Regularization methods in Banach spaces

Cover......Page 1
Title......Page 4
Copyright......Page 5
Dedications......Page 6
Preface......Page 8
Contents......Page 10
I Why to use Banach spaces in regularization theory?......Page 14
1.1 X-ray diffractometry......Page 17
1.2 Two phase retrieval problems......Page 19
1.3 A parameter identification problem for an elliptic partial differential equation......Page 22
1.4 An inverse problem from finance......Page 26
1.5 Sparsity constraints......Page 31
II Geometry and mathematical tools of Banach spaces......Page 38
2.1 Basic mathematical tools......Page 41
2.2.1 The subgradient of convex functionals......Page 44
2.2.2 Duality mappings......Page 47
2.3 Geometry of Banach space norms......Page 49
2.3.1 Convexity and smoothness......Page 50
2.3.2 Bregman distance......Page 57
3.1 Operator equations and the ill-posedness phenomenon......Page 62
3.1.1 Linear problems......Page 63
3.1.2 Nonlinear problems......Page 65
3.1.3 Conditional well-posedness......Page 68
3.2 Mathematical tools in regularization theory......Page 69
3.2.1 Regularization approaches......Page 70
3.2.2 Source conditions and distance functions......Page 76
3.2.3 Variational inequalities......Page 80
3.2.4 Differences between the linear and the nonlinear case......Page 82
III Tikhonov-type regularization......Page 90
4.1.1 Existence and stability of regularized solutions......Page 94
4.1.2 Convergence of regularized solutions......Page 97
4.2 Error estimates and convergence rates......Page 102
4.2.1 Error estimates under variational inequalities......Page 103
4.2.2 Convergence rates for the Bregman distance......Page 108
4.2.3 Tikhonov regularization under convex constraints......Page 112
4.2.4 Higher rates briefly visited......Page 114
4.2.5 Rate results under conditional stability estimates......Page 116
4.2.6 A glimpse of rate results under sparsity constraints......Page 118
5.1 Source conditions......Page 121
5.2.1 A priori parameter choice......Page 126
5.2.2 Morozov’s discrepancy principle......Page 128
5.2.3 Modified discrepancy principle......Page 129
5.3 Minimization of the Tikhonov functionals......Page 135
5.3.1 Primal method......Page 136
5.3.2 Dual method......Page 148
IV Iterative regularization......Page 154
6 Linear operator equations......Page 157
6.1.1 Noise-free case......Page 159
6.1.2 Regularization properties......Page 165
6.2 Sequential subspace optimization methods......Page 170
6.2.1 Bregman projections......Page 171
6.2.2 The method for exact data (SESOP)......Page 176
6.2.3 The regularization method for noisy data (RESESOP)......Page 178
6.3 Iterative solution of split feasibility problems (SFP)......Page 190
6.3.1 Continuity of Bregman and metric projections......Page 192
6.3.2 A regularization method for the solution of SFPs......Page 196
7.1.1 Conditions on the spaces......Page 206
7.1.2 Variational inequalities......Page 207
7.1.3 Conditions on the forward operator......Page 208
7.2.1 Convergence of the Landweber iteration with the discrepancy principle......Page 212
7.2.2 Convergence rates for the iteratively regularized Landweber iteration with a priori stopping rule......Page 216
7.3 The iteratively regularized Gauss-Newton method......Page 225
7.3.1 Convergence with a priori parameter choice......Page 228
7.3.2 Convergence with a posteriori parameter choice......Page 238
7.3.3 Numerical illustration......Page 243
V The method of approximate inverse......Page 246
8 Setting of the method......Page 249
9.1 The case X = Lp(Ω)......Page 252
9.2 The case X = C(K)......Page 257
9.3 An application to X-ray diffractometry......Page 261
10 A glimpse of semi-discrete operator equations......Page 266
Bibliography......Page 278
Index......Page 293