  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 .
