Two-level Nystr\"om--Schur preconditioner for sparse symmetric positive definite matrices

TL;DR

This paper introduces a novel two-level preconditioner for large sparse SPD matrices using Nyström's method, improving efficiency and robustness in solving large linear systems.

Contribution

It develops a new Nyström--Schur preconditioner that leverages randomized low-rank approximations for enhanced performance in large-scale sparse systems.

Findings

Preconditioner effectively reduces iteration counts.

Inner Schur complement system can be solved efficiently.

Large convergence tolerances do not compromise preconditioner quality.

Abstract

Randomized methods are becoming increasingly popular in numerical linear algebra. However, few attempts have been made to use them in developing preconditioners. Our interest lies in solving large-scale sparse symmetric positive definite linear systems of equations where the system matrix is preordered to doubly bordered block diagonal form (for example, using a nested dissection ordering). We investigate the use of randomized methods to construct high quality preconditioners. In particular, we propose a new and efficient approach that employs Nystr\"om's method for computing low rank approximations to develop robust algebraic two-level preconditioners. Construction of the new preconditioners involves iteratively solving a smaller but denser symmetric positive definite Schur complement system with multiple right-hand sides. Numerical experiments on problems coming from a range of application areas demonstrate that this inner system can be solved cheaply using block conjugate gradients and that using a large convergence tolerance to limit the cost does not adversely affect the quality of the resulting Nystr\"om--Schur two-level preconditioner.