Linear convergence rates for Tikhonov regularization with positively homogeneous functionals

Abstract

The goal of this paper is the formulation of an abstract setting that can be used for the derivation of linear convergence rates for a large class of sparsity promoting regularization functionals for the solution of ill-posed linear operator equations. Examples where the proposed setting applies include joint sparsity and group sparsity, but also (possibly higher order) discrete total variation regularization. In all of these cases, a range condition together with some kind of restricted injectivity imply linear convergence rates. That is, the error in the approximate solution, measured in terms of the regularization functional, is of the same order as the noise level.