Regularization of Inverse ProblemsSpringer Science & Business Media, 31 Mar 2000 - 322 sayfa In the last two decades, the field of inverse problems has certainly been one of the fastest growing areas in applied mathematics. This growth has largely been driven by the needs of applications both in other sciences and in industry. In Chapter 1, we will give a short overview over some classes of inverse problems of practical interest. Like everything in this book, this overview is far from being complete and quite subjective. As will be shown, inverse problems typically lead to mathematical models that are not well-posed in the sense of Hadamard, i.e., to ill-posed problems. This means especially that their solution is unstable under data perturbations. Numerical meth ods that can cope with this problem are the so-called regularization methods. This book is devoted to the mathematical theory of regularization methods. For linear problems, this theory can be considered to be relatively complete and will be de scribed in Chapters 2 - 8. For nonlinear problems, the theory is so far developed to a much lesser extent. We give an account of some of the currently available results, as far as they might be of lasting value, in Chapters 10 and 11. Although the main emphasis of the book is on a functional analytic treatment in the context of operator equations, we include, for linear problems, also some information on numerical aspects in Chapter 9. |
İçindekiler
Examples of Inverse Problems | 3 |
11 Differentiation as an Inverse Problem | 4 |
12 Radon Inversion XRay Tomography | 7 |
13 Examples of Inverse Problems in Physics | 10 |
14 Inverse Problems in Signal and Image Processing | 12 |
15 Inverse Problems in Heat Conduction | 18 |
16 Parameter Identification | 23 |
17 Inverse Scattering | 25 |
7 The Conjugate Gradient Method | 177 |
72 Stability and Convergence | 181 |
73 The Discrepancy Principle | 186 |
74 The Number of Iterations | 191 |
8 Regularization With Differential Operators | 197 |
82 Regularization with Seminorms | 202 |
83 Examples | 207 |
84 Hilbert Scales | 210 |
2 IllPosed Linear Operator Equations | 31 |
21 The MoorePenrose Generalized Inverse | 32 |
Singular Value Expansion | 36 |
23 Spectral Theory and Functional Calculus | 42 |
3 Regularization Operators | 49 |
32 Order Optimality | 55 |
33 Regularization by Projection | 63 |
4 Continuous Regularization Methods | 71 |
42 Saturation and Converse Results | 80 |
43 The Discrepancy Principle | 83 |
44 Improved Aposteriori Rules | 89 |
45 Heuristic Parameter Choice Rules | 100 |
46 Mollifier Methods | 112 |
5 Tikhonov Regularization | 117 |
52 Regularization with Projection | 126 |
53 Maximum Entropy Regularization | 134 |
54 Convex Constraints | 140 |
6 Iterative Regularization Methods | 154 |
62 Accelerated Landweber Methods | 160 |
63 The vMethods | 166 |
85 Regularization in Hilbert Scales | 215 |
9 Numerical Realization | 221 |
92 Reduction to Standard Form | 224 |
93 Implementation of Tikhonov Regularization | 228 |
94 Updating the Regularization Parameter | 233 |
95 Implementation of Iterative Methods | 237 |
10 Tikhonov Regularization of Nonlinear Problems | 241 |
102 Convergence Analysis | 243 |
103 Aposteriori Parameter Choice Rules | 249 |
104 Regularization in Hilbert Scales | 253 |
105 Applications | 256 |
106 Convergence of Maximum Entropy Regularization | 262 |
11 Iterative Methods for Nonlinear Problems | 277 |
112 Newton Type Methods | 285 |
A Appendix | 289 |
A2 Orthogonal Polynomials | 291 |
A3 Christoffel Functions | 295 |
Bibliography | 299 |
319 | |
Diğer baskılar - Tümünü görüntüle
Regularization of Inverse Problems Heinz Werner Engl,Martin Hanke,Günther Neubauer Önizleme Yok - 1996 |
Sık kullanılan terimler ve kelime öbekleri
Algorithm applied approximation approximation error assume bounded CGNE compact operator compute condition conjugate gradient conjugate gradient method consider continuous convergence rate convex corresponding data error defined definition denote differentiation discrepancy principle estimate finite-dimensional follows function H. W. ENGL hence Hilbert scales holds ill-posed problems implies inequality integral equation Inverse Problems iterative methods L-curve L²(N Landweber iteration least-squares solution Lemma linear operator Math matrix maximum entropy minimizer Moore-Penrose generalized inverse nonlinear problems Note nullspace numerical obtain optimal order order-optimal orthogonal projector parameter choice rule proof of Theorem Proposition regularization method regularization parameter regularized solutions residual polynomials respect right-hand side Section selfadjoint sequence singular value solving spectral family spectral theory stopping rule subspace T¹y Tikhonov regularization unique v-method weakly yields zero θα