A new preconditioner that exploits low-rank approximations to factorization error

Abstract

We consider ill-conditioned linear systems Ax = b that are to be solved iteratively, and assume that a low accuracy LU factorization A ≈ LÛ is available for use in a preconditioner. We have observed that for ill-conditioned matrices A arising in practice, A-1 tends to be numerically low rank, that is, it has a small number of large singular values. Importantly, the error matrix E = Û-1L -1A-I tends to have the same property. To understand this phenomenon we give bounds for the distance from E to a low-rank matrix in terms of the corresponding distance for A-1. We then design a novel preconditioner that exploits the low-rank property of the error to accelerate the convergence of iterative methods. We apply this new preconditioner in three different contexts fitting our general framework: low oating-point precision (e.g., half precision) LU factorization, incomplete LU factorization, and block low-rank LU factorization. In numerical experiments with GMRES-based iterative refinement we show that our preconditioner can achieve a significant reduction in the number of iterations required to solve a variety of real-life problems.