Samarskii A.A., Vabishchevich P.N. Numerical Methods for Solving Inverse Problems of Mathematical Physics

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Inverse and Ill-Posed Problems Series

Managing Editor

Sergey I. Kabanikhin, Novosibirsk, Russia /Almaty, Kazakhstan

Alexander A. Samarskii

Peter N. Vabishchevich

Numerical Methods for

Solving Inverse Problems of

Mathematical Physics

≥

Walter de Gruyter · Berlin · New York

Keywords:

Inverse problems, mathematical physics, boundary value problems, ordinary differential equations,

elliptic equations, parabolic equations, right-hand side identification, evolutionary inverse problems,

ill-posed problems, regularization methods, Tikhonov regularization, conjugate gradient method,

discrepancy principle, finite difference methods, finite element methods.

Mathematics Subject Classification 2000:

65-02, 65F22, 65J20, 65L09, 65M32, 65N21

앪

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to ensure permanence and durability.

ISBN 978-3-11-019666-5

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Printed in Germany

Cover design: Martin Zech, Bremen.

Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen.

Preface

Applied problems often require solving boundary value problems for partial differen-

tial equations. Elaboration of approximate solution methods for such problems rests

on the development and examination of numerical methods for boundary value prob-

lems formulated for basic (fundamental, model) mathematical physics equations. If

one considers second-order equations, then such equations are elliptic, parabolic and

hyperbolic equations.

The solution of a boundary value problem is to be found from the equation and

from some additional conditions. For time-independent equations, to be speciﬁed are

boundary conditions, and for time-dependent equations, in addition, initial conditions.

Such classical problems are treated in all tutorials on mathematical physics equations

and partial differential equations.

The above boundary value problems belong to the class of direct mathematical

physics problems. A typical inverse problem is the problem in which it is required

to ﬁnd equation coefﬁcients from some additional information about the solution; in

the latter case, the problem is called a coefﬁcient inverse problem. In boundary inverse

problems, to be reconstructed are unknown boundary conditions, and so on.

Inverse mathematical physics problems often belong to the class of classically ill-

posed problems. First of all, ill-posedness here is a consequence of lacking continuous

dependence of solution on input data. In this case, one has to narrow the class of ad-

missible solutions and, to determine a stable solution, apply some special regularizing

procedures.

Numerical solution of direct mathematical physics problems is presently a well-

studied matter. In solving multi-dimensional boundary value problems, difference

methods and the ﬁnite element method are widely used. At present, tutorials and

monographs on numerical solution methods for inverse problems are few in number.

The latter was the primary motivation behind writing the present book.

By no means being a comprehensive guide, this book treats some particular in-

verse problems for time-dependent and time-independent equations often encountered

in mathematical physics. Rather a complete and closed consideration of basic difﬁcul-

ties in approximate solution of inverse problems is given. A minimum mathematical

apparatus is used, related with some basic properties of operators in ﬁnite-dimensional

spaces.

A predominant contribution to the scope of problems dealt with in the theory and so-

lution practice of inverse mathematical physics problems was made by Russian math-

ematicians, and the pioneer here was Andrei Nikolaevich Tikhonov. His ideas, under-

lying the modern applied mathematics, are now developed by his numerous disciples.

Our work pays tribute to A. N. Tikhonov.

Main deﬁnitions and notations

A, B, C, D, S — difference operators;

E — unit (identity) operator;

∗

— adjoint operator;

−1

— operator inverse to A;

A > 0 — positive operator (( Ay , y)>0ify = 0);

A ≥ 0 — non-negative operator (( Ay, y) ≥ 0);

A ≥ δ E, δ>0 — positive deﬁnite operator;

= (A + A

∗

)/2 — self-adjoint part of A;

= (A − A

∗

)/2 — skew-symmetric part of A;

H — Hilbert space of mesh functions;

(·, ·) — scalar product in H ;

·— norm in H ;

(y,v)

= (Ay ,v)— scalar product in H

(A = A

∗

> 0);

·

— norm in H

;

(ω) — Hilbert space of mesh functions;

·— norm in L

;

A — norm of difference operator A;

M, M

— positive constants;

 — computation domain;

∂ — boundary of ;

ω — set of internal nodes;

∂ω — set of boundary nodes;

h, h

— mesh size in space;

τ — time step;

σ — weighting parameter of difference scheme;

y(x + h) − y(x)

— right-hand difference derivative at the point x;

viii Main deﬁnitions and notations

¯x

y(x) − y(x − h)

— left-hand difference derivative at the point x;

◦

= (y

+ y

¯x

)/2 — central difference derivative at the point x;

¯xx

− y

¯x

— second difference derivative at the point x;

y = y

= y(x, t

) — magnitude of mesh function at the point x at the time t

= nτ ,

n = 0, 1,...;

α — regularization parameter;

δ — typical input-data inaccuracy;

Contents

Preface v

Main deﬁnitions and notations vii

1 Inverse mathematical physics problems 1

1.1 Boundary value problems ........................ 1

1.1.1 Stationary mathematical physics problems ........... 1

1.1.2 Nonstationary mathematical physics problems ........ 2

1.2 Well-posed problems for partial differential equations ......... 4

1.2.1 The notion of well-posedness ................. 4

1.2.2 Boundary value problem for the parabolic equation ...... 4

1.2.3 Boundary value problem for the elliptic equation ....... 8

1.3 Ill-posed problems ........................... 9

1.3.1 Example of an ill-posed problem ................ 10

1.3.2 The notion of conditionally well-posed problems ....... 11

1.3.3 Condition for well-posedness of the inverted-time problem . . 11

1.4 Classiﬁcation of inverse mathematical physics problems ....... 13

1.4.1 Direct and inverse problems .................. 13

1.4.2 Coefﬁcient inverse problems .................. 14

1.4.3 Boundary value inverse problems ............... 15

1.4.4 Evolutionary inverse problems ................. 16

1.5 Exercises ................................ 16

2 Boundary value problems for ordinary differential equations 19

2.1 Finite-difference problem ........................ 19

2.1.1 Model differential problem ................... 19

2.1.2 Difference scheme ....................... 20

2.1.3 Finite element method schemes ................ 23

2.1.4 Balance method ......................... 25

2.2 Convergence of difference schemes .................. 26

2.2.1 Difference identities ...................... 27

2.2.2 Properties of the operator A .................. 28

2.2.3 Accuracy of difference schemes ................ 30

2.3 Solution of the difference problem ................... 31

2.3.1 The sweep method ....................... 32

2.3.2 Correctness of the sweep algorithm .............. 33

2.3.3 The Gauss method ....................... 34

2.4 Program realization and computational examples ........... 35

2.4.1 Problem statement ....................... 35