A Study on Ill-Posed Equations and Iterative Methods

Abstract

Many problems that arise in various fields of study can be modeled into equations that are well-posed/ill-posed (linear or nonlinear). Especially in science and engineer ing, most of the inverse problems are ill-posed. The first half of the thesis focuses on f inite dimensional realization of regularization methods for ill-posed problems. The second half deals with iterative methods for solving well-posed nonlinear equations. It is proved in the literature that the Fractional Tikhonov regularization method (FTR) reduces the over smoothing of the solution compared to the usual Tikhonov reg ularization method for ill-posed problems. In Chapter 2 of the thesis, the FTR method in the finite dimensional setting is studied. The regularization parameter is chosen using Raus and Gfrerer type discrepancy principle in this Chapter. The choice of regularization parameter and suitable source condition plays an im portant role in a regularization method. In Chapter 3, an efficient new parameter choice strategy is introduced. The advantage is that this parameter choice strategy computes the regularization parameter before computing the approximate solution and is depen dent on the given data of the problem. This new parameter choice also provide the optimal order. The proposed parameter choice strategy is depending on a new source condition. Higher order iterative methods are used to solve nonlinear equations. The conver gence order of these methods uses Taylor’s expansion and assumptions on the higher order Fréchet derivative of the operator. In Chapter 4 and Chapter 5, we have elimi nated the use of Taylor’s expansion and hence assumptions on the higher order Fréchet derivatives of the operator in the problem. Moreover, the desired convergence order of the iterative method is obtained without using assumptions on the higher order Fréchet derivatives and hence the applicability of these iterative methods are extended to prob lem which were not posible using earlier studies. These iterative methods are also applied to solve nonlinear ill-posed problems.