Convergence rates for Morozov's discrepancy principle using variational inequalities

Abstract

We derive convergence rates for Tikhonov-type regularization with convex penalty terms, where the regularization parameter is chosen according to Morozov's discrepancy principle and variational inequalities are used to generalize classical source and nonlinearity conditions. Rates are obtained first with respect to the Bregman distance and a Taylor-type distance and those results are combined to derive rates in the norm and the penalty term topology. For the special case of the sparsity promoting weighted ℓp-norms as penalty terms and for a searched-for solution, which is known to be sparse, the above results give convergence rates of up to linear order.