Regularization Methods in Banach Spaces (eBook)

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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.

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