Newton Type Methods for Lavrentiev Regularization of Nonlinear Ill-Posed Operator Equations

Abstract

In this thesis we consider nonlinear ill-posed operator equations of the form F(x) = f; that arise from the study of nonlinear inverse problems, where F : X ! X is a nonlinear monotone operator defined on a real Hilbert space X: In applications, instead of f; usually only noisy data fδ are available. Then the problem of recovery of the exact solution ^ x from noisy equation F(x) = fδ is ill-posed, in the sense that a small perturbation in the data can cause large deviation in the solution. Thus the computation of a stable approximation for ^ x from the solution of F(x) = fδ; becomes an important issue in ill-posed problems, and the regularization techniques have to be taken into account. Approximation methods are an attractive choice since they are straightforward to implement, for getting the numerical solution of nonlinear ill-posed problems. Thus in the last few years more emphasis was put on the investigation of iterative regularization methods. We consider Newton type iterative regularization methods and their finite dimensional realizations, for obtaining approximation for ^ x in the Hilbert space and Hilbert scales settings. We use the adaptive scheme of Pereverzyev and Schock (2005), for choosing the regularization parameter.