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Numerical methods for inverse problems PDF

234 Pages·2016·2.536 MB·English
by  KernMichel
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Numerical Methods for Inverse Problems To my wife Elisabeth, to my children David and Jonathan Series Editor Nikolaos Limnios Numerical Methods for Inverse Problems Michel Kern First published 2016 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd John Wiley & Sons, Inc. 27-37 St George’s Road 111 River Street London SW19 4EU Hoboken, NJ 07030 UK USA www.iste.co.uk www.wiley.com © ISTE Ltd 2016 The rights of Michel Kern to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2016933850 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-818-5 Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Part1.IntroductionandExamples . . . . . . . . . . . . . . . . . . . . . . 1 Chapter1.OverviewofInverseProblems . . . . . . . . . . . . . . . . . 3 1.1.Directandinverseproblems . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.Well-posedandill-posedproblems . . . . . . . . . . . . . . . . . . . . 4 Chapter2.ExamplesofInverseProblems . . . . . . . . . . . . . . . . . 9 2.1.Inverseproblemsinheattransfer . . . . . . . . . . . . . . . . . . . . . . 10 2.2.Inverseproblemsinhydrogeology . . . . . . . . . . . . . . . . . . . . . 13 2.3.Inverseproblemsinseismicexploration. . . . . . . . . . . . . . . . . . 16 2.4.Medicalimaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5.Otherexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Part2.LinearInverseProblems . . . . . . . . . . . . . . . . . . . . . . . . 29 Chapter3.IntegralOperatorsandIntegralEquations . . . . . . . . . 31 3.1.Definitionandfirstproperties. . . . . . . . . . . . . . . . . . . . . . . . 31 3.2.Discretizationofintegralequations . . . . . . . . . . . . . . . . . . . . 36 3.2.1.Discretizationbyquadrature–collocation . . . . . . . . . . . . . . . 36 3.2.2.DiscretizationbytheGalerkinmethod . . . . . . . . . . . . . . . . 39 3.3.Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Chapter4.LinearLeastSquaresProblems–SingularValue Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.1.Mathematicalpropertiesofleastsquaresproblems. . . . . . . . . . . . 45 4.1.1.Finitedimensionalcase . . . . . . . . . . . . . . . . . . . . . . . . . 50 vi NumericalMethodsforInverseProblems 4.2.Singularvaluedecompositionformatrices . . . . . . . . . . . . . . . . 52 4.3.Singularvalueexpansionforcompactoperators . . . . . . . . . . . . . 57 4.4.ApplicationsoftheSVDtoleastsquaresproblems . . . . . . . . . . . 60 4.4.1.Thematrixcase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.4.2.Theoperatorcase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.5.Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Chapter5.RegularizationofLinearInverseProblems . . . . . . . . . 71 5.1.Tikhonov’smethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.1.1.Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.1.2.Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.1.3.TheL-curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.2.ApplicationsoftheSVE . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2.1.SVEandTikhonov’smethod . . . . . . . . . . . . . . . . . . . . . . 84 5.2.2.RegularizationbytruncatedSVE . . . . . . . . . . . . . . . . . . . 85 5.3.Choiceoftheregularizationparameter . . . . . . . . . . . . . . . . . . 88 5.3.1.Morozov’sdiscrepancyprinciple. . . . . . . . . . . . . . . . . . . . 88 5.3.2.TheL-curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.3.3.Numericalmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.4.Iterativemethods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.5.Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Part3.NonlinearInverseProblems . . . . . . . . . . . . . . . . . . . . . 103 Chapter6.NonlinearInverseProblems–Generalities . . . . . . . . . 105 6.1.Thethreefundamentalspaces . . . . . . . . . . . . . . . . . . . . . . . 106 6.2.Leastsquaresformulation. . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.2.1.Difficultiesofinverseproblems . . . . . . . . . . . . . . . . . . . . 114 6.2.2.Optimization,parametrization,discretization. . . . . . . . . . . . . 114 6.3.Methodsforcomputingthegradient–theadjointstatemethod . . . . 116 6.3.1.Thefinitedifferencemethod . . . . . . . . . . . . . . . . . . . . . . 116 6.3.2.Sensitivityfunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.3.3.Theadjointstatemethod . . . . . . . . . . . . . . . . . . . . . . . . 119 6.3.4.ComputationoftheadjointstatebytheLagrangian . . . . . . . . . 120 6.3.5.Theinnerproducttest . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.4.Parametrizationandgeneralorganization . . . . . . . . . . . . . . . . . 123 6.5.Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Chapter7.SomeParameterEstimationExamples . . . . . . . . . . . 127 7.1.Ellipticequationinonedimension . . . . . . . . . . . . . . . . . . . . . 127 7.1.1.Computationofthegradient . . . . . . . . . . . . . . . . . . . . . . 128 7.2.Stationarydiffusion: ellipticequationintwodimensions . . . . . . . . 129 Contents vii 7.2.1.Computationofthegradient: applicationofthegeneral method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 7.2.2.ComputationofthegradientbytheLagrangian . . . . . . . . . . . 134 7.2.3.Theinnerproducttest . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.2.4.Multiscaleparametrization . . . . . . . . . . . . . . . . . . . . . . . 135 7.2.5.Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.3.Ordinarydifferentialequations . . . . . . . . . . . . . . . . . . . . . . . 137 7.3.1.Anapplicationexample . . . . . . . . . . . . . . . . . . . . . . . . . 144 7.4.Transientdiffusion: heatequation . . . . . . . . . . . . . . . . . . . . . 147 7.5.Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Chapter8.FurtherInformation . . . . . . . . . . . . . . . . . . . . . . . . 155 8.1.Regularizationinothernorms . . . . . . . . . . . . . . . . . . . . . . . 155 8.1.1.Sobolevsemi-norms. . . . . . . . . . . . . . . . . . . . . . . . . . . 155 8.1.2.Boundedvariationregularizationnorm . . . . . . . . . . . . . . . . 157 8.2.Statisticalapproach: Bayesianinversion . . . . . . . . . . . . . . . . . 157 8.2.1.Leastsquaresandstatistics . . . . . . . . . . . . . . . . . . . . . . . 158 8.2.2.Bayesianinversion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 8.3.Othertopics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 8.3.1.Theoreticalaspects: identifiability . . . . . . . . . . . . . . . . . . . 163 8.3.2.Algorithmicdifferentiation . . . . . . . . . . . . . . . . . . . . . . . 163 8.3.3.Iterativemethodsandlarge-scaleproblems . . . . . . . . . . . . . . 164 8.3.4.Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Appendix1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Appendix2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Appendix3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

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