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.
Author(s): M. Thamban Nair
Publisher: World Scientific Publishing Company
Year: 2009
Language: English
Pages: 264
Tags: Математика;Функциональный анализ;
Contents......Page 13
Preface......Page 9
1.1 General Introduction......Page 16
1.2 Well-Posedness and Ill-Posedness......Page 17
1.2.1 Examples of well-posed equations......Page 18
1.3 What Do We Do in This Book?......Page 19
2.1.1 Spaces......Page 22
2.1.2 Bounded operators......Page 29
2.1.3 Compact operators......Page 36
2.2.1 Uniform boundedness principle......Page 44
2.2.2 Closed graph theorem......Page 49
2.2.3 Hahn-Banach theorem......Page 51
2.2.4 Projection and Riesz representation theorems......Page 53
2.2.4.1 Adjoint of an operator......Page 55
2.3.1 Invertibility of operators......Page 57
2.3.2 Spectral notions......Page 59
2.3.3 Spectrum of a compact operator......Page 63
2.3.4 Spectral Mapping Theorem......Page 64
2.3.5 Spectral representation for compact self adjoint operators......Page 65
2.3.6 Singular value representation......Page 66
2.3.7 Spectral theorem for self adjoint operators......Page 68
Problems......Page 69
3.1 Introduction......Page 72
3.2 Convergence and Error Estimates......Page 75
3.3 Conditions for Stability......Page 80
3.3.1 Norm approximation......Page 81
3.3.2 Norm-square approximation......Page 85
3.4.1 Interpolatory projections......Page 86
3.4.2 A projection based approximation......Page 89
3.5 Quadrature Based Approximation......Page 91
3.5.1 Quadrature rule......Page 92
3.5.2 Interpolatory quadrature rule......Page 93
3.5.3 Nystrom approximation......Page 94
3.5.4 Collectively compact approximation......Page 97
3.6 Norm-square Approximation Revisited......Page 99
3.7 Second Kind Equations......Page 103
3.7.1 Iterated versions of approximations......Page 104
3.8 Methods for Second Kind Equations......Page 105
3.8.1 Projection methods......Page 106
3.8.2 Computational aspects......Page 120
3.8.3 Error estimates under smoothness assumptions......Page 123
3.8.4 Quadrature methods for integral equations......Page 133
3.8.5 Accelerated methods......Page 137
3.9.1 Uniform and arbitrarily slow convergence......Page 139
3.9.2 Modification of ASC-methods......Page 142
Problems......Page 145
4.1 Ill-Posedness of Operator Equations......Page 150
4.1.1 Compact operator equations......Page 151
4.1.2 A backward heat conduction problem......Page 152
4.2.1 LRN solution......Page 155
4.2.2 Generalized inverse......Page 160
4.2.3 Normal equation......Page 163
4.2.4 Picard criterion......Page 165
4.3.1 Regularization family......Page 166
4.3.2 Regularization algorithm......Page 168
4.4 Tikhonov Regularization......Page 170
4.4.1 Source conditions and order of convergence......Page 177
4.4.2 Error estimates with inexact data......Page 181
4.4.3 An illustration of the source condition......Page 186
4.4.4 Discrepancy principles......Page 187
4.4.5 Remarks on general regularization......Page 198
4.5 Best Possible Worst Case Error......Page 200
4.5.1 Estimates for (M, )......Page 203
4.5.2 Illustration with di erentiation problem......Page 205
4.5.3 Illustration with backward heat equation......Page 207
4.6.1 Why do we need a general source condition?......Page 208
4.6.2 Error estimates for Tikhonov regularization......Page 210
4.6.3 Parameter choice strategies......Page 211
4.6.4 Estimate for (M , , )......Page 216
Problems......Page 218
5.1 Introduction......Page 222
5.2.1 Convergence and general error estimates......Page 224
5.2.2 Error estimates under source conditions......Page 228
5.2.3 Error estimates for Ritz method......Page 229
5.3 Using an Approximation (An) of T*T......Page 232
5.3.1 Results under norm convergence......Page 233
5.4 Methods for Integral Equations......Page 235
5.4.1 A degenerate kernel method......Page 237
5.4.2 Regularized Nystrom method......Page 245
Problems......Page 253
Bibliography......Page 256
Index......Page 262
