# Convergence Rates via Basis Smoothness in Sparsity Regularization

Stephan Anzengruber*, Bernd Hofmann and Ronny Ramlau

Tikhonov regularization with sparsity promoting $\ell^q$ constraints ($0<q \leq 1$) has attracted considerable attention over the last decade. Its regularizing and convergence properties are well understood, but results on convergence rates often depend on the sparsity of the unknown solution and on a structural assumption linking the underlying basis to the range of the adjoint of the operator. Both of these assumptions have been considered somewhat restrictive. In this talk, we first illustrate by means of the Radon transform how the latter assumption is related to the smoothness of the basis, and then present convergence rates results that do not require sparsity of the unknown solution even in the non-convex case $q<1$.

Mathematics Subject Classification: 47A52 65J20

Keywords: Tikhonov regularization; sparsity; convergence rates;

Minisymposion: Iterative Methods for Ill-Posed Problems