cone point (X > . Under the given assumptions the function i s continuous in the conoid with i t s f i r s t derivatives ( £ p ) x c » {(rt^) ± . But to estimate we need the continuity of the function (rj, and i t s f i r s t derivatives only a t the point i - T ' j I X ^ e S L . Under t h i s r e s t r i c t i o n i t i s easy to s e e , t h a t the values of the function G^ ()i l X, 0 are used only s t r i c t l y i n - , side the dependency conoids, where Q ^ i s a smooth function.

220

It is easy to find, that *

oz(7)

*

llfllm)

u *

{j

C s ^ H j ^ a )

,

(16)

where

cs-

• i

[ ( i f y ' - ^ c p ^

%

H I ..J.A tV * S* ]J * } SI n J '


ficiently smooth' boundary as. We take, generally, D = D(t,x,u) > 0 and f = f(t,x,u) but will primarily be concerned with the linear case: D = D(t,x), f = q(t,x)u so (l1)

il =

VD(t,x)Vu - q(t,x)u

=:

Lu

in

Q.

Associated with this are initial conditions (2)

u(0, •) = u Q

in

a

and boundary conditions (3)

au + puv =: Bu = cp

on

S :» (0,T) x dffi

in-which uv = Dn*Vu is the outward D-normal derivative on as take a2 +p 2 = 1 , assuming that (locally on aa) either p = 0

and we

*Department of Mathematics, University of Maryland Baltimore County, Baltimore, Md. 21228, U.S.A.

(Dirichlet condition) or is defined to be (4)

p ^ 0.

\|f := Bu

:=

The complementary boundary data for

-flu + au^

on

If r is a specified 'boundary patch' ( r c a s ) restriction of §u to Sp := (0,T) x r c S.

u

S.

we will let Sj,u be the

We will consider the following types of problems: CONTROL (given UQ, find