Inverse Problems: Mathematical And Analytical Techniques With Applications To Engineering

TeAM YYePG Digitally signed by TeAM YYePG DN: cn=TeAM YYePG, c=US, o=TeAM YYePG, ou=TeAM YYePG, [email protected] Reason: I attest to the accuracy and integrity of this document Date: 2005.08.11 16:11:34 +08'00' INVERSE PROBLEMS MATHEMATICAL AND ANALYTICAL TECHNIQUES WITH APPLICATIONS TO ENGINEERING MATHEMATICAL AND ANALYTICAL TECHNIQUES WITH APPLICATIONS TO ENGINEERING Alan Jeffrey, Consulting Editor Published: Inverse Problems A. G. Ramm Singular Perturbation Theory R. S. Johnson Forthcoming: Methods for Constructing Exact Solutions of Partial Differential Equations with Applications S. V. Meleshko The Fast Solution of Boundary Integral Equations S. Rjasanow and O. Steinbach Stochastic Differential Equations with Applications R. Situ INVERSE PROBLEMS MATHEMATICAL AND ANALYTICAL TECHNIQUES WITH APPLICATIONS TO ENGINEERING ALEXANDER G. RAMM Springer eBook ISBN: Print ISBN: 0-387-23218-4 0-387-23195-1 ©2005 Springer Science + Business Media, Inc. Print ©2005 Springer Science + Business Media, Inc. Boston All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Springer's eBookstore at: and the Springer Global Website Online at: http://ebooks.springerlink.com http://www.springeronline.com To Luba and Olga This page intentionally left blank CONTENTS Foreword Preface xv xvii 1. Introduction 1 1.1 Why are inverse problems interesting and practically important? 1 1.2 Examples of inverse problems 2 1.2.1 Inverse problems of potential theory 2 1.2.2 Inverse spectral problems 2 1.2.3 Inverse scattering problems in quantum physics; finding the potential from the impedance function 2 1.2.4 Inverse problems of interest in geophysics 3 1.2.5 Inverse problems for the heat and wave equations 3 1.2.6 Inverse obstacle scattering 4 1.2.7 Finding small subsurface inhomogeneities from the measurements of the scattered field on the surface 5 1.2.8 Inverse problem of radiomeasurements 5 1.2.9 Impedance tomography (inverse conductivity) problem 5 1.2.10 Tomography and other integral geometry problems 5 1.2.11 Inverse problems with “incomplete data” 6 1.2.12 The Pompeiu problem, Schiffer’s conjecture, and inverse problem of plasma theory 7 1.2.13 Multidimensional inverse potential scattering 8 1.2.14 Ground-penetrating radar 8 1.2.15 A geometrical inverse problem 9 1.2.16 Inverse source problems 10 viii Contents Identification problems for integral-differential equations 12 Inverse problem for an abstract evolution equation 12 Inverse gravimetry problem 12 Phase retrieval problem (PRP) 12 Non-overdetermined inverse problems 12 Image processing, deconvolution 13 Inverse problem of electrodynamics, recovery of layered medium from the surface scattering data 13 1.2.24 Finding ODE from a trajectory 13 1.2.17 1.2.18 1.2.19 1.2.20 1.2.21 1.2.22 1.2.23 1.3 Ill-posed problems 14 1.4 Examples of Ill-posed problems 15 1.4.1 Stable numerical differentiation of noisy data 15 1.4.2 Stable summation o