Modern regularization methods for inverse problems

Variable-smoothing Regularization Methods for Inverse Problems

Many inverse problems of practical interest are ill-posed in the sense that solutions do not depend continuously on data. To eeectively solve such problems, regularization methods are typically used. One problem associated with classical regularization methods is that the solution may be oversmoothed in the process. We present an alternative \local regularization" approach in which a decomposit...

Iterated Regularization Methods for Solving Inverse Problems

Newton-type regularization methods for nonlinear inverse problems

Inverse problems arise whenever one searches for unknown causes based on observation of their effects. Such problems are usually ill-posed in the sense that their solutions do not depend continuously on the data. In practical applications, one never has the exact data; instead only noisy data are available due to errors in the measurements. Thus, the development of stable methods for solving in...

Saturation spaces for regularization methods in inverse problems

The aim of this article is to characterize the saturation spaces that appear in inverse problems. Such spaces are defined for a regularization method and the rate of convergence of the estimation part of the inverse problem depends on their definition. Here we prove that it is possible to define these spaces as regularity spaces, independent of the choice of the approximation method. Moreover, ...

Global Saturation of Regularization Methods for Inverse Ill-Posed Problems

In this article the concept of saturation of an arbitrary regularization method is formalized based upon the original idea of saturation for spectral regularization methods introduced by Neubauer [5]. Necessary and sufficient conditions for a regularization method to have global saturation are provided. It is shown that for a method to have global saturation the total error must be optimal in t...