Enhancing Performance and Robustness of ILU Preconditioners by Blocking and Selective Transposition

Abstract

Incomplete factorization is one of the most effective general-purpose preconditioning methods for Krylov subspace solvers for large sparse systems of linear equations. Two techniques for enhancing the robustness and performance of incomplete LU factorization for sparse unsymmetric systems are described. A block incomplete factorization algorithm based on the Crout variation of LU factorization is presented. The algorithm is suitable for incorporating threshold-based dropping as well as unrestricted partial pivoting, and it overcomes several limitations of existing incomplete LU factorization algorithms with and without blocking. It is shown that blocking has a threepronged impact: it speeds up the computation of incomplete factors and the solution of the associated triangular systems, it permits denser and more robust factors to be computed economically, and it permits a trade-off with the restart parameter of GMRES to further improve the overall speed and robustness. A highly effective heuristic for improving the quality of preconditioning and subsequent convergence of the associated iterative method is presented. The choice of the Crout variant as the underlying factorization algorithm enables efficient implementation of this heuristic, which has the potential to improve both incomplete and complete sparse LU factorization of matrices that require pivoting for numerical stability.

Enhancement of Algebraic Block Multi-Color Ordering for ILU Preconditioning and Its Performance Evaluation in Preconditioned GMRES Solver

“…The proposed blocking framework has been proved to lead to faster and more robust preconditioning in Ref. [27]. Bahi et al proposed an efficient parallel implementation of GMRES for GPU clusters and verified the high data-parallel nature of GPUs in Ref.…”

Javelin: A Scalable Implementation for Sparse Incomplete LU Factorization

“…Additionally most packages that offer traditional ILU use only ILU(k, τ ) and does not provide an interface to ILU(k). Therefore, we will use this double method to compare to the commercial package Watson Sparse Matrix Package (WSMP) [18], but will use ILU(k) for scalable performance analysis for reasons stated before. WSMP is given the matrix in the level ordering used by Javelin with k=0 and τ is set to be a value so that nonzeros are similar to that of ILU(0), and we do not allow pivoting.…”