A posteriori parameter choice strategies for some Newton type methods for the regularization of nonlinearill-posed problems

Abstract

This paper treats a class of Newton type methods for the approximate solution of nonlinear ill-posed operator equations, that use so-called filter functions for regularizing the linearized equation in each Newton step. For noisy data we derive an aposteriori stopping rule that yields convergence of the iterates to asolution, as the noise level goes to zero, under certain smoothness conditions on the nonlinear operator. Appropriate closeness and smoothness assumptions on the starting value and the solution are shown to lead to optimal convergence rates. Moreover we present an application of the Newton type methods under consideration to a parameter identification problem, together with some numerical results.