A Projection-Regularized Newton Method for Nonlinear Ill-Posed Problems and Its Application to Parameter Identification Problems with Finite Element Discretization

Abstract

This paper is concerned with nonlinear ill-posed operator equations F(a)=y (e.g., parameter identification problems) and their approximate solution by a Newton-type method that is regularized by projecting the linearized equation in each Newton step onto a finite-dimensional space (e.g., a finite element space) and by stopping the Newton iteration at an appropriate index. We prove convergence as the iteration index n goes to infinity in the noise-free case and convergence as the data noise level $\delta$ goes to zero in the case of noisy data, as well as convergence rates under certain additional conditions.