Format:
Online-Ressource
Edition:
1. Aufl.
Edition:
Reproduktion 2011
ISBN:
3110255723
,
3110255243
,
9781283627924
Series Statement:
Radon Series on Computational and Applied Mathematics 10
Content:
Regularization methods aimed at finding stable approximate solutions are a necessary tool to tackle inverse and ill-posed problems. Usually the mathematical model of an inverse problem consists of an operator equation of the first kind and often the associated forward operator acts between Hilbert spaces. However, for numerous problems the reasons for using a Hilbert space setting seem to be based rather on conventions than on an approprimate and realistic model choice, so often a Banach space setting would be closer to reality. Furthermore, sparsity constraints using general Lp-norms or the BV-norm have recently become very popular. Meanwhile the most well-known methods have been investigated for linear and nonlinear operator equations in Banach spaces. Motivated by these facts the authors aim at collecting and publishing these results in a monograph. 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.
Note:
Includes bibliographical references and index
,
Preface; I Why to use Banach spaces in regularization theory?; 1 Applications with a Banach space setting; 1.1 X-ray diffractometry; 1.2 Two phase retrieval problems; 1.3 A parameter identification problem for an elliptic partial differential equation; 1.4 An inverse problem from finance; 1.5 Sparsity constraints; II Geometry and mathematical tools of Banach spaces; 2 Preliminaries and basic definitions; 2.1 Basic mathematical tools; 2.2 Convex analysis; 2.2.1 The subgradient of convex functionals; 2.2.2 Duality mappings; 2.3 Geometry of Banach space norms; 2.3.1 Convexity and smoothness
,
2.3.2 Bregman distance3 Ill-posed operator equations and regularization; 3.1 Operator equations and the ill-posedness phenomenon; 3.1.1 Linear problems; 3.1.2 Nonlinear problems; 3.1.3 Conditional well-posedness; 3.2 Mathematical tools in regularization theory; 3.2.1 Regularization approaches; 3.2.2 Source conditions and distance functions; 3.2.3 Variational inequalities; 3.2.4 Differences between the linear and the nonlinear case; III Tikhonov-type regularization; 4 Tikhonov regularization in Banach spaces with general convex penalties; 4.1 Basic properties of regularized solutions
,
4.1.1 Existence and stability of regularized solutions4.1.2 Convergence of regularized solutions; 4.2 Error estimates and convergence rates; 4.2.1 Error estimates under variational inequalities; 4.2.2 Convergence rates for the Bregman distance; 4.2.3 Tikhonov regularization under convex constraints; 4.2.4 Higher rates briefly visited; 4.2.5 Rate results under conditional stability estimates; 4.2.6 A glimpse of rate results under sparsity constraints; 5 Tikhonov regularization of linear operators with power-type penalties; 5.1 Source conditions; 5.2 Choice of the regularization parameter
,
5.2.1 A priori parameter choice5.2.2 Morozov's discrepancy principle; 5.2.3 Modified discrepancy principle; 5.3 Minimization of the Tikhonov functionals; 5.3.1 Primal method; 5.3.2 Dual method; IV Iterative regularization; 6 Linear operator equations; 6.1 The Landweber iteration; 6.1.1 Noise-free case; 6.1.2 Regularization properties; 6.2 Sequential subspace optimization methods; 6.2.1 Bregman projections; 6.2.2 The method for exact data (SESOP); 6.2.3 The regularization method for noisy data (RESESOP); 6.3 Iterative solution of split feasibility problems (SFP)
,
6.3.1 Continuity of Bregman and metric projections6.3.2 A regularization method for the solution of SFPs; 7 Nonlinear operator equations; 7.1 Preliminaries; 7.1.1 Conditions on the spaces; 7.1.2 Variational inequalities; 7.1.3 Conditions on the forward operator; 7.2 Gradient type methods; 7.2.1 Convergence of the Landweber iteration with the discrepancy principle; 7.2.2 Convergence rates for the iteratively regularized Landweber iteration with a priori stopping rule; 7.3 The iteratively regularized Gauss-Newton method; 7.3.1 Convergence with a priori parameter choice
,
7.3.2 Convergence with a posteriori parameter choice
,
In English
Additional Edition:
ISBN 3110255243
Additional Edition:
ISBN 9781283627924
Additional Edition:
ISBN 9783110255249
Additional Edition:
ISBN 9783112204504
Additional Edition:
Erscheint auch als Druck-Ausgabe
Additional Edition:
Erscheint auch als Druck-Ausgabe ISBN 978-1-283-62792-4
Additional Edition:
Erscheint auch als Druck-Ausgabe ISBN 978-3-11-220450-4
Additional Edition:
Erscheint auch als Druck-Ausgabe Regularization methods in Banach spaces Berlin [u.a.] : De Gruyter, 2012 ISBN 3110255243
Additional Edition:
ISBN 9783110255249
Language:
English
Subjects:
Mathematics
Keywords:
Regularisierung
;
Banach-Raum
;
Regularisierung
;
Banach-Raum
DOI:
10.1515/9783110255720
URL:
Volltext
(lizenzpflichtig)
Author information:
Schuster, Thomas 1971-
Author information:
Hofmann, Bernd 1953-
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