Preconditioned normal equations for solving discretised partial differential equations

TL;DR

This paper introduces a novel preconditioning strategy for normal equations in PDE discretization, improving convergence speed and stability in solving non-symmetric linear systems with numerical experiments validating its effectiveness.

Contribution

It proposes the concept of 'normal' preconditioning and a method to construct preconditioners based on the associated 'normal' PDE, advancing iterative solution techniques.

Findings

Enhanced convergence speed in convection-diffusion problems

Stable solutions achieved with the proposed preconditioning

Numerical experiments confirm effectiveness

Abstract

This paper explores preconditioning the normal equation for non-symmetric square linear systems arising from PDE discretization, focusing on methods like CGNE and LSQR. The concept of ``normal'' preconditioning is introduced and a strategy to construct preconditioners studying the associated ``normal'' PDE is presented. Numerical experiments on convection-diffusion problems demonstrate the effectiveness of this approach in achieving fast and stable convergence.