ebook ebooks e-book e-books downloaden bei MyEbooks.ch downloaden

Regularization Methods in Banach Spaces

:	Thomas Schuster, Barbara Kaltenbacher, Bernd Hofmann, Kamil S. Kazimierski
:	Regularization Methods in Banach Spaces
:	Walter de Gruyter GmbH& Co.KG
:	9783110255720
:	Radon Series on Computational and Applied MathematicsISSN
:	1
:	CHF 143.40
:

:	Analysis
:	English

:	294
:	Wasserzeichen/DRM
:	PC/MAC/eReader/Tablet
:	PDF

< >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 generalL^p 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.

< r />
< >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