Recent Developments in Inverse Problems - Abstract

Hofmann, Bernd

On $ell^1$-regularization in light of Nashed's ill-posedness concept

Based on the powerful tool of variational inequalities, in recent papers convergence rates results on $ell^1$-regularization for ill-posed inverse problems have been formulated in infinite dimensional spaces under the condition that the sparsity assumption slightly fails, but the solution is still in $ell^1$. In this talk, there are presented some new ideas including the improvement of convergence rates results and their extension to non-compact linear forward operators with closed and non-closed range. The results are also applied to the Cesáro operator equation in $ell^2$ and to specific denoising problems. Moreover, one part of the talk is devoted to the relationships between Nashed's types of ill-posedness and mapping properties of the forward operator like compactness and strict singularity. This is joint work with Jens Flemming and Ivan Veselić (TU Chemnitz). The research is partially supported by the Deutsche Forschungsgemeinschaft (DFG) under grant HO1454/8-2.