Preconditioned normal equations for solving discretised partial differential equations

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 associatednormal'' PDE is presented. Numerical experiments on convection-diffusion problems demonstrate the effectiveness of this approach in achieving fast and stable convergence.