Newton-type regularization methods for nonlinear inverse problems

Abstract

Inverse problems arise whenever one searches for unknown causes based on observation of their effects. Such problems are usually ill-posed in the sense that their solutions do not depend contin- uously on the data. In practical applications, one never has the exact data; instead only noisy data are available due to errors in the measurements. Thus, the development of stable methods for solving inverse problems is an important topic. In the last two decades, many methods have been developed for solving nonlinear inverse problems. Due to their straightforward implementation and fast convergence property, more and more attention has been paid on Newton-type regularization methods including the general iteratively regularized Gaus-newton methods and the inexact Newton regularization methods. The iteratively regularized Gauss-Newton method was proposed by Bakushinski for solving nonlinear inverse problems in Hilbert spaces, and the method was quickly generalized to its general form. These methods produce all the iterates in some trust regions centered around the initial guess. The regularization property was explored under either a priori or a posteriori stopping rules. We will present our recent convergence results when the discrepancy principle is used to terminate the iteration. The inexact Newton regularization methods was initiated by Hanke and then generalized by Rieder to solve nonlinear inverse problems in Hilbert spaces. In contrast to the iteratively regularized Gauss- Newton methods, such methods produce the next iterate in a trust region centered around the current iterate by regularizing local linearized equations. An approximate solution is output by a discrepancy principle. Although numerical simulation indicates that they are quite efficient, for a long time it has been an open problem whether the inexact Newton methods are order optimal. We will report our recent work and confirm that the methods indeed are order optimal. In some situations, regularization methods formulated in Hilbert space setting may not produce good results since they tend to smooth the solutions and thus destroy the special feature in the exact solution. On the other hand, many inverse problems can be more naturally formulated in Banach spaces than in Hilbert spaces. Therefore, it is necessary to develop regularization methods in the framework of Banach spaces. By making use of duality mappings and Bregman distance we will indicate how to formulate some Newton-type methods in Banach space setting and present the corresponding convergence results.