Approximate Inverse Preconditioners for General Sparse Matrices

The standard Incomplete LU (ILU) preconditioners often fail for general sparse indeenite 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 each column to maintain sparsity in the approximate inverse. Compared to previous methods, this is a natural way to determine the sparsity pattern of the approximate inverse. This paper discusses options such as Newton and`global' iteration, self-preconditioning, dropping strategies, and factorized forms. The performance of the options are compared on standard problems from the Harwell-Boeing collection. Theoretical results on general approximate inverses and the convergence behavior of the algorithms are derived. Finally, some ideas and experiments with practical variations and applications are presented.