GMRES using pseudoinverse for range symmetric singular systems

Abstract

Consider solving large sparse range symmetric singular linear systems Ax = 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. R(A) = R(A T ), where R(A) is the range space of A.However, in practice, GMRES may not converge due to numerical instability. In order to improve the convergence, we propose using the pseudo-inverse for the solution of the severely ill-conditioned Hessenberg systems in GMRES. Numerical experiments on semi-definite inconsistent systems indicate that the method is efficient and robust. Finally, we further improve the convergence of the method, by reorthogonalizing the Modified Gram-Schmidt procedure.