An Introduction to the Mathematical Theory of Inverse Problems
This book introduces the reader to the area of inverse problems. The study of inverse problems is of vital interest to many areas of science and technology such as geophysical exploration, system identification, nondestructive testing and ultrasonic tomog
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Andreas Kirsch
An Introduction to the Mathematical Theory of Inverse Problems Second edition
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Andreas Kirsch Department of Mathematics Karlsruhe Institute of Technology (KIT) Kaiserstrasse 89, 76133 Karlsruhe, Germany [email protected]
ISSN 0066-5452 ISBN 978-1-4419-8473-9 e-ISBN 978-1-4419-8474-6 DOI 10.1007/978-1-4419-8474-6 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011924472 Mathematics Subject Classification (2010): 31A25, 35J05, 35P25, 35R25, 35R30, 45A05, 45B05, 45Q05, 47A52, 65R20, 65R30, 65R32, 78A45, 78A46 c Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface of the First Edition
Following Keller [136] we call two problems inverse to each other if the formulation of each of them requires full or partial knowledge of the other. By this definition, it is obviously arbitrary which of the two problems we call the direct and which we call the inverse problem. But usually, one of the problems has been studied earlier and, perhaps, in more detail. This one is usually called the direct problem, whereas the other is the inverse problem. However, there is often another more important difference between these two problems. Hadamard (see [103]) introduced the concept of a well-posed problem, originating from the philosophy that the mathematical model of a physical problem has to have the properties of uniqueness, existence, and stability of the solution. If one of the properties fails to hold, he called the problem ill-posed. It turns out that many interesting and important inverse problems in science lead to ill-posed problems, whereas the corresponding direct problems are well-posed. Often, existence and uniqueness can be forced by enlarging or reducing the solution space (the space of “models”). For restoring stability, however, one has to change the topology of the spaces, which is in many cases impossible because of the presence of measurement errors. At first glance, it seems to be impossible to compute the solution of a problem numerically if the solution of the problem does not depend continuously on the data, that is, for the case of ill-posed problems. Under
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