Inverse and Ill-Posed Problems PDF
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Inverse and Ill-Posed Problems Edited by Heinz W. Engl Institut fttr Mathematik Johannes-Kepler- Universität Linz, Austria C.W. Groetsch Department of Mathematical Sciences University of Cincinnati Cincinnati y Ohio ACADEMIC PRESS, INC. Harcourt Brace Jovanovich, Publishers Boston Orlando San Diego New York Austin London Sydney Tokyo Toronto Copyright © 1987 by Academic Press, Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. ACADEMIC PRESS, INC. Orlando, Florida 32887 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24-28 Oval Road, London NW1 7DX Library of Congress Cataloging-in-Publication Data Inverse and ill-posed problems. (Notes and reports in mathematics in science and engineering; v. 4) Papers presented at the Alpine-U.S. Seminar on Inverse and Ill-Posed Problems, held June 1986 in St. Wolfgang, Austria. Bibliography: p. 1. Inverse problems (Differential equations)— Congresses. 2. Boundary value problems—Improperly posed problems—Congresses. I. Engl, Heinz W. II. Groetsch, C.W. III. Alpine-U.S. Seminar on Inverse and Ill-Posed Problems (1986: Sankt Wolfgang im Salzkammergut, Austria) IV. Series. QA370.I586 1987 515.3'5 87-14452 ISBN 0-12-239040-7 Printed in the United States of America 87 88 89 90 987654321 Contributors Numbers in parentheses refer to the pages on which authors' contributions begin. Karen A. Ames (443), Department of Mathematics, Iowa State University, Ames, Iowa R.S. Anderssen (19), CSIRO Division of Mathematics and Statistics, and Centre for Mathematical Analysis, Australian National University, Canberra, Australia Victor Barcilon (385), Department of Geophysical Sciences, University of Chicago, Chicago, Illinois J. Baumeister (325), Fachbereich Mathematik, Universität Frankfurt/Main, Feder al Republic of Germany M. Bertero (291), Dipartimento di Fisica and I.N.F.N., Università di Genova, Geno va, Italy Helmut Brakhage (165), Fachbereich Mathematik, Universität Kaiserslautern, Kaiserslautern, Federal Republic of Germany John Rozier Cannon (315), Mathematics Department, Washington State University, Pullman, Washington 99164-2930 C. Chicone (513), Department of Mathematics, University of Missouri, Columbia, Missouri David Colton (261), Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716 M. Costabel (369), Mathematisches Institut, Technische Hochschule Darmstadt, Schlossgartenstrasse 7, D-6100 Darmstadt, Federal Republic of Germany Franz-Jürgen Delvos (541), Fachbereich 6 Mathematik I, Universität- Gesamthochschule Siegen, Siegen, Federal Republic of Germany C. De Mol (291), Département de Mathématique, Université Libre de Bruxelles, Bruxelles, Belgium C.R. Dietrich (19), Centre for Resource and Environmental Studies, Australian National University, Canberra, Australia Lars Eiden (345), Department of Mathematics, Linkòping University, Linköping, Sweden IX X Contributors Heinz W. Engl (97), Institut ßr Mathematik, Johannes-Kepler-Universität, Linz, Austria Richard Ewing (483), Cooperative Research Center for Mathematical Modeling, University of Wyoming, Laramie, Wyoming Richard Falk (483), Department of Mathematics, Rutgers University, New Bruns wick, New Jersey Jürgen Gerlach (513), Department of Mathematics, University of Missouri, Colum bia, Missouri Helmut Gfrerer (127), Institut ßr Mathematik, Johannes-Kepler-Universität, Linz, Austria Rudolf Gorenflo (195), Fachbereich Mathematik, Freie Universität, Berlin, Fed eral Republic of Germany Michael Hanke (523), Sektion Mathematik, Humboldt-Universität, Berlin, GDR G.C. Hsiao (461), Department of Mathematical Sciences, University of Delaware, Newark, Delaware Andreas Kirsch (279), Institut ßr Numerische und Angewandte Mathematik, Univer sität Göttingen, Göttingen, Federal Republic of Germany Peter Knabner (351), Institut ßr Mathematik, Universität Augsburg, Federal Repub lic of Germany Rainer Kress (279, 417), Institut ßr Numerische und Angewandte Mathematik, Universität Göttingen, Göttingen, Federal Republic of Germany Karl Kunisch (499), Institut ßr Mathematik, Technische Universität Graz, Koper- nikusgasse 24, A-80I0 Graz, Austria Howard A. Levine (443, 451), Department of Mathematics, Iowa State University, Ames, Iowa Tao Lin (483), Cooperative Research Center for Mathematical Modeling, Universi ty of Wyoming, Laramie, Wyoming Peter Linz (29), Division of Computer Science, University of California, Davis, California Alfred K. Louis (177), Fachbereich Mathematik, Technische Universität Berlin, D-1000 Berlin 12, Federal Republic of Germany Roswitha März (523), Sektion Mathematik, Humboldt-Universität, Berlin, GDR Peter Monk (261), Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716 J.D. Murphy (425), NASA-Ames Research Center, Moffett Field, California 94035 M.Z. Nashed (53), Department of Mathematical Sciences, University of Delaware, Newark, Delaware Frank Natterer (247), Institut ßr Numerische und Instrumentelle Mathematik, Universität Münster, Münster, Federal Republic of Germany Andreas Neubauer (97, 523), Institut ßr Mathematik, Johannes-Kepler-Universität, Linz, Austria K. Onishi (369), Department of Applied Mathematics, Fukuoka University, Jonan- ku, Nanakuma, Fukuoka 814-01, Japan Contributors xi L.E. Payne (399, 443), Department of Mathematics, Cornell University, Ithaca, New York 14853 E.R. Pike (291), Physics Department, King*s College, London, United Kingdom P.M. Prenter (425), Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523 H.-J. Reinhardt (325), Battelle-Institut E.V., Frankfurt/Main, Federal Republic of Germany P.C. Sabatier (1), Laboratoire de Physique Mathématique, Université des Sciences et Techniques du Languedoc, Montpellier, France Eberhard Schock (185), Fachbereich Mathematik, Universität Kaiserslautern, Kaiserslautern, Federal Republic of Germany Gennadi Vainikko (77), Department of Mathematics, State University of Tartu, 202400 Tartu, USSR V. V. Vasin (211), Institute of Mathematics and Mechanics, Ural Scientific Center of the USSR Academy of Sciences, Sverdlovsk, USSR Sergio Vessella (351), Istituto Analisi Globale e Applicazioni, CNR, Firenze, Italy G.A. Viano (151), Dipartimento di Fisica, Università di Genova, Genova, Italy Curtis R. Vogel (231), Department of Mathematical Sciences, Montana State Univer sity, Bozeman, Montana 59717 Grace Wahba (37), Department of Statistics, University of Wisconsin-Madison, Madi son, Wisconsin W.L. Wendland (369, 461), Mathematisches Institut, Universität Stuttgart, Pfaffen- waldring 57, D-7000 Stuttgart 80, Federal Republic of Germany Preface Back to their source the holy rivers turn their tide. Order and the universe are being reversed. Euripedes, Medea Inverse problems of mathematical physics may be broadly described as problems of determining the internal structure or past state of a system from indirect meas urements. Such problems would include for example the determination of diffusivi- ties, conductivities, densities, sources, geometries of scatterers and absorbers, and prior temperature distributions to name just a few typical applications. Inverse problems appeared as early as 1765 with Daniel Bernoulli's study of the inhomogene- ous vibrating string; however, only recently has a systematic treatment of such problems begun to emerge. Developments in this area have been spurred by the needs of modern technology for reliable means of indirect measurement. At the same time, the main tool necessary for the solution of inverse problems has been supplied by modern technology in the form of the digital computer. Many inverse problems can be modeled abstractly as Kx = g, where K is a given operator between appropriate function spaces, g is given by measured "external" parameters, and the desired solution x represents "internal" parameters, which are inaccessible to direct measurement. In the important special case when £ is a com pact linear operator, it is well known that the problem of determining x is generally ill-posed in the sense of Hadamard, and it is this ill-posedness that engenders peculiar problems in the numerical approximation and interpretation of solutions. The papers in this volume were presented at the Alpine-U.S. Seminar on Inverse and Ill-Posed Problems held in St. Wolfgang, Austria in June of 1986. Major fund ing for the seminar was provided by the U.S. National Science Foundation, the Aus trian Bundesministerium für Wissenschaft und Forschung, and the Linzer Hochschulfonds. The purpose of the seminar was to bring together applied mathema ticians and scientists working on inverse problems in various disciplines with mathematicians involved with the theory of inverse and ill-posed problems, such as those arising in the solution of integral equations of the first kind. The papers have been loosely arranged in this volume according to subject area. The first group of papers is concerned with general considerations for ill-posed problems. In the initial paper, Sabatier considers the notion of well-posed questions for ill-posed problems and their use in providing a geometrical description of the set of possible solutions. Related to the concept of well-posed questions is the linear functional strategy for ill-posed problems. The linear functional strategy with appli cations to the aquifer transmissivity problem is studied in the paper by Anderssen xin xiv Preface and Dietrich. Linz provides a general formalism for assessing the uncertainty in a constrained approximate solution of an ill-posed problem. Wahba discusses three topics on ill-posed problems: the imposition of specified types of discontinuities on solutions of ill-posed problems, the use of generalized cross validation as a data- based termination rule for iterative methods, and a parameter estimation problem in reservoir modeling. A general scheme for classification of ill-posed problems and a new notion of approximate regularization of ill-posed operator equations in linear spaces are given in the paper of Nashed. The next group of papers deals with various regularization methods for integral and operator equations of the first kind. The paper by Vainikko concerns optimal regularization methods for linear and nonlinear equations of the first kind. A posteriori parameter choice strategies for (iterated) Tikhonov regularization which lead to asymptotically optimal convergence rates are provided in the papers of Engl and Neubauer, and Gfrerer. A statistical method for determining the truncation level in eigenfunction expansions for Fredholm equations of the first kind in which the data are contaminated by error is investigated in the paper of Viano. New higher order convergence rates for the method of conjugate gradients applied to linear ill- posed equations are provided in the paper of Brakhage. Louis uses a different method to give improved rates for the conjugate gradient method applied to compact opera tors. Schock studies semi-iterative methods for linear ill-posed problems. Gorenflo analyzes discretization methods for approximating nonsmooth solutions of Abel equa tions. Vasin studies weak and strong convergence of iterative methods for nonlinear equations of the first kind subject to general constraints. A survey of some iterative methods for nonlinear ill-posed problems based on optimization techniques is given by Vogel. The next five papers are concerned with applications in tomography, inverse scat tering, detection of radiation sources, and optics. The severely ill-posed Radon in version problems in computerized tomography that result from incomplete data are investigated in the contribution of Natterer. Colton and Monk discuss a new method based on a projection theorem for the inverse scattering problem for acoustic waves. Another inverse acoustic scattering method that places special emphasis on the choice of a regularizing norm is proposed by Kirsch and Kress. Bertero, DeMol, and Pike investigate the use of singular function expansions in the inversion of severely ill- posed problems arising in confocal scanning microscopy, particle sizing, and velocim- etry. Cannon surveys some typical inverse problems related to the determination of an unknown source term in the wave equation. Attention then turns to inverse problems in heat conduction and other parabolic problems. Baumeister and Reinhardt investigate a computational method for an in verse heat conduction problem in two space dimensions. Eiden discusses various stabilization methods for a non-characteristic Cauchy problem for the heat equation in a quarter plane. Stability estimates for non-characteristic Cauchy problems for parabolic equations in one space dimension are studied by Knabner and Vessella. Costabel, Onishi, and Wendland investigate a boundary element collocation method with optimal convergence rate for the heat equation with Neumann boundary con ditions. Preface xv Additional inverse problems in partial differential equations and mechanics are addressed in the next gorup of contributions. Barcilon discusses problems associat ed with determining the density and flexural rigidity of a beam from modal infor mation. Payne reports on some recent results on the stabilization of certain classes of ill-posed Cauchy problems in which errors are made in characterizing the do main geometry and/or the modeling equation. Kress describes a least squares method for solving an ill-posed initial boundary value problem arising in solar magnetogra- phy. Murphy and Prenter discuss some ill-posed linear operator equations that arise in invicid two-dimensional fluid flow and some parameter estimation problems as sociated with boundary layer equations for separated flow. A modification of the weighted energy method which leads to improved continuous dependence results for evolutionary equations is given in the paper of Ames, Levine, and Payne. Le vine presents some recent results concerning a nonlinear wave equation in a half strip. Wendland and Hsiao study an asymptotic expansion and boundary element method for a problem in two-dimensional elasticity. Parameter estimation problems are then taken up. Ewing and Falk address the problem of estimation of distributed parameters in parabolic problems with special emphasis on the interplay between increasing modeling accuracy and the correspond ing increase in the difficulty of the numerical estimation procedure. A convergence rate for a Galerkin method for the estimation of the coefficient in an elliptic problem, involving discretization of both the state equation and the parameter space, is provided in the paper of Kunisch. The paper by Chicone and Gerlach deals with the identifia- bility problem for the coefficient in an elliptic problem. Finally there are papers that address various questions related to ill-posed problems. Hanke, März, and Neubauer study an ill-posed problem for an algebraic differen tial equation and show that it is tractable by regularization methods. Delvos inves tigates some theoretical aspects of abstract spline projections. Professors Cannon and Vasin were prevented by circumstances from attending the seminar, nevertheless their contributions appear in this volume. Unfortunately, we did not receive the papers by Professors Hoffmann, Davies, Sprekels, and McLaughlin which were presented at the seminar. The preliminary editing of this volume was done while the editors held visiting fellowships at the Centre for Mathematical Analysis of the Australian National Univer sity. They wish to thank the Centre, and particularly Bob Anderssen, for their cor diality. The final editorial work was done during H. Engl's stay in Cincinnati (sponsored by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung, project S32/03). The editors express their appreciation to the members of the scien tific advisory committee for the seminar, K.H. Hoffmann, J.T. Marti, M.Z. Nashed, and L.E. Payne, and they gratefully acknowledge financial support provided by the agencies mentioned above. Finally, they sincerely thank the administration and staff of the Bundesinstitut für Erwachsenenbildung, St. Wolfgang for their warm hospi tality which perfectly complemented the natural beauty and charm of the Salzkam mergut. A FEW GEOMETRICAL FEATURES OF INVERSE AND ILL-POSED PROBLEMS P.C. SABATIER Laboratoire de Physique Mathématique Université des Sciences et Techniques du Languedoc Montpellier France I. INTRODUCTION We would like to describe here the main ideas that have underlined studies of inverse problems in our group of Montpellier University during the last ten years. These are very simple, and, to some extent, naive, geometrical remarks on the structure of the problems. Yet, they prove to be fruitful, and since they are simple, they apply to reasonably wide classes of problems. Needless to say, several other geometrical remarks have ins pired other groups. They are of much interest and the only reason for which I do not cite them here is that they are not my cup of tea. Throughout the lecture, we shall consider problems described by a mapping M of a set C called the set of parameters into a set E called the set of results. The physical model being described by M and "physical" parameters in M, our definition means that if the physical model is not "certain", this uncertainty has been described by labelling M with other "mathematical" parameters. C should contain all parameters, and is defined as a metric space, where several criterions or constraints enable one to choose admissible parameters. E should contain altogether calculated results (ieM{Cj) and results of all possible measurements, and is defined as a metri" space. We assume henceforth that M is continuous, although this is not Inverse and Ill-Posed Probiems 1 Copyright © 1987 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-239040-7 2 P.C. Sabatier always necessary. We call "solution" of the inverse problem (a) either a true solution, when M(C) = E and the inverse (multi) mapping is (sufficiently) continuous for a (reasonable) distance in E : thus the only ill-posedness there may be due to the existence of sets X(e) of "equivalent" parameters x that correspond to the same result e (underdetermined problem) (b) or a "solution" obtained after regularizing, ie after defining in a proper way a mapping or a multimapping M from E to C, sufficien tly continuous for a reasonable distance. This may be done by considering ε-solutions (d(e,Mx) < ε) or quasisolutions (x = M~ [Proj(e)J) or approximate M(C) solutions (obtained by minimizing a weighted combination of the fit-shift in E and the choice criterions in C). There is obviously much room for underdetermination in this processing of solutions. Thus our study is limited to well-posed problems or to problems whose illposedness has been reduced to underdetermination only by a preliminary work. We try to solve difficulties connected with geometrical descriptions of sets of equivalent solutions (we try that it makes sense in applications) ; joint description of routes in C and E (we try to solve the inverse problem or to understand the nature of illposedness) ; joint description of tra jectories in C and E (we try to improve the so-called "invers emethods").
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