Accelerated Newton-Landweber Iterations for Regularizing Nonlinear Inverse Problems

In this paper, we investigate the convergence behaviour of a class of regularized Newton methods for the solution of nonlinear inverse problems. In order to keep the overall numerical effort as small as possible, we propose to solve the linearized equations by certain semiiterative regularization methods, in particular, iterations with optimal speed of convergence. Our convergence rate analysis of this class of accelerated Newton-Landweber methods contributes to the analysis of Newton-type regularization methods in two ways: first, we show that under standard assumptions, accelerated Newton-Landweber iterations yield optimal convergence rates under appropriate a priori stopping criteria. Secondly, we prove inproved convergence rates for μ > 1/2 under an adequate a posteriori stopping rule, thus extending existing results. Our theory naturally applies to a wider class of Newton-type regularization methods. We conclude with several examples and numerical tests confirming the theoretical results, including a comparison to the Gauß-Newton method and the NewtonLandweber iteration.