Parameter estimation and inverse problems.pdf

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1 INTRODUCTION

1.1 CLASSIFICATION OF INVERSE PROBLEMS

1.2 EXAMPLES OF PARAMETER ESTIMATION PROBLEMS

1.3 EXAMPLES OF INVERSE PROBLEMS

1.4 WHY INVERSE PROBLEMS ARE HARD

1.5 EXERCISES

1.6 NOTES AND FURTHER READING

2 LINEAR REGRESSION

2.1 INTRODUCTION TO LINEAR REGRESSION

2.2 STATISTICAL ASPECTS OF LEAST SQUARES

2.3 UNKNOWN MEASUREMENT STANDARD DEVIATIONS

2.4 L1 REGRESSION

2.5 MONTE CARLO ERROR PROPAGATION

2.6 EXERCISES

2.7 NOTES AND FURTHER READING

3 DISCRETIZING CONTINUOUS INVERSE PROBLEMS

3.1 INTEGRAL EQUATIONS

3.2 QUADRATURE METHODS

3.3 EXPANSION IN TERMS OF REPRESENTERS

3.4 EXPANSION IN TERMS OF ORTHONORMAL BASIS FUNCTIONS

3.5 THE METHOD OF BACKUS AND GILBERT

3.6 EXERCISES

3.7 NOTES AND FURTHER READING

4 RANK DEFICIENCYAND ILL-CONDITIONING

4.1 THE SVD AND THE GENERALIZED INVERSE

4.2 COVARIANCE AND RESOLUTION OF THE GENERALIZED INVERSE SOLUTION

4.3 INSTABILITY OF THE GENERALIZED INVERSE SOLUTION

4.4 AN EXAMPLE OF A RANK-DEFICIENT PROBLEM

4.5 DISCRETE ILL-POSED PROBLEMS

4.6 EXERCISES

4.7 NOTES AND FURTHER READING

5 TIKHONOV REGULARIZATION

5.1 SELECTING A GOOD SOLUTION

5.2 SVD IMPLEMENTATION OF TIKHONOV REGULARIZATION

5.3 RESOLUTION, BIAS, AND UNCERTAINTY IN THE TIKHONOV SOLUTION

5.4 HIGHER-ORDER TIKHONOV REGULARIZATION

5.5 RESOLUTION IN HIGHER-ORDER TIKHONOV REGULARIZATION

5.6 THE TGSVD METHOD

5.7 GENERALIZED CROSS VALIDATION

5.8 ERROR BOUNDS

5.9 EXERCISES

5.10 NOTES AND FURTHER READING

6 ITERATIVE METHODS

6.1 INTRODUCTION

6.2 ITERATIVE METHODS FOR TOMOGRAPHY PROBLEMS

6.3 THE CONJUGATE GRADIENT METHOD

6.4 THE CGLS METHOD

6.5 EXERCISES

6.6 NOTES AND FURTHER READING

7 ADDITIONAL REGULARIZATION TECHNIQUES

7.1 USING BOUNDS AS CONSTRAINTS

7.2 MAXIMUM ENTROPY REGULARIZATION

7.3 TOTAL VARIATION

7.4 EXERCISES

7.5 NOTES AND FURTHER READING

8 FOURIER TECHNIQUES

8.1 LINEAR SYSTEMS IN THE TIME AND FREQUENCY DOMAINS

8.2 DECONVOLUTION FROMA FOURIER PERSPECTIVE

8.3 LINEAR SYSTEMS IN DISCRETE TIME

8.4 WATER LEVEL REGULARIZATION

8.5 EXERCISES

8.6 NOTES AND FURTHER READING

9 NONLINEAR REGRESSION

9.1 NEWTON’S METHOD

9.2 THE GAUSS–NEWTON AND LEVENBERG–MARQUARDT METHODS

9.3 STATISTICAL ASPECTS

9.4 IMPLEMENTATION ISSUES

9.5 EXERCISES

9.6 NOTES AND FURTHER READING

10 NONLINEAR INVERSE PROBLEMS

10.1 REGULARIZING NONLINEAR LEAST SQUARES PROBLEMS

10.2 OCCAM’S INVERSION

10.3 EXERCISES

10.4 NOTES AND FURTHER READING

11 BAYESIAN METHODS

11.1 REVIEW OF THE CLASSICAL APPROACH

11.2 THE BAYESIAN APPROACH

11.3 THE MULTIVARIATE NORMAL CASE

11.4 MAXIMUM ENTROPY METHODS

11.5 EPILOGUE

11.6 EXERCISES

11.7 NOTES AND FURTHER READING

Appendix A REVIEW OF LINEAR ALGEBRA

A.1 SYSTEMS OF LINEAR EQUATIONS

A.2 MATRIX AND VECTOR ALGEBRA

A.3 LINEAR INDEPENDENCE

A.4 SUBSPACES OF Rn

A.5 ORTHOGONALITY AND THE DOT PRODUCT

A.6 EIGENVALUES AND EIGENVECTORS

A.7 VECTOR AND MATRIX NORMS

A.8 THE CONDITION NUMBER OF A LINEAR SYSTEM

A.9 THE QR FACTORIZATION

A.10 LINEAR ALGEBRA IN SPACES OF FUNCTIONS

A.11 EXERCISES

A.12 NOTES AND FURTHER READING

Appendix B REVIEW OF PROBABILITYAND STATISTICS

B.1 PROBABILITY AND RANDOM VARIABLES

B.2 EXPECTED VALUE AND VARIANCE

B.3 JOINT DISTRIBUTIONS

B.4 CONDITIONAL PROBABILITY

B.5 THE MULTIVARIATE NORMAL DISTRIBUTION

B.6 THE CENTRAL LIMIT THEOREM

B.7 TESTING FOR NORMALITY

B.8 ESTIMATING MEANS AND CONFIDENCE INTERVALS

B.9 HYPOTHESIS TESTS

B.10 EXERCISES

B.11 NOTES AND FURTHER READING

Appendix C REVIEW OF VECTOR CALCULUS

C.1 THE GRADIENT, HESSIAN, AND JACOBIAN

C.2 TAYLOR’S THEOREM

C.3 LAGRANGE MULTIPLIERS

C.4 EXERCISES

C.5 NOTES AND FURTHER READING

Appendix D GLOSSARY OF NOTATION

BIBLIOGRAPHY

INDEX

International Geophysics Series

About the CD-ROM

Parameter Estimation and Inverse Problems

This is Volume 90 in the INTERNATIONAL GEOPHYSICS SERIES A series of monographs and textbooks Edited by RENATA DMOWSKA, JAMES R. HOLTON, and H. THOMAS ROSSBY A complete list of books in this series appears at the end of this volume.

Parameter Estimation and Inverse Problems Richard C. Aster, Brian Borchers, and Clifford H. Thurber Amsterdam • Boston Heidelberg London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo

Acquisition Editor Project Manager Editorial Coordinator Marketing Manager Cover Design Composition Cover Printer Interior Printer Frank Cynar Kyle Sarofeen Jennifer Helé Linda Beattie Suzanne Rogers Cepha Imaging Private Ltd. Phoenix Color Corporation Maple-Vail Book Manufacturing Group Elsevier Academic Press 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA 84 Theobald’s Road, London WC1X 8RR, UK This book is printed on acid-free paper. Copyright © 2005, Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: permissions@elsevier.com.uk. You may also complete your request online via the Elsevier homepage (http://elsevier.com), by selecting “Customer Support” and then “Obtaining Permissions.” Library of Congress Cataloging-in-Publication Data Application submitted British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 0-12-065604-3 For information on all Elsevier Academic Press Publications visit our Web site at www.books.elsevier.com Printed in the United States of America 04 05 06 07 08 09 9 8 7 6 5 4 3 2 1

Contents Preface xi 1 INTRODUCTION 1 Examples of Parameter Estimation Problems Examples of Inverse Problems 1.1 Classiﬁcation of Inverse Problems 1.2 1.3 1.4 Why Inverse Problems Are Hard 1.5 1.6 Notes and Further Reading Exercises 14 14 1 7 11 2 LINEAR REGRESSION 15 15 Introduction to Linear Regression Statistical Aspects of Least Squares 2.1 2.2 17 2.3 Unknown Measurement Standard Deviations 2.4 L1 Regression 2.5 Monte Carlo Error Propagation 2.6 2.7 Notes and Further Reading Exercises 36 30 40 35 4 26 3 DISCRETIZING CONTINUOUS INVERSE 41 PROBLEMS 3.1 3.2 Quadrature Methods Integral Equations 41 41 v

vi Contents Expansion in Terms of Representers 46 Expansion in Terms of Orthonormal Basis Functions 47 3.3 3.4 3.5 3.6 3.7 Notes and Further Reading Exercises 52 54 The Method of Backus and Gilbert 48 4 RANK DEFICIENCY AND ILL-CONDITIONING 55 55 The SVD and the Generalized Inverse Instability of the Generalized Inverse Solution 4.1 4.2 Covariance and Resolution of the Generalized Inverse Solution 4.3 4.4 An Example of a Rank-Deﬁcient Problem 4.5 Discrete Ill-Posed Problems 4.6 4.7 Notes and Further Reading Exercises 64 73 85 87 67 62 5 TIKHONOV REGULARIZATION 89 89 Selecting a Good Solution SVD Implementation of Tikhonov Regularization 5.1 5.2 91 5.3 Resolution, Bias, and Uncertainty in the Tikhonov Solution 5.4 Higher-Order Tikhonov Regularization 5.5 Resolution in Higher-Order Tikhonov Regularization 5.6 5.7 Generalized Cross Validation 5.8 5.9 5.10 Notes and Further Reading The TGSVD Method Error Bounds Exercises 106 109 105 114 117 98 95 103 6 ITERATIVE METHODS 119 6.1 6.2 6.3 6.4 Introduction 119 Iterative Methods for Tomography Problems 120 The Conjugate Gradient Method 126 The CGLS Method 131

Contents vii 6.5 Exercises 135 6.6 Notes and Further Reading 136 7 ADDITIONALREGULARIZATION TECHNIQUES 139 7.1 Using Bounds as Constraints 139 7.2 Maximum Entropy Regularization 143 7.3 7.4 Total Variation 146 Exercises 151 7.5 Notes and Further Reading 152 8 FOURIER TECHNIQUES 153 8.1 Linear Systems in the Time and Frequency Domains 153 8.2 Deconvolution from a Fourier Perspective 158 8.3 Linear Systems in Discrete Time 161 8.4 Water Level Regularization 164 8.5 Exercises 168 8.6 Notes and Further Reading 170 9 NONLINEAR REGRESSION 171 9.1 Newton’s Method 171 9.2 9.3 9.4 9.5 The Gauss–Newton and Levenberg–Marquardt Methods 174 Statistical Aspects 177 Implementation Issues 181 Exercises 186 9.6 Notes and Further Reading 189 10 NONLINEAR INVERSE PROBLEMS 10.1 Regularizing Nonlinear Least Squares Problems 191 191 10.2 Occam’s Inversion 195 10.3 Exercises 199 10.4 Notes and Further Reading 199

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