Proxy-GMRES: Preconditioning via GMRES in Polynomial Space [journal]
SIAM Journal on Matrix Analysis and Applications - August 5, 2021
Xin Ye (postdoctoral associate), Yuanzhe Xi (postdoctoral associate), Yousef Saad (professor)
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.