Additive and Multiplicative Two-Level Spectral Preconditioning for General Linear Systems

Abstract

In this paper we introduce new preconditioning techniques for the solution of general symmetric and unsymmetric linear systems $A x = b$. These approaches borrow some ideas of the multigrid philosophy designed for the solution of linear systems arising from the discretization of elliptic partial differential equations. We attempt to improve the convergence rate of a prescribed preconditioner $M_1$. In a two-grid framework, this preconditioner is viewed as a smoother and the coarse space is spanned by the eigenvectors associated with the smallest eigenvalues of $M_1A$. We derive both additive and multiplicative variants of the resulting iterated two-level preconditioners for unsymmetric linear systems that can also be adapted for Hermitian positive definite problems. We show that these two-level preconditioners shift the smallest eigenvalues to one and tend to better cluster around one those eigenvalues that $M_1$ already succeeded in moving into the neighborhood of one. We illustrate the behavior of our method through extensive numerical experiments on a set of general linear systems. Finally, we show the effectiveness of these approaches on two challenging real applications; the first comes from a nonoverlapping domain decomposition method in semiconductor device modeling, the second from industrial electromagnetism applications.