An analysis of Tikhonov regularization for nonlinear ill-posed problems under a general smoothness assumption

Abstract

In this paper we introduce an adaptive regularization scheme based on algorithms for minimization of the Tikhonov functional to reconstruct the solution x† of nonlinear ill-posed problem F(x) = y, where the right-hand side is replaced by noisy data yδ ∊ Y with ‖y − yδ‖ ⩽ δ, and F : D(F) ⊂ X → Y is a nonlinear operator between Hilbert spaces X and Y. Under the assumption that Fréchet derivative F′ is Lipschitz continuous, a choice of the regularization parameter and the stopping criteria for minimization algorithms are presented. We prove that under a general source condition given in terms of a nonlinear operator F the error ‖x† − xk‖ between the regularized approximation xk and the solution x† is of optimal order.