Low-rank correction methods for algebraic domain decomposition preconditioners

TL;DR

This paper introduces a parallel preconditioning technique for distributed sparse linear systems that combines domain decomposition with low-rank corrections, improving efficiency and robustness when used with Krylov subspace methods.

Contribution

It proposes a novel preconditioning framework that integrates low-rank corrections via Sherman-Morrison-Woodbury and Lanczos procedures within domain decomposition, enhancing solver performance.

Findings

Preconditioner is efficient and robust with Krylov methods.

Numerical experiments show competitive performance against existing methods.

The approach is suitable for symmetric sparse linear systems.

Abstract

This paper presents a parallel preconditioning method for distributed sparse linear systems, based on an approximate inverse of the original matrix, that adopts a general framework of distributed sparse matrices and exploits the domain decomposition method and low-rank corrections. The domain decomposition approach decouples the matrix and once inverted, a low-rank approximation is applied by exploiting the Sherman-Morrison-Woodbury formula, which yields two variants of the preconditioning methods. The low-rank expansion is computed by the Lanczos procedure with reorthogonalizations. Numerical experiments indicate that, when combined with Krylov subspace accelerators, this preconditioner can be efficient and robust for solving symmetric sparse linear systems. Comparisons with other distributed-memory preconditioning methods are presented.