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Stephan W. Anzengruber , Bernd Hofmann, Peter Mathé: Regularization properties of the discrepancy principle for Tikhonov regularization in Banach spaces

Stephan W. Anzengruber , Bernd Hofmann, Peter Mathé: Regularization properties of the discrepancy principle for Tikhonov regularization in Banach spaces

Author(s):
Stephan W. Anzengruber
Bernd Hofmann
Peter Mathé
Title:
Stephan W. Anzengruber , Bernd Hofmann, Peter Mathé: Regularization properties of the discrepancy principle for Tikhonov regularization in Banach spaces
Electronic source:
application/pdf
Preprint series:
Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 12, 2012
Mathematics Subject Classification:
65J20 []
47J06 []
47A52 []
49J40 []
Abstract:
The stable solution of ill-posed non-linear operator equations in Banach space requires regularization. One important approach is based on Tikhonov regularization, in which case a one-parameter family of regularized solutions is obtained. It is crucial to choose the parameter appropriately. Here, a variant of the discrepancy principle is analyzed. In many cases such parameter choice exhibits the feature, called regularization property below, that the chosen parameter tends to zero as the noise tends to zero, but slower than the noise level. Here we shall show such regularization property under two natural assumptions. First, exact penalization must be excluded, and secondly, the discrepancy principle must stop after a finite number of iterations. We conclude this study with a discussion of some consequences for convergence rates obtained by the discrepancy principle under the validity of some kind of variational inequality, a recent tool for the analysis of inverse problems.
Keywords:
Inverse problems, Tikhonov-type regularization, discrepancy principle, parameter choice properties
Language:
English
Publication time:
11/2012