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

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 a posteriori stopping rule that yields convergence of the iterates to a solution, 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 convergence rates. Moreover, we present an application of the Newton type methods under consideration to a parameter identification problem, together with some numerical results.