Regularization Methods For Nonlinear Problems Research Articles

An iterative regularization method for nonlinear problems based on Bregman projections

In this paper, we present an iterative method for the regularization of ill-posed, nonlinear problems. The approach is based on the Bregman projection onto stripes the width of which is controlled by both the noise level and the structure of the operator. In our investigations, we follow (Lorenz et al 2014 SIAM J. Imaging Sci.7 1237–62) and extend the respective method to the setting of nonlinear operators. Furthermore, we present a proof for the regularizing properties of the method.

Convergence of projected iterative regularization methods for nonlinear problems with smooth solutions

This paper is concerned with two aspects in the convergence analysis of regularization methods for nonlinear problems. Firstly, if a solution of the inverse problem (or its difference to an initial guess) is sufficiently smooth, the conditions on the nonlinearity of the forward operator become rather weak. Here we prove a convergence result for general regularized Newton methods that, among others, includes regularization by discretization. Secondly, constraints given by a closed and convex set can be realized by projection onto this set within each step. It is also shown that for this projected version of the regularized Newton method, convergence and optimal convergence rates are obtained.