Inverse Problems in the Mathematical Sciences
Classical applied mathematics is dominated by the Laplacian paradigm of known causes evolving continuously into uniquely determined effects. The classical direct problem is then to find the unique effect of a given cause by using the appropriate law of ev
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Inverse Problems in the Mathematical Sciences
Charles W. Groetsch Inverse Problems in the Mathematical Sciences
Charles W. Groetsch
Inverse Problems in the Mathematical Sciences With 38 Illustrations
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Die Deutsche Bibliothek- CIP-Einheitsaufnahme Groetsch, Charles W.: Inverse problems in the mathematical sciences 1 Charles W. Groetsch. - Braunschweig; Wiesbaden: Vieweg, 1993 ISBN 3-528-06545-1
Professor Charles W. Groetsch Department of Mathematical Sciences University of Cincinnati Cincinnati, Ohio 45221-0025 USA
Mathematical Subject Classification: OOA69, 45B05, 65R30, 45B05
AII rights reserved © Springer Fachmedien Wiesbaden 1993
Originally pub1ished by Friedr. Vieweg & Sohn Ver1agsgesellschaft mbH, Braunschweig/Wiesbaden, in 1993. Vieweg is a subsidiary company of the Bertelsmann Publishing Group International.
No part of this publication may be reproduced, stored in a retrieval system or transmitted, mechanical, photocopying or otherwise, without prior permission of the copyright holder.
Cover design: Klaus Birk, Wiesbaden Printed on acid-free paper
ISBN 978-3-322-99204-8 ISBN 978-3-322-99202-4 (eBook) DOI 10.1007/978-3-322-99202-4
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Preface Classical applied mathematics is dominated by the Laplacian paradigm of known causes evolving continuously into uniquely determined effects. The classical direct problem is then to find the unique effect of a given cause by using the appropriate law of evolution. It is therefore no surprise that traditional teaching in mathematics and the natural sciences emphasizes the point of view that problems have a solution, this solution is unique, and the solution is insensitive to small changes in the problem. Such problems are called well-posed and they typically arise from the so-called direct problems of natural science. The demands of science and technology have recently brought to the fore many problems that are inverse to the classical direct problems, that is, problems which may be interpreted as finding the cause of a given effect or finding the law of evolution given the cause and effect. Included among such problems are many questions of remote sensing or indirect measurement such as the determination of internal characteristics of an inaccessible region from measurements on its boundary, the determination of system parameters from inputoutput measurements, and the reconstruction of past events from measurements of the present state. Inverse problems of this type are often ill-posed in the sense that distinct causes can account for the same effect and small changes in a perceived effect can correspond to very large changes in a given cause. Very frequently such inverse problems are modeled by integral equations of the first kind. The level of research activity in integral equations of the first kind, inverse problems and ill-posed problems have been very high in recent years, however, the rank-and-file teaching faculty in undergraduate institutions is largely unaware of this exciting and important area of research. This is a double tragedy beca
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