Sharp converse results for the regularization error using distance functions
Authors
- Flemming, Jens
- Hofmann, Bernd
- Mathé, Peter
ORCID: 0000-0002-1208-1421
2010 Mathematics Subject Classification
- 47A52 65J20
Keywords
- regularization, smoothness, distance function, distribution function
DOI
Abstract
In the analysis of ill-posed inverse problems the impact of solution smoothness on accuracy and convergence rates plays an important role. For linear ill-posed operator equations in Hilbert spaces and with focus on the linear regularization schema we will establish relations between the different kinds of measuring solution smoothness in a point-wise or integral manner. In particular we discuss the interplay of distribution functions, profile functions that express the regularization error, index functions generating source conditions, and distance functions associated with benchmark source conditions. We show that typically the distance functions and the profile functions carry the same information as the distribution functions, and that this is not the case for general source conditions. The theoretical findings are accompanied with examples exhibiting applications and limitations of the approach.
Appeared in
- Inverse Problems, 27 (2011) pp. 025006/1--025006/18.
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