Incomplete Inverse Matrices

The solution of large sparse linear systems is required in many scientific fields such as computational fluid dynamics, computational electromagnetism, computational finance, etc. The computation of the solution of these systems is performed with preconditioned iterative methods, which rely on effective preconditioning schemes. A new class of approximate inverses is proposed, namely incomplete inverse matrices, which are computed using a recursive Schur complement-based approach. This class of approximate inverses is based on a priori knowledge of a sparsity pattern. In order to have finer control over the density of the proposed approximate inverse, especially in the case of three-dimensional problems, on-the-fly filtration is used, resulting in substantial reduction in the number of nonzero elements. Implementation details and analysis for computing the proposed scheme are given. Numerical results depicting the effectiveness and applicability of the proposed scheme are also provided.