The Method of Approximate Inverse: Theory and Applications
Inverse problems arise whenever one tries to calculate a required quantity from given measurements of a second quantity that is associated to the first one. Besides medical imaging and non-destructive testing, inverse problems also play an increasing role
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Thomas Schuster
The Method of Approximate Inverse: Theory and Applications
1906
Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris
1906
Thomas Schuster
The Method of Approximate Inverse: Theory and Applications
ABC
Author Thomas Schuster Department of Mechanical Engineering Helmut Schmidt University Holstenhofweg 85 22043 Hamburg Germany e-mail: [email protected]
Library of Congress Control Number: 2007922352 Mathematics Subject Classification (2000): 15A29, 35R30, 45Q05, 65J22, 65N21 ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN-10 3-540-71226-7 Springer Berlin Heidelberg New York ISBN-13 978-3-540-71226-8 Springer Berlin Heidelberg New York DOI 10.1007/978-3-540-71227-5 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting by the author and SPi using a Springer LATEX macro package Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper
SPIN: 12027460
VA41/3100/SPi
543210
Dedicated to Petra, for her patience, understanding, and love
Preface
Many questions and applications in natural science, engineering, industry or medical imaging lead to inverse problems, that is: given some measured data one tries to recover a searched for quantity. These problems are of growing interest in all these disciplines and thus there is a great need for modern and stable solvers for these problems. A prominent example of an inverse problem is the problem of computerized tomography: From measured X-ray attenuation coefficients one has to calculate densities in human tissue. Mathematically inverse problems often are described as operator equations of first kind Af = g ,
(0.1)
where A : X → Y is a bounded operator acting on appropriate topological spaces X and Y . In case of 2D computerized tomography the mapping A is given by the Radon transform. Typically these operators have unbounded inverses A−1 , if they are invertible at all. For instance if A is compact with infinite dimensional range, then A−1 is not continuous. In case of Hilbert spaces X and Y the generalized inverse A† exists and has a dense domain. But A† is bounded if
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