Optimal parameter choice for Tikhonov regularization in Hilbert scales

Abstract

In this paper we consider the method of Tikhonov regularization in Hilbert scales for finding solutions q^* of ill-posed linear and nonlinear operator equations A(q) = z, which consists in minimizing the functional J_α(q) = ‖A(q) - z^δ‖² + α ‖q − −q‖ ²₅ . Here z^δ are the available perturbed data with ‖z − z^δ ≤ δ, q is a suitable approximation of q^* and ‖ · ‖, is the norm in a Hilbert scale Q_s. Assuming ‖A(q) − A(q^*)‖ [ ≈ ‖q − q^*‖_−a and q^* − q ∈ Q_2γ for some a ≥ 0 and γ ≥ 0 we discuss questions of the appropriate choice of the norm ‖ · ‖, and the regularization parameter α > 0. We prove that the choice of the regularization parameter by Morozov's discrepancy principle leads to optimal convergence rates if 2γ ∈ [s, 2s+ a] and discuss convergence rate results for the case 2γ ∉[s, 2s + a].