Factorization-based sparse solvers and preconditioners

Abstract

Efficient solution of large-scale, ill-conditioned and highly-indefinite algebraic equations often relies on high quality preconditioners together with iterative solvers. Because of their robustness, the factorization-based algorithms could play a significant role when they are combined with iterative methods, particularly in the development of scalable solvers. We present our recent work in using the direct solver SuperLU code base to develop a new supernode-based ILU preconditioner and a domain-decomposition hybrid solver. Our ILU preconditioner is a modification of the classic ILUTP approach, incorporating a number of techniques to improve robustness and performance, which include new dropping strategies that accommodate the use of supernodal structure in the factored matrix. Our hybrid solver is based on the Schur complement method. We use parallel graph partitioning to obtain hierarchical interface/domain decomposition, and multiple parallel direct solvers to solve the subdomain problems simultaneously, and parallel preconditioned iterative solvers to solve the interface problem. We will demonstrate the effectiveness of our new techniques by applying them to two SciDAC applications, modeling next-generation particle accelerators and fusion devices.