On The Implementation of Regularization Methods for Nonlinear Ill-Posed Operator Equations

Abstract

In this thesis we consider nonlinear ill-posed operator equations of the form F(x) = y; where F : X ! Y is a nonlinear operator between Hilbert spaces X and Y: Many problems from computational sciences and other disciplines can be brought to the form F(x) = y: In practical applications, usually noisy data yδ are available instead of y: The problem of recovery of the exact solution x^ from noisy equation F(x) = yδ is ill posed, in the sense that a small perturbation in the data can cause large deviation in the solution and the solutions of these equations are usually unknown in the closed form. Thus the computation of a stable approximation for x^ from the solution of F(x) = yδ; becomes an important issue in the ill-posed problems, and most methods for solving these equations are iterative. We consider iterative regularization methods and their finite dimensional realization, for obtaining an approximation for x^ in the Hilbert space. The choice of regularization parameter plays an important role in the convergence of regularization methods. We use the adaptive scheme of Pereverzev and Schock (2005), for choosing the regularization parameter. The error bounds obtained are of optimal order with respect to a general source condition.