A New Preconditioner that Exploits Low-Rank Approximations to Factorization Error

Nicholas J. Higham (1) , Théo Mary (1)

1 School of Mathematics - Department of Mathematics [Manchester]

We consider ill-conditioned linear systems Ax = b that are to be solved iteratively, and assume that a low accuracy LU factorization A ≈ L U 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 = U −1 L −1 A−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 floating-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.

Mots clés: matrix factorization; preconditioning; low-rank approximations; mixed precision iterative refinement; GMRES

Domaines: Analyse numérique [math.NA]

Nicholas J. Higham, Théo Mary. A New Preconditioner that Exploits Low-Rank Approximations to Factorization Error. SIAM Journal on Scientific Computing, 2019, 41 (1), pp.A59-A82. ⟨10.1137/18M1182802⟩. ⟨hal-02311256⟩