Absolute value preconditioning for symmetric indefinite linear systems

TL;DR

This paper introduces a new preconditioning strategy called absolute value preconditioning for symmetric indefinite systems, demonstrating its effectiveness through a geometric multigrid example that improves robustness and efficiency.

Contribution

The paper proposes a novel absolute value preconditioning approach for symmetric indefinite matrices and provides a practical multigrid implementation that enhances iterative solver performance.

Findings

The new preconditioner converges rapidly, often in two steps.

Numerical tests show the preconditioner is robust and memory-efficient.

The approach is effective for shifted negative Laplacian problems.

Abstract

We introduce a novel strategy for constructing symmetric positive definite (SPD) preconditioners for linear systems with symmetric indefinite matrices. The strategy, called absolute value preconditioning, is motivated by the observation that the preconditioned minimal residual method with the inverse of the absolute value of the matrix as a preconditioner converges to the exact solution of the system in at most two steps. Neither the exact absolute value of the matrix nor its exact inverse are computationally feasible to construct in general. However, we provide a practical example of an SPD preconditioner that is based on the suggested approach. In this example we consider a model problem with a shifted discrete negative Laplacian, and suggest a geometric multigrid (MG) preconditioner, where the inverse of the matrix absolute value appears only on the coarse grid, while operations on finer grids are based on the Laplacian. Our numerical tests demonstrate practical effectiveness of the new MG preconditioner, which leads to a robust iterative scheme with minimalist memory requirements.