Interval regularization for imprecise linear algebraic equations

TL;DR

This paper introduces an interval regularization method for solving ill-conditioned linear systems by embedding them into interval systems to enhance stability and find robust solutions.

Contribution

It presents a novel approach of interval regularization for imprecise linear systems, improving solution stability and proposing computational methods for pseudo-solution determination.

Findings

Interval regularization yields better conditioned matrices.

Enhanced stability in solutions of ill-conditioned systems.

Proposed algorithms effectively find pseudo-solutions.

Abstract

In this paper, we consider the solution of ill-conditioned systems of linear algebraic equations that can be determined imprecisely. To improve the stability of the solution process, we "immerse" the original imprecise linear system in an interval system of linear algebraic equations of the same structure and then consider its tolerable solution set. As the result, the "intervalized" matrix of the system acquires close and better conditioned matrices for which the solution of the corresponding equation system is more stable. As a pseudo-solution of the original linear equation system, we take a point from the tolerable solution set of the intervalized linear system or a point that provides the largest tolerable compatibility (consistency). We propose several computational recipes to find such pseudo-solutions.