Regularization Methods in Banach Spaces
Description
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.
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Content
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]
