Iteratively regularized Gauss-Newton method for nonlinear inverse problems with random noise

Abstract

We study the convergence of regularized Newton methods applied to nonlinear operator equations in Hilbert spaces if the data are perturbed by random noise. It is shown that the expected square error is bounded by a constant times the minimax rates of the corresponding linearized problem if the stopping index is chosen using a priori knowledge of the smoothness of the solution. For unknown smoothness the stopping index can be chosen adaptively based on Lepskii's balancing principle. For this stopping rule we establish an oracle inequality, which implies order optimal rates for deterministic errors, and optimal rates up to a logarithmic factor for random noise. The performance and the statistical properties of the proposed method are illustrated by Monte Carlo simulations.