Robust Parallel Smoothing for Multigrid Via Sparse Approximate Inverses

• Authors: Oliver Bröker;Marcus J. Grote;Carsten Mayer;Arnold Reusken
• Venue: SIAM Journal on Scientific Computing
• Year: 2001

Sparse approximate inverses are considered as smoothers for multigrid. They are based on the SPAI-Algorithm [M. J. Grote and T. Huckle, SIAM J. Sci. Comput., 18 (1997), pp. 838--853], which constructs a sparse approximate inverse M of a matrix A by minimizing I -MA in the Frobenius norm. This yields a new hierarchy of smoothers: SPAI-0, SPAI-1, SPAI$(\varepsilon)$. Advantages of SPAI smoothers over classical smoothers are inherent parallelism, possible local adaptivity, and improved robustness. The simplest smoother, SPAI-0, is based on a diagonal matrix M. It is shown to satisfy the smoothing property for symmetric positive definite problems. Numerical experiments show that SPAI-0 smoothing is usually preferable to damped Jacobi smoothing. For the SPAI-1 smoother the sparsity pattern of M is that of A; its performance is typically comparable to that of Gauss--Seidel smoothing; however, both the computation and the application of the smoother remain inherently parallel. In more difficult situations, where the simpler SPAI-0 and SPAI-1 smoothers are not adequate, the SPAI$(\varepsilon)$ smoother provides a natural procedure for improvement where needed. Numerical examples illustrate the usefulness of SPAI smoothing.