Dynamical systems gradient method for solving ill-conditioned linear algebraic systems

TL;DR

This paper introduces a dynamical systems approach for solving ill-conditioned linear systems, providing stopping rules and an algorithm that can serve as an alternative to variational regularization, especially when spectral decomposition is feasible.

Contribution

The paper develops a DSM-based method with justified stopping rules and an algorithm utilizing spectral decomposition, offering an alternative to existing regularization techniques.

Findings

Numerical results demonstrate effectiveness of the method.

The method is advantageous when spectral decomposition is available or inexpensive.

It provides an alternative to variational regularization for ill-conditioned systems.

Abstract

A version of the Dynamical Systems Method (DSM) for solving ill-conditioned linear algebraic systems is studied in this paper. An {\it a priori} and {\it a posteriori} stopping rules are justified. An algorithm for computing the solution using a spectral decomposition of the left-hand side matrix is proposed. Numerical results show that when a spectral decompositon of the left-hand side matrix is available or not computationally expensive to obtain the new method can be considered as an alternative to the Variational Regularization.