Approximate inverse preconditioners via sparse-sparse iterations

Edmond Chow, Yousef Saad

Research output: Contribution to journalArticlepeer-review

169 Scopus citations

Abstract

The standard incomplete LU (ILU) preconditioners often fail for general sparse indefinite matrices because they give rise to "unstable" factors L and U. In such cases, it may be attractive to approximate the inverse of the matrix directly. This paper focuses on approximate inverse preconditioners based on minimizing ∥I -AM∥F, where AM is the preconditioned matrix. An iterative descent-type method is used to approximate each column of the inverse. For this approach to be efficient, the iteration must be done in sparse mode, i.e., with "sparse-matrix by sparse-vector" operations. Numerical dropping is applied to maintain sparsity; compared to previous methods, this is a natural way to determine the sparsity pattern of the approximate inverse. This paper describes Newton, "global," and column-oriented algorithms, and discusses options for initial guesses, self-preconditioning, and dropping strategies. Some limited theoretical results on the properties and convergence of approximate inverses are derived. Numerical tests on problems from the Harwell-Boeing collection and the FIDAP fluid dynamics analysis package show the strengths and limitations of approximate inverses. Finally, some ideas and experiments with practical variations and applications are presented.

Original languageEnglish (US)
Pages (from-to)995-1023
Number of pages29
JournalSIAM Journal of Scientific Computing
Volume19
Issue number3
DOIs
StatePublished - May 1998

Keywords

  • Approximate inverse
  • Krylov subspace methods
  • Preconditioning
  • Threshold dropping strategies

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