Analysis of profile functions for general linear regularization methods

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Date
2006
Volume
1107
Issue
Journal
Series Titel
WIAS Preprints
Book Title
Publisher
Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
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Abstract

The stable approximate solution of ill-posed linear operator equations in Hilbert spaces requires regularization. Tight bounds for the noise-free part of the regularization error are constitutive for bounding the overall error. Norm bounds of the noise-free part which decrease to zero along with the regularization parameter are called profile functions and are subject of our analysis. The interplay between properties of the regularization and certain smoothness properties of solution sets, which we shall describe in terms of source-wise representations is crucial for the decay of associated profile functions. On the one hand, we show that a given decay rate is possible only if the underlying true solution has appropriate smoothness. On the other hand, if smoothness fits the regularization, then decay rates are easily obtained. If smoothness does not fit, then we will measure this in terms of some distance function. Tight bounds for these allow us to obtain profile functions. Finally we study the most realistic case when smoothness is measured with respect to some operator which is related to the one governing the original equation only through a link condition. In many parts the analysis is done on geometric basis, extending classical concepts of linear regularization theory in Hilbert spaces ...

Description
Keywords
linear ill-posed problems, regularization, distance function, convergence rates, index function, source condition, qualification, range inclusion
Citation
Mathé, P., & Hofmann, B. (2006). Analysis of profile functions for general linear regularization methods (Vol. 1107). Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik.
License
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