Schuster / Kaltenbacher / Hofmann Regularization Methods in Banach Spaces

1. Auflage 2012
ISBN: 978-3-11-025572-0
Verlag: De Gruyter
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 10, 294 Seiten

Reihe: Radon Series on Computational and Applied MathematicsISSN

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

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.