Methods for Solving Inverse Problems in Mathematical Physics - Prilepko, Orlovskiy.pdf - Mathematical Physics - realmodeler

MATHEMATICALPHYSICS

I. Prilepko

Moscow State University

Moscow,Russia

DmitryG. Orlovsky

Igor A. Vasin

Moscow

MARCELDEKKER,INC.

NEWYORK- BASEL

DEKKER

METHODSFOR SOLVING

INVERSE PROBLEMS

Aleksey

State Institute of Engineering Physics

Moscow,Russia

Library of CongressCataloging-in-Publication

for solving inverse problemsin mathematicalphysics / AlekseyI.

Prilepko, DmitryG. Orlovsky,Igor A. Vasin.

p. crn. -- (Monographs

and textbooks in pure and applied mathematics;

solutions.

Mathematicalphysics. I. Orlovsky, DmitryG. II. Vasin, Igor A. III.

Title. IV. Series.

QC20.7.DSP73 1999

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Prilepko, A. I. (Aleksei Ivanovich)

Methods

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1. Inverse problems(Differential equations)~Numerical

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Preface

The theory of inverse problemsfor differential equations is being ex-

tensively developed within the framework of mathematical physics. In the

study of the so-called direct problems the solution of a given differential

equation or system of equations is realised by meansof supplementary con-

ditions, while in inverse problems the equation itself is also unknown.The

determination of both the governing equation and its solution necessitates

imposing more additional conditions than in related direct problems.

The sources of the theory of inverse problems maybe found late in the

19th century or early 20th century. Theyinclude the problemof equilibrium

figures for the rotating fluid, the kinematic problems in seismology, the

inverse Sturm-Liuville problem and more. Newton’s problem of discovering

forces makingplanets movein accordance with Kepler’s laws was one of the

first inverse problemsin dynamicsof mechanical systems solved in the past.

Inverse problems in potential theory in which it is required to determine

the body’s position, shape and density from available values of its potential

have a geophysical origin. Inverse problems of electromagnetic exploration

were caused by the necessity to elaborate the theory and methodology of

electromagnetic fields in investigations of the internal structure of Earth’s

crust.

The influence of inverse problems of recovering mathematical physics

equations, in which supplementary conditions help assign either the values

of solutions for fixed values of someor other argumentsor the values of cer-

tain functionals of a solution, began to spread to more and more branches

as they gradually took on an important place in applied problems arising

in "real-life" situations. Froma classical point of view, the problemsunder

consideration are, in general, ill-posed. A unified treatment and advanced

theory of ill-posed and conditionally well-posed problems are connected

with applications of various regularization methods to such problems in

mathematical physics. In manycases they include the subsidiary infor-

mation on the structure of the governing differential equation, the type of

its coefficients and other parameters. Quite often the unique solvability

of an inverse problem is ensured by the surplus information of this sort.

A definite structure of the differential equation coefficients leads to an in-

verse problem being well-posed from a commonpoint of view. This book

treats the subject of such problems containing a sufficiently complete and

systematic theory of inverse problems and reflecting a rapid growth and

iii

Preface

development over recent years. It is based on the original works of the

authors and involves an experience of solving inverse problems in many

branches of mathematical physics: heat and mass transfer, elasticity the-

ory, potential theory, nuclear physics, hydrodynamics, etc. Despite a great

generality of the presented research, it is of a constructive nature and gives

the reader an understanding of relevant special cases as well as providing

one with insight into what is going on in general.

In mastering the challenges involved, the monographincorporates the

well-known classical results for direct problems of mathematical physics

and the theory of differential equations in Banachspaces serving as a basis

for advanced classical theory of well-posed solvability of inverse problems

for the equations concerned. It is worth noting here that plenty of inverse

problems are intimately connected or equivalent to nonlocal direct problems

for differential equations of somecombinedtype, the newproblems arising

in momentumtheory and the theory of approximation, the new .types of

¯ linear and nonlinear integral and integro-differential equations of the first

and second kinds. In such cases the well-posed solvability of inverse prob-

lem entails the newtheorems on unique solvability for nonclassical direct

problems we have mentioned above. Also, the inverse problems under con-

sideration can be treated as problems from the theory of control of systems

with distributed or lumped parameters.

It mayhappen that the well-developed methods for solving inverse

problems permit, one to establish, under certain constraints on the input

data, the property of having fixed sign for source functions, coefficients and

solutions themselves. If so, the inverse problems from control theory are

in principal difference with classical problemsof this theory. These special

inverse problems from control theory could be more appropriately referred

to as problems of the "forecast-monitoring" type. The property of having

fixed sign for a solution of "forecast-monitoring" problemswill be of crucial

importance in applications to practical problems of heat and mass transfer,

the theory of stochastic diffusion equations, mathematical economics, var-

ious problems of ecology, automata control and computerized tomography.

In manycases the well-posed solvability of inverse problems is established

with the aid of the contraction mapping principle, the Birkhoff-Tarsky

principle, the NewtonvKantorovich method and other effective operator

methods, making it possible to solve both linear and nonlinear problems

following constructive iterative procedures.

The monographcovers the basic types of equations: elliptic, parabolic

and hyperbolic. Special emphasis is given to the Navier-Stokes equations as

well as to the well-knownkinetic equations: Bolzmanequation and neutron

transport equation.

Being concerned with equations of parabolic type, one of the wide-

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