A two-level iterative scheme for general sparse linear systems based on approximate skew-symmetrizers

Abstract

We propose a two-level iterative scheme for solving general sparse linear systems. The proposed scheme consists of a sparse preconditioner that increases the norm of the skew-symmetric part relative to the rest and makes the main diagonal of the coefficient matrix as close to the identity as possible so that the preconditioned system is as close to a shifted skew-symmetric matrix as possible. The preconditioned system is then solved via a particular Minimal Residual Method for Shifted Skew-Symmetric Systems (MRS). This leads to a two-level (inner and outer) iterative scheme where the MRS has short-term recurrences and satisfies an optimality condition. A preconditioner for the inner system is designed via a skew-symmetry-preserving deflation strategy based on the skew-Lanczos process. We demonstrate the robustness of the proposed scheme on sparse matrices from various applications.

Structured shifts for skew-symmetric matrices

“…If Q T SQ = T with Q orthogonal and T tridiagonal and skew-symmetric, then Q T (S + αI)Q = T + αI. Iterative linear solvers have been based on shifts and have been proven to be effective; see [21,22,23,29,33].…”