Executive Summary : | By the term inverse problem, we mean to say that a problem belongs to a set of two problems whose formulations depend on each other. Generally, inverse problems are ill-posed in nature in the sense of no continuous dependence on the data. Therefore, regularization methods (either Variational or Iterative) are incorporated to find the stable solution of inverse problems. For regularization methods, one of the most important tasks is to deduce the speed of the convergence or convergence rates at which the approximate solutions obtained via regularization methods converge to the exact solution. The convergence rates are deduced by imposing some extra condition on the exact solution. The first aim of the proposed research is to show that for various iterative methods, convergence and convergence rates can be obtained through a solo condition known as Holder stability estimates. There is no requirement of any extra condition. The next goal is to discover novel fast and efficient iterative methods. These iterative methods must reach the solution to ill-posed inverse problems very fast as compared to other methods. Further, another important goal is to numerically simulate the iterative method and derive the numerical results for some ill-posed inverse problem. This shows the demonstration of the methods as well as their applications. One of the major missions of the proposed project is to show that the convergence analysis and convergence rates of iterative methods which can be obtained through Holder stability estimates even in the presence of noisy data. This is a challenging task as it is in contrast to the standard analysis that requires different conditions to study the convergence and convergence rates. Final task of the proposed work is to develop the convergence of various iterative methods using conditional stability estimates for solving non-smooth inverse problems. |