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

TL;DR

This paper introduces accelerated iterative regularization methods for large-scale ill-posed linear systems resulting from inverse problems, demonstrating improved stability and efficiency over traditional methods through numerical experiments.

Contribution

The paper proposes a new class of accelerated iterative regularization methods with early stopping and discrepancy principle, outperforming conventional techniques like Landweber, v-method, and Nesterov in stability and speed.

Findings

Accelerated methods show better stability in numerical experiments.

Early termination with discrepancy principle effectively regularizes solutions.

Proposed methods outperform traditional regularization techniques.

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.