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

We consider ill-conditioned linear systems $Ax =$ b that are to be solved iteratively, and assume that a low accuracy LU factorization $A \approx \widehat{L}\widehat{U}$ 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 $E = \widehat{U}^{-1}\widehat{L}^{-1}A - I$ 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.