Optimized sparse approximate inverse smoothers for solving Laplacian linear systems

TL;DR

This paper introduces new sparse approximate inverse smoothers for multigrid methods that significantly improve convergence rates for 2D and 3D Laplacian systems, with demonstrated efficiency and parallelism.

Contribution

The paper proposes novel SPAI smoothers that outperform existing methods in reducing smoothing factors for Laplacian systems in 2D and 3D.

Findings

Achieves smaller smoothing factors than previous SPAI smoothers.

Provides better convergence than weighted Jacobi in 3D cases.

Numerical results confirm high efficiency and effectiveness.

Abstract

In this paper we propose and analyze new efficient sparse approximate inverse (SPAI) smoothers for solving the two-dimensional (2D) and three-dimensional (3D) Laplacian linear system with geometric multigrid methods. Local Fourier analysis shows that our proposed SPAI smoother for 2D achieves a much smaller smoothing factor than the state-of-the-art SPAI smoother studied in [Bolten, M., Huckle, T.K. and Kravvaritis, C.D., 2016. Sparse matrix approximations for multigrid methods. Linear Algebra and its Applications, 502, pp.58-76.]. The proposed SPAI smoother for 3D cases provides smaller optimal smoothing factor than that of weighted Jacobi smoother. Numerical results validate our theoretical conclusions and illustrate the high-efficiency and high-effectiveness of our proposed SPAI smoothers. Such SPAI smoothers have the advantage of inherent parallelism. The MATLAB codes for implementing our proposed algorithms are publicly available online at http://github.com/junliu2050/SPAI-MG-Laplacian .