Solving Severely Ill-Posed Linear Systems with Time Discretization Based Iterative Regularization Methods

Abstract: Recently, inverse problems have attracted more and more attention in computational mathematics and become increasingly important in engineering applications. After the discretization, many of inverse problems are reduced to linear systems. Due to the typical ill-posedness of inverse problems, the reduced linear systems are often ill-posed, especially when their scales are large. This brings great computational difficulty. Particularly, a small perturbation in the right side of an ill-posed linear system may cause a dramatical change in the solution. Therefore, regularization methods should be adopted for stable solutions. In this paper, a new class of accelerated iterative regularization methods is applied to solve this kind of large-scale ill-posed linear systems. An iterative scheme becomes a regularization method only when the iteration is early terminated. And a Morozov's discrepancy principle is applied for the stop criterion. Compared with the conventional Landweber iteration, the new methods have acceleration effect, and can be compared to the well-known accelerated v-method and Nesterov method. From the numerical results, it is observed that using appropriate discretization schemes, the proposed methods even have better behavior when comparing with v-method and Nesterov method.