Linear Operator Equations: Approximation And Regularization

by M Thamban Nair

Overview

Many problems in science and engineering have their mathematical formulation as an operator equation Tx=y, where T is a linear or nonlinear operator between certain function spaces. In practice, such equations are solved approximately using numerical methods, as their exact solution may not often be possible or may not be worth looking for due to physical constraints. In such situations, it is desirable to know how the so-called approximate solution approximates the exact solution, and what the error involved in such procedures would be.This book is concerned with the investigation of the above theoretical issues related to approximately solving linear operator equations. The main tools used for this purpose are basic results from functional analysis and some rudimentary ideas from numerical analysis. To make this book more accessible to readers, no in-depth knowledge on these disciplines is assumed for reading this book.

Product Details

ISBN-13:	9789812835642
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	05/27/2009
Pages:	264
Product dimensions:	6.10(w) x 9.00(h) x 0.70(d)

Table of Contents

Preface vii

1 Introduction 1

1.1 General Introduction 1

1.2 Well-Posedness and Ill-Posedness 2

1.2.1 Examples of well-posed equations 3

1.2.2 Examples of ill-posed equations 4

1.3 What Do We Do in This Book? 4

2 Basic Results from Functional Analysis 7

2.1 Spaces and Operators 7

2.1.1 Spaces 7

2.1.2 Bounded operators 14

2.1.3 Compact operators 21

2.2 Some Important Theorems 29

2.2.1 Uniform boundedness principle 29

2.2.2 Closed graph theorem 34

2.2.3 Hahn-Banach theorem 36

2.2.4 Projection and Riesz representation theorems 38

2.3 Spectral Results for Operators 42

2.3.1 Invertiability of operators 42

2.3.2 Spectral notions 44

2.3.3 Spectrum of a compact operator 48

2.3.4 Spectral Mapping Theorem 49

2.3.5 Spectral representation for compact self adjoint operators 50

2.3.6 Singular value representation 51

2.3.7 Spectral theorem for self adjoint operators 53

Problems 54

3 Well-Posed Equations and Their Approximations 57

3.1 Introduction 57

3.2 Convergence and Error Estimates 60

3.3 Conditions for Stability 65

3.3.1 Norm approximation 66

3.3.2 Norm-square approximation 70

3.4 Projection Based Approximation 71

3.4.1 Interpolatory projections 71

3.4.2 A projection based approximation 74

3.5 Quadrature Based Approximation 76

3.5.1 Quadrature rule 77

3.5.2 Interpolatory quadrature rule 78

3.5.3 Nyströet;m approximation 79

3.5.4 Collectively compact approximation 82

3.6 Norm-square Approximation Revisited 84

3.7 Second Kind Equations 88

3.7.1 Iterated versions of approximations 89

3.8 Methods for Second Kind Equations 90

3.8.1 Projection methods 91

3.8.2 Computational aspects 105

3.8.3 Error estimatesunder smoothness assumptions 108

3.8.4 Quadrature methods for integral equations 118

3.8.5 Accelerated methods 122

3.9 Qualitative Properties of Convergence 124

3.9.1 Uniform and arbitrarily slow convergence 124

3.9.2 Modification of ASC-methods 127

Problems 130

4 Ill-Posed Equations and Their Regularizations 135

4.1 Ill-Posedness of Operator Equations 135

4.1.1 Compact operator equations 136

4.1.2 A backward heat conduction problem 137

4.2 LRN Solution and Generalized Inverse 140

4.2.1 LRN solution 140

4.2.2 Generalized inverse 145

4.2.3 Normal equation 148

4.2.4 Picard criterion 150

4.3 Regularization Procedure 151

4.3.1 Regularization family 151

4.3.2 Regularization algorithm 153

4.4 Tikhonov Regularization 155

4.4.1 Source conditions and order of convergence 162

4.4.2 Error estimates with inexact data 166

4.4.3 An illustration of the source condition 171

4.4.4 Discrepancy principles 172

4.4.5 Remarks on general regularization 183

4.5 Best Possible Worst Case Error 185

4.5.1 Estimates for ω (M, ○) 188

4.5.2 Illustration with differentiation problem 190

4.5.3 Illustration with backward heat equation 192

4.6 General Source Conditions 193

4.6.1 Why do we need a general source condition? 193

4.6.2 Error estimates for Tikhonov regularization 195

4.6.3 Parameter choice strategies 196

4.6.4 Estimate for ω (Mϕ&rho, ○) 201

Problems 203

5 Regularized Approximation Methods 207

5.1 Introduction 207

5.2 Using an Approximation (Tn) of T 209

5.2.1 Convergence and general error estimates 209

5.2.2 Error estimates under source conditions 213

5.2.3 Error estimates for Ritz method 214

5.3 Using an Approximation (An) of T*T 217

5.3.1 Results under norm convergence 218

5.4 Methods for Integral Equations 220

5.4.1 A degenerate kernel method 222

5.4.2 Regularized Nyströet;m method 230

Problems 238

Bibliography 241

Index 247