Sparse approximate inverse smoothers for geometric and algebraic multigrid

  • Authors:

  • Oliver Bröker;Marcus J. Grote

  • Affiliations:

  • Departement Informatik, ETH Zürich, CH-8092 Zürich, Switzerland;Departement Mathematik, ETH Zürich, CH-8092 Zürich, Switzerland

  • Venue:
  • Applied Numerical Mathematics - Developments and trends in iterative methods for large systems of equations—in memoriam Rüdiger Weiss
  • Year:
  • 2002

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Abstract

Sparse approximate inverses are considered as smoothers for geometric and algebraic multigrid methods. They are based on the SPAI-Algorithm [MJ. Grote, T. Huckle, SIAM J. Sci. Comput. 18 (1997) 838-853], which constructs a sparse approximate inverse M of a matrix A, by minimizing I - MA in the Frobenius norm. This leads to a new hierarchy of inherently parallel smoothers: SPAI-0, SPAI-1, and SPAI(ε). For geometric multigrid, the performance of SPAI-1 is usually comparable to that of Gauss-Seidel smoothing. In more difficult situations, where neither Gauss-Seidel nor the simpler SPAI-0 or SPAI-1 smoothers are adequate, further reduction of ε automatically improves the SPAI(ε) smoother where needed. When combined with an algebraic coarsening strategy [J.W. Ruge, K. Stüben, in: S.F. McCormick (Ed.), Multigrid Methods, SIAM, 1987, pp. 73-130] the resulting method yields a robust, parallel, and algebraic multigrid iteration, easily adjusted even by the non-expert. Numerical examples demonstrate the usefulness of SPAI smoothers, both in a sequential and a parallel environment.Essential advantages of the SPAI-smoothers are: improved robustness, inherent parallelism, ordering independence, and possible local adaptivity.