Approximate Inverse Preconditioners via Sparse-Sparse Iterations

The standard incomplete LU (ILU) preconditioners often fail for general sparse in-deenite matrices because they give rise tòunstable' 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 kI ? AMk 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 eecient, 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 uid dynamics analysis package show the strengths and limitations of approximate inverses. Finally, some ideas and experiments with practical variations and applications are presented.