Iterative regularization for ill-posed operator equations in Hilbert scales

Abstract

In this paper we report on a method for regularizing a nonlinear ill-posed operator equation in Hilbert scales. The proposed  method is a combination of Lavrentiev regularization method and a Modified Newton's method in Hilbert scales . Under the assumptions that the operator F is continuously differentiable with a Lipschitz-continuous first derivative and that the solution of (\ref{eq:1}) fulfils a general source condition, we  give an optimal order convergence rate result with respect to the general source function.