Proxy-GMRES: Preconditioning via GMRES in Polynomial Space

Xin Ye, Yuanzhe Xi, Yousef Saad

Abstract:

This paper proposes a class of polynomial preconditioners for solving non-Hermitian
linear systems of equations. The polynomial is obtained from a least-squares approximation in polynomial space instead of a standard Krylov subspace. The process for building the polynomial relies
on an Arnoldi-like procedure in a small dimensional polynomial space and is equivalent to performing
GMRES in polynomial space. It is inexpensive and produces the desired polynomial in a numerically
stable way. A few improvements to the basic scheme are discussed including the development of a
short-term recurrence and the use of compound preconditioners. Numerical experiments, including a
test with challenging nonnormal three-dimensional Helmholtz equations and a few publicly available
sparse matrices, are provided to illustrate the performance of the proposed preconditioners.