Dynamical Systems Method for solving ill-conditioned linear algebraic systems

TL;DR

This paper introduces a new iterative scheme based on the Dynamical Systems Method (DSM) for effectively solving ill-conditioned linear algebraic systems, with justified stopping rules and proven convergence.

Contribution

It proposes a novel iterative scheme for ICLAS based on DSM, including new a posteriori stopping rules and convergence proofs.

Findings

The iterative scheme converges to the true solution.

A posteriori stopping rules are effective for the proposed method.

The method is applicable to a wide class of ill-posed problems.

Abstract

A new method, the Dynamical Systems Method (DSM), justified recently, is applied to solving ill-conditioned linear algebraic system (ICLAS). The DSM gives a new approach to solving a wide class of ill-posed problems. In this paper a new iterative scheme for solving ICLAS is proposed. This iterative scheme is based on the DSM solution. An a posteriori stopping rules for the proposed method is justified. This paper also gives an a posteriori stopping rule for a modified iterative scheme developed in A.G.Ramm, JMAA,330 (2007),1338-1346, and proves convergence of the solution obtained by the iterative scheme.