On enhanced convergence rates for Tikhonov regularization of nonlinear ill-posed problems in Banach spaces

Abstract

It is well known for the Tikhonov regularization of ill-posed problems in Hilbert spaces that the best possible convergence rate is O(α) in the noise-free case and in the noisy case if the exact solution x† satisfies the source condition . We will show under what conditions the best possible rates with respect to α may be obtained for nonlinear ill-posed problems in Banach spaces. Moreover, we present convergence rate results with respect to the Bregman distance and show special cases where these rates are best possible.