2. M.Tech Research Reports

Permanent URI for this collectionhttps://idr.nitk.ac.in/handle/123456789/16917

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Author: search.filters.author.George, SanthoshStart date: 2023

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    Iterative Methods and their Applications for Solving Non-Linear Ill-Posed Equations
    (National Institute of Technology Karnataka, Surathkal., 2023) K., Muhammed Saeed; George, Santhosh; P., Jidesh
    This thesis deals with iterative methods and their convergence for solving non-linear equations in Banach Spaces. As an application, it also deals with solving non-linear ill posed equations in a Hilbert space setting. Under various assumptions, local and semi local convergence analyses of some iterative schemes are studied. We have established the desired order of convergence using weaker assumptions than those available in the literature. We have also extended some of the methods efficiently. Computable radii of convergence and dynamics analysis using the basin of attractions are other highlights. The first contribution of the thesis is the convergence analysis of a fifth-order it erative method using conditions only on the first Fréchet derivative. This increased the applicability of the method. In our second work, we used the iterative method for solving the regularized equation corresponding to a non-linear ill-posed equation. We introduced a new source condition and parameter choice strategy for the desired results. Thirdly, using Lipschitz-type assumptions on first and second derivatives instead of Taylor series expansion, we established third-order convergence of an iterative Home ier method. We further extended this method to the fifth and sixth order. Lastly, we studied another iterative method introduced by Traub. We established third-order con vergence without using Taylor series expansion. We extended this method to the fifth and sixth order.
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    A Study on Ill-Posed Equations and Iterative Methods
    (National Institute of Technology Karnataka, Surathkal., 2024) R, Krishnendu; George, Santhosh; P, Jidesh
    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.