Simultaneous preconditioning and symmetrization of non-symmetric linear systems

TL;DR

This paper introduces a novel approach that simultaneously preconditions and symmetrizes large non-symmetric linear systems, enhancing the efficiency of iterative methods especially for ill-conditioned problems.

Contribution

The paper proposes a new scheme that combines preconditioning and symmetrization, improving the solution process for non-symmetric linear systems.

Findings

Effective for ill-conditioned, highly non-symmetric systems

Integrates with existing iterative methods

Shows improved convergence in experiments

Abstract

Motivated by the theory of self-duality which provides a variational formulation and resolution for non self-adjoint partial differential equations \cite{G1, G2}, we propose new templates for solving large non-symmetric linear systems. The method consists of combining a new scheme that simultaneously preconditions and symmetrizes the problem, with various well known iterative methods for solving linear and symmetric problems. The approach seems to be efficient when dealing with certain ill-conditioned, and highly non-symmetric systems.