Iterative regularization techniques for nonlinear ill posed operator equations with and without Hilbert scales

Abstract

newline Abstract newline An inverse problem focuses on estimating data that cannot be obtained through direct newline measurement. Such phenomenon occurs commonly in science and engineering fields newline such as ultrasonic imaging, electrical impedance tomography, inverse scattering prob newlinelems, inverse source problems, microwave tomography, and parameter identification newline problems in partial differential equations. A key characteristic of inverse problems is newline their ill-posedness, meaning their solutions do not depend continuously on the data. To newline address this issue, regularization techniques are employed to produce stable approxi newlinemate solutions. This thesis explores ill-posed operator equations represented by newline F(u) = y newline (1) newline where F : U YdenotesanonlinearoperatordefinedbetweenHilbertspaces UandY. newline In the literature on ill-posed operator equations, various regularization techniques have newline been developed for obtaining stable regularized solutions. The main goal of this thesis newline is to examine the convergence behavior and obtain the convergence rates of some itera newlinetive regularization methods for nonlinear ill-posed operator equations with and without newline Hilbert scale framework. This study explores modified variants of existing iterative newline methods, as discussed in the existing literature, on nonlinear ill-posed operator equa newlinetions within these setting. Work done in the thesis will highlight advantages of these newline modified methods considered in the various chapters of the thesis. newline The first chapter of the thesis is introductory in nature. The chapter begins with intro newlineducing ill-posed operator equations which follows by examples of linear and nonlinear newline ill-posed operator equations. The chapter also includes review of fundamental defini newlinetions, results, and concepts from Functional Analysis and Operator Theory which will newline be used throughout the thesis. newline ix newlinex newline In the second chapter of the thesis, we consider a modified version of the simplified newline Landweber iterative regularization method considered by Jose and Rajan (2017). In the newline modified vari