GMRES using pseudoinverse for range symmetric singular systems

TL;DR

This paper investigates GMRES for large sparse range symmetric singular systems, deriving conditions for convergence, and proposes using pseudoinverse and reorthogonalization techniques to enhance robustness and convergence.

Contribution

It introduces a pseudoinverse-based approach within GMRES for singular systems and provides necessary and sufficient conditions for convergence in range symmetric cases.

Findings

Pseudoinverse improves GMRES convergence on ill-conditioned systems.

Reorthogonalization enhances the robustness of the method.

Numerical experiments confirm the effectiveness of the proposed techniques.

Abstract

Consider solving large sparse range symmetric singular linear systems $ A {\bf x}= {\bf b} $ which arise, for instance, in the discretization of convection diffusion equations with periodic boundary conditions, and partial differential equations for electromagnetic fields using the edge-based finite element method. In theory, the Generalized Minimal Residual (GMRES) method converges to the least squares solution for inconsistent systems if the coefficient matrix $A$ is range symmetric, i.e. $ {\rm R}(A)= {\rm R}(A^{ \rm T } )$, where $ {\rm R}(A)$ is the range space of $A$. We derived the necessary and sufficient conditions for GMRES to determine a least squares solution of inconsistent and consistent range symmetric systems assuming exact arithmetic except for the computation of the elements of the Hessenberg matrix. In practice, GMRES may not converge due to numerical instability. In order to improve the convergence, we propose using the pseudoinverse for the solution of the severely ill-conditioned Hessenberg systems in GMRES. Numerical experiments on inconsistent systems indicate that the method is effective and robust. Finally, we further improve the convergence of the method by reorthogonalizing the Modified Gram-Schmidt procedure.