Regularization Methods for Nonlinear Ill-Posed Hammerstein Type Operator Equations

Abstract

This thesis is devoted for obtaining a stable approximate solution for nonlinear ill-posed Hammerstein type operator equations KF (x) = f. Here K : X → Y is a bounded linear operator, F : X → X is a non-linear operator, X and Y are Hilbert spaces. It is assumed throughout that the available data is fδ with kf − fδk ≤ δ. Many problems from computational sciences and other disciplines can be brought in a form similar to equation KF (x) = y using mathematical modelling (Engl et al. (1990), Scherzer, Engl and Anderssen (1993), Scherzer (1989)). The solutions of these equations can rarely be found in closed form. That is why most solution methods for these equations are iterative. The study about convergence matter of iterative procedures is usually based on two types: semi-local and local convergence analysis. The semi-local convergence matter is, based on the information around an initial point, to give conditions ensuring the convergence of the iterative procedure; while the local one is, based on the information around a solution, to find estimates of the radii of convergence balls. We aim at approximately solving the non-linear ill-posed Hammerstein type operator equations KF (x) = f using a combination of Tikhonov regularization with Newton-type Method in Hilbert spaces and in Hilbert Scales. Also we consider a combination of Tikhonov regularization with Dynamical System Method in Hilbert spaces. Precisely in the methods discussed in this thesis we considered two cases of the operator F : in the first case it is assumed that F ′(.)−1 exist (F ′(.) denotes the Fre´chet derivative of F ) and in the second case it is assumed that F ′(.)−1 does not exist but F is a monotone operator. The choice of regularization parameter plays an important role in the convergence of regularization method. We use the adaptive scheme suggested by Pereverzev and Schock (2005) for the selection of regularization parameter. The error bounds obtained are of optimal order with respect to a general source condition. Algorithms to implement the method is suggested and the computational results provided endorse the reliability and effectiveness of our methods.