Ill-Posed Problems: Theory and Applications

267 Pages·1994·23.59 MB·English

by  A. Bakushinsky, A. Goncharsky (auth.)

lll-Posed Problems: Theory and Applications
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 301
III-Posed Problems:
Theory and Applications
by
A. Bakushinsky
Institute jfor System Studies,
Russian Academy ojf Sciences,
Moscow, Russia
and
A. Goncharsky
Department ojfComputational Mathematics and Cybemetics,
Moscow State University,
Moscow, Russia
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
A C I. .P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-4447-9 ISBN 978-94-011-1026-6 (eBook)
DOI 10.1007/978-94-011-1026-6
This monograph is a new and original work based 01)
two books by the same authors previously published in Russian:
Iterative Melhods for Solving /ll-Posed Problems,
Moscow, Nauka © 1989, and
/ll-Posed Problems, Numerical Melhods and IIS Applicalions,
Moscow, Moscow State University Press © 1989.
Printed on acid-free paper
All Rights Reserved
© 1994 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1994
Softcover reprint ofthe hardcover 1st edition 1994
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical.
inc1uding photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
Contents
Preface ix
Introduction 1
1 General problems of regularizability 4
1.1 Definition of regularizing algorithm (RA) 4
1.2 General theorems on regularizability and principles of con-
structing the regularizing algorithms . . . . . . . . . . . .. 7
1.3 Estimates of approximation error in solving the ill-posed prob-
lems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4 Comparison of RA. The concept of optimal algorithm 19
2 Regularizing algorithms on compacta 23
2.1 The normal solvability of operator equations. 24
2.2 Theorems on stability of the inverse mappings. 26
2.3 Quasisolutions of the ill-posed problems . . . . 28
2.4 Properties of 6-quasisolutions on the sets with special structure 36
2.5 Numerical algorithms for approximate solving the ill-posed
problem on the sets with special structure . . . . . . . . . .. 40
3 Tikhonov's scheme for constructing regularizing algorithms 43
3.1 RA in Tikhonov's scheme with a priori choice of the regular-
ization parameter . . . . . . . . . . . . . . . . . . . . . . . .. 43
3.2 A choice of regularization parameter with the use of the gen-
eralized discrepancy 47
3.3 Application of Tikhonov's scheme to Fredholm integral equa-
tions of the first kind . . . . . . . . . . . . . . . . . . . . . .. 57
3.4 Tikonov's scheme for nonlinear operator equations . . . . .. 61
3.5 Numerical implementation of Tikhonov's scheme for solving
operator equation . . . . . . . . . . . . . . . . . . . . . . . .. 68
v
vi
4 General technique for constructing linear RA for linear prob-
lems in Hilbert space 73
4.1 General scheme for constructing RA for linear problems with
completely continuous operator . . . . . . . . . . . . . . . .. 74
4.2 General case of constructing the approximating families and
RA 77
4.3 Error estimates for solutions of the ill-posed problems. The
optimal algorithms . . . . . . . . . . . . . . . . . . . . . . .. 90
4.4 Regularization in case of perturbed operator. . . . . . . . . . 100
4.5 Construction of linear approximating families and RA in Ba-
nach space 114
4.6 Stochastic errors. Approximation and regularization of the
solution of linear problems in case of stochastic errors .. . . 122
5 Iterative algorithms for solving non-linear ill-posed problems
with monotonic operators. Principle of iterative
regularization 127
5.1 Variational inequalities as a way of formulating non-linear
problems 128
5.2 Equivalent transforms of variational inequalities. . . . . . . . 131
5.3 Browder-Tikhonov approximation for the solutions of varia-
tional inequalities. . . . . . . . . . . . . . . . . . . . . . . 136
5.4 Principle of iterative regularization . . . . . . . . . . . . . . . 141
5.5 Iterative regularization based on the zero-order techniques . . 142
5.6 Iterative regularization based on the first-order technique (re-
gularized Newton technique) 150
5.7 RA for solving variational inequalities 155
5.8 Estimates of convergence rate of the iterative regularizing
algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6 Applications of the principle of iterative regularization 164
6.1 Algorithms for minimizing convex functionals. Solving the
non-linear equations with monotonic operators .. . . . . . . 164
6.2 Algorithms for minimizing quadratic functionals. Non-linear
procedures for solving linear problems .. . . . . . . . . . . . 168
6.3 Iterative algorithms for solving general problems of mathe-
matical programming. . . . . . . . . . . . . . . . . . . . . . . 172
6.4 Algorithms to find the saddle points and equilibrium
points in games . . . . . . . . . . . . . . . . . . . . . . . . . . 178
vii
7 Iterative methods for solving non-linear ill-posed operator
equations with non-monotonic operators 185
7.1 Iteratively regularized Gauss - Newton technique for operator
equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
7.2 The other ways of constructing iterative algorithms for general
ill-posed operator equations . . . . . . . . . . . . . . . . . . . 195
8 Application of regularizing algorithms to solving practical
problems 199
8.1 Inverse problems of image processing . . . . . 200
8.2 Reconstructive computerized tomography . . 208
8.3 Computerized tomography of layered objects 213
8.4 Tomographic examination of objects with focused radiation 218
8.5 Seismic tomography in engineering geophysics 222
8.6 Inverse problems of acoustic sounding in wave approximation 227
8.7 Inverse problems of gravimetry . 236
8.8 Problems of linear programming 238
Bibliography 242
Index 254
Preface
Recent years have been characterized by the increasing amount of publications in
the field of so-called ill-posed problems. This is easily understandable because we
observe the rapid progress of a relatively young branch of mathematics, of which
the first results date back to about 30 years ago.
By now, impressive results have been achieved both in the theory of solving
ill-posed problems and in the applications of algorithms using modem computers.
To mention just one field, one can name the computer tomography which could
not possibly have been developed without modem tools for solving ill-posed
problems.
When writing this book, the authors tried to define the place and role of ill-
posed problems in modem mathematics. In a few words, we define the theory of
ill-posed problems as the theory of approximating functions with approximately
given arguments in functional spaces. The difference between well-posed and ill-
posed problems is concerned with the fact that the latter are associated with
discontinuous functions.
This approach is followed by the authors throughout the whole book. We
hope that the theoretical results will be of interest to researchers working in
approximation theory and functional analysis.
As for particular algorithms for solving ill-posed problems, the authors paid
general attention to the principles of constructing such algorithms as the methods
for approximating discontinuous functions with approximately specified
arguments. In this way it proved possible to define the limits of applicability of
regularization techniques.
The possibility of constructing iterative procedures for approximating
solutions of the ill-posed problems is thoroughly investigated for a wide range of
linear and non-linear problems.
The authors have acquired extensive experience in applying the methods of
solving ill-posed problems. This allowed them to demonstrate the efficiency of
algorithms using model and applied practical problems of mathematical
programming, linear algebra, functional optimization, solving the operator
equations.
Some new approaches concerned with computer tomography in the
frameworks of both geometrical optics and wave approximation are presented in
a monograph for the first time ever. We hope that these sections of the book will
attract the interest of researchers specializing in computer tomography of seismic
geophysics or those applying tomographical approaches in industry.
The presentation of the material in this book was strongly affected by many
years of scientific contacts with colleagues from the USA, Japan, France and Italy.
It was also influenced by numerous discussions with Academician A. Tikhonov
and scientists of his school, with which the authors associate themselves.
ix
x
This book is based on two monographs of the same authors which have been
published in Russian earlier (Bakushinsky et aI., 1989a, 1989b). Compared to the
Russian edition, the text of this book is substantially changed and extended to
include some of the most recent results. Chapters 7 and 8 have been specially
prepared for the present edition. The list of references was substantially extended.
The authors are grateful to Dr. LV. Kochikov for translating this book into
English, and to E.O Drevyatnikova for wordprocessing the book.
A. Bakushinsky
A. Goncharsky
Introduction
The impressive and steadily increasing opportunities provided by modern
computers stimulate a continuous growth of the fields of human activities where
mathematical models are applied.
In some of these models the algorithms for approximate solution of the
correspondent mathematical problems can be formulated within the framework of
traditional computational mathematics. In this case it is often possible to prove the
formal convergence of algorithms and to evaluate the approximations errors. The
technical problems related to numerical implementation of such algorithms are
associated with rounding errors, data representation, etc. and do not usually lead
to significant difficulties.
However. it is often the case when data available to a researcher can only be
interpreted with a formal model which does not allow the application of traditional
computational algorithms.
Such models usually lead to formulation of the ill-posed problems for which
there exist no theorems on solvability in some natural functional spaces.
Moreover, these problems lack stability (in the classical sense) of the solution with
respect to errors of input data. This is significant since we almost never deal with
the absolutely exact values of input parameters.
The theory of solving ill-posed problems is a relatively new branch of
computer science which began to separate into an independent science shortly
after the publications by A. Tikhonov (Tikhonov. 1963a, 1963b). In these
publications the fundamental concept of regularizing algorithm was introduced.
Here we formulate this concept for a mathematical model represented by
operator equation Az = U where operator A acts from a metric space Z into a
metric space U. The peculiarity of the ill-posed problems is that operator A is not
assumed to be continuously invertible (in local or in global sense). Let us assume
that for the "exact" values of uand A there exists the "exact" solution zwhich we
are interested in.
Let. for the sake of clarity, operator A be known exactly, and instead of uwe
are given its approximation uf, E U such that p(Ul)' u) ::; O. Here 0 is a numeric
parameter characterizing the errors of input data ul). It is natural to require that
a numerical algorithm for solving the operator equation should have the following
main property: the less is the error 8, the more close approximation to zcan be
obtained. This approach underlies the formal definition of regularizing algorithm.
The regularizing algorithm(RA) is the operator R which puts into
correspondence to any pair (Uf,,<i) the element Zl) E Z such that zf, ~ z (in the
metrics of Z) as 8 ~ O. For a given set of input data, R(uf,,8) can be treated as
the approximate solution of the problem.