A general convergence analysis of some Newton-type methods for nonlinear inverse problems

Abstract

We consider the methods x(n+1)(delta) - x(n)(delta) - g(alpha n) (F'(x(n)(delta))* F'(x(n)(delta)))F'(x(n)(delta))*(F(x(n)(delta)) - y(delta)) for solving nonlinear ill-posed inverse problems F(x) = y using the only available noise data y(delta) satisfying parallel to y(delta) - y parallel to <= delta with a given small noise level delta > 0. We terminate the iteration by the discrepancy principle parallel to F(x(n delta)(delta))-y(delta)parallel to <= tau delta < parallel to F(x(n)(delta))-y(delta)parallel to, 0 <= n < n(delta), with a given number tau > 1. Under certain conditions on {alpha(n)} and F, we prove for a large class of spectral filter functions {g(alpha)} the convergence of x(n delta)(delta) to a true solution as delta -> 0. Moreover, we derive the order optimal rates of convergence when certain Holder source conditions hold. Numerical examples are given to test the theoretical results.