Error estimates for tikhonov regularization in hilbert scales

Abstract

In this paper we consider the method of Tikhonov regularization for finding solutions q^* of ill-posed linear and nonlinear operator equations A(q) = z, which consists in minimizing the functional Here z ^ρ are the available perturbed data and is the norm in a Hilbert scale Q _s. Assuming for some a ≥0 and γ≥0 we prove that the choice of the regularization parameter α by Morozov's discrepancy principle leads to optimal convergence rates if . Furthermore we discuss convergence rate results for the case and apply our results to a special integral equation of the first kind.