Schuster / Kaltenbacher / Hofmann Regularization Methods in Banach Spaces

1. Auflage 2012
ISBN: 978-3-11-025572-0
Verlag: De Gruyter
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, Band 10, 294 Seiten

Reihe: Radon Series on Computational and Applied Mathematics

Regularization methods aimed at finding stable approximate solutions are a necessary tool to tackle inverse and ill-posed problems. Inverse problems arise in a large variety of applications ranging from medical imaging and non-destructive testing via finance to systems biology. Many of these problems belong to the class of parameter identification problems in partial differential equations (PDEs) and thus are computationally demanding and mathematically challenging. Hence there is a substantial need for stable and efficient solvers for this kind of problems as well as for a rigorous convergence analysis of these methods.This monograph consists of five parts. Part I motivates the importance of developing and analyzing regularization methods in Banach spaces by presenting four applications which intrinsically demand for a Banach space setting and giving a brief glimpse of sparsity constraints. Part II summarizes all mathematical tools that are necessary to carry out an analysis in Banach spaces. Part III represents the current state-of-the-art concerning Tikhonov regularization in Banach spaces. Part IV about iterative regularization methods is concerned with linear operator equations and the iterative solution of nonlinear operator equations by gradient type methods and the iteratively regularized Gauß-Newton method. Part V finally outlines the method of approximate inverse which is based on the efficient evaluation of the measured data with reconstruction kernels.

Zielgruppe

Researchers, Lecturers, and Graduate Students in Mathematics; Academic Libraries

Autoren/Hrsg.

Schuster, Thomas

Kaltenbacher, Barbara

Hofmann, Bernd

Kazimierski, Kamil S.

Fachgebiete

• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen
• Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis

Inhaltsverzeichnis

1;Preface;7
2;I Why to use Banach spaces in regularization theory?;13
2.1;1 Applications with a Banach space setting;16
2.1.1;1.1 X-ray diffractometry;16
2.1.2;1.2 Two phase retrieval problems;18
2.1.3;1.3 A parameter identification problem for an elliptic partial differential equation;21
2.1.4;1.4 An inverse problem from finance;25
2.1.5;1.5 Sparsity constraints;30
3;II Geometry and mathematical tools of Banach spaces;37
3.1;2 Preliminaries and basic definitions;40
3.1.1;2.1 Basic mathematical tools;40
3.1.2;2.2 Convex analysis;43
3.1.2.1;2.2.1 The subgradient of convex functionals;43
3.1.2.2;2.2.2 Duality mappings;46
3.1.3;2.3 Geometry of Banach space norms;48
3.1.3.1;2.3.1 Convexity and smoothness;49
3.1.3.2;2.3.2 Bregman distance;56
3.2;3 Ill-posed operator equations and regularization;61
3.2.1;3.1 Operator equations and the ill-posedness phenomenon;61
3.2.1.1;3.1.1 Linear problems;62
3.2.1.2;3.1.2 Nonlinear problems;64
3.2.1.3;3.1.3 Conditional well-posedness;67
3.2.2;3.2 Mathematical tools in regularization theory;68
3.2.2.1;3.2.1 Regularization approaches;69
3.2.2.2;3.2.2 Source conditions and distance functions;75
3.2.2.3;3.2.3 Variational inequalities;79
3.2.2.4;3.2.4 Differences between the linear and the nonlinear case;81
4;III Tikhonov-type regularization;89
4.1;4 Tikhonov regularization in Banach spaces with general convex penalties;93
4.1.1;4.1 Basic properties of regularized solutions;93
4.1.1.1;4.1.1 Existence and stability of regularized solutions;93
4.1.1.2;4.1.2 Convergence of regularized solutions;96
4.1.2;4.2 Error estimates and convergence rates;101
4.1.2.1;4.2.1 Error estimates under variational inequalities;102
4.1.2.2;4.2.2 Convergence rates for the Bregman distance;107
4.1.2.3;4.2.3 Tikhonov regularization under convex constraints;111
4.1.2.4;4.2.4 Higher rates briefly visited;113
4.1.2.5;4.2.5 Rate results under conditional stability estimates;115
4.1.2.6;4.2.6 A glimpse of rate results under sparsity constraints;117
4.2;5 Tikhonov regularization of linear operators with power-type penalties;120
4.2.1;5.1 Source conditions;120
4.2.2;5.2 Choice of the regularization parameter;125
4.2.2.1;5.2.1 A priori parameter choice;125
4.2.2.2;5.2.2 Morozov’s discrepancy principle;127
4.2.2.3;5.2.3 Modified discrepancy principle;128
4.2.3;5.3 Minimization of the Tikhonov functionals;134
4.2.3.1;5.3.1 Primal method;135
4.2.3.2;5.3.2 Dual method;147
5;IV Iterative regularization;153
5.1;6 Linear operator equations;156
5.1.1;6.1 The Landweber iteration;158
5.1.1.1;6.1.1 Noise-free case;158
5.1.1.2;6.1.2 Regularization properties;164
5.1.2;6.2 Sequential subspace optimization methods;169
5.1.2.1;6.2.1 Bregman projections;170
5.1.2.2;6.2.2 The method for exact data (SESOP);175
5.1.2.3;6.2.3 The regularization method for noisy data (RESESOP);177
5.1.3;6.3 Iterative solution of split feasibility problems (SFP);189
5.1.3.1;6.3.1 Continuity of Bregman and metric projections;191
5.1.3.2;6.3.2 A regularization method for the solution of SFPs;195
5.2;7 Nonlinear operator equations;205
5.2.1;7.1 Preliminaries;205
5.2.1.1;7.1.1 Conditions on the spaces;205
5.2.1.2;7.1.2 Variational inequalities;206
5.2.1.3;7.1.3 Conditions on the forward operator;207
5.2.2;7.2 Gradient type methods;211
5.2.2.1;7.2.1 Convergence of the Landweber iteration with the discrepancy principle;211
5.2.2.2;7.2.2 Convergence rates for the iteratively regularized Landweber iteration with a priori stopping rule;215
5.2.3;7.3 The iteratively regularized Gauss-Newton method;224
5.2.3.1;7.3.1 Convergence with a priori parameter choice;227
5.2.3.2;7.3.2 Convergence with a posteriori parameter choice;237
5.2.3.3;7.3.3 Numerical illustration;242
6;V The method of approximate inverse;245
6.1;8 Setting of the method;248
6.2;9 Convergence analysis in Lp (O) and C (K);251
6.2.1;9.1 The case X = Lp(O);251
6.2.2;9.2 The case X = C (K);256
6.2.3;9.3 An application to X-ray diffractometry;260
6.3;10 A glimpse of semi-discrete operator equations;265
7;Bibliography;277
8;Index;292

Über Autor(innen)

Thomas Schuster, Carl von Ossietzky Universität Oldenburg, Germany; Barbara Kaltenbacher, University of Stuttgart, Germany; Bernd Hofmann, Chemnitz University of Technology, Germany; Kamil S. Kazimierski, University of Bremen, Germany.