Parallel implementation and practical use of sparse approximate inverse preconditioners with a priori sparsity patterns

This paper describes and tests a parallel message-passing code for construc ting sparse approximate inverse preconditioners using Frobenius norm minimi zation. The sparsity patterns of the preconditioners are chosen as patterns of powers of sparsified matrices. Sparsification is necessary when powers of a matrix have a large number of nonzeros, making the approximate inverse computation expensive. For our test problems, the minimum solution time is achieved with approximate inverses with less than twice the number of nonz eros of the original matrix. Additional accuracy is not compensated by the increased cost per iteration. The results lead to further understanding of how to use these methods and how well these methods work in practice. in ad dition, this paper describes programming techniques required for high perfo rmance, including one-sided communication, local coordinate numbering, and load repartitioning.