Approximate Inverse Preconditioners via Sparse-Sparse Iterations

  • Authors:

  • Edmond Chow;Yousef Saad

  • Affiliations:

  • -;-

  • Venue:
  • SIAM Journal on Scientific Computing
  • Year:
  • 1998

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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.