Error estimates for non-quadratic regularization and the relation to enhancement

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Abstract
In this paper error estimates for non-quadratic regularization of non- linear ill-posed problems in Banach spaces are derived. Our analysis is based on a few novel features: In comparison with the classical analy- sis of regularization methods for inverse and ill-posed problems where a Lipschitz continuity for the Frechet-derivative is required, we use a dier- entiability condition with respect to the Bregman distance. Also, a sta- bility result for the regularized solutions in terms of Bregman distances is proven. Moreover, a source wise representation of the solution as used in standard theory is interpreted in terms of data enhancement. It is also shown that total variation Bregman distance regularization for image analysis, as developed recently, can be considered a two step regulariza- tion method consisting of a combination of total variation regularization and additional enhancement.

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