Regularization Methods in Banach Spaces

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