Incomplete Inverse Preconditioners

Incomplete LU factorization is a valuable preconditioning approach for sparse iterative solvers. An “ideal” but inefficient preconditioner for the iterative solution of Ax = b is A−1 itself. This paper describes a preconditioner based on sparse approximations to partitioned representations of A−1, in addition to the results of implementation of the proposed method in a shared memory parallel environment. The partitioned inverses are normally somewhat sparse. Their sparsity can be enhanced with suitable ordering and partitioning algorithms. Sparse approximations to these partitioned inverse representations can be obtained either by discarding selected nonzero entries of these inverses or by precluding the creation of some inversion fills. Experimental results indicate that the use of these partitioned incomplete inverses as preconditioners results in excellent highly parallel preconditioners.