Adaptive Techniques for Improving the Performance of Incomplete Factorization Preconditioning

Three techniques for improving the robustness and performance of iterative solvers for sparse systems with symmetric positive definite or mildly indefinite coefficient matrices are introduced. The primary contribution is new block algorithms for incomplete factorization that results in an improvement in the performance of both the preconditioner generation and the iterative solution phases. One of the algorithms applied to matrices that have a natural block structure in their original form, and the other one applied to matrices without natural dense blocks. Additionally, two relatively simple but highly effective techniques are introduced. These include selecting the solver based on the definiteness properties of the preconditioner and automatic selection and tuning of incomplete factorization parameters. All three techniques have adaptive componenets; i.e., the preconditioner-solver combination chooses parameters or algorithmic components based on the properties of the coefficient matrix and its incomplete factors. Two of the three techniques are applicable to incomplete LU factorization for unsymmetric systems as well.