High Performance Interactive Solvers

点击翻译按钮获取中文摘要

The goal of this project is to develop high performance iterative techniques for solving general sparse linear systems of equations. Here, high-performance is meant in the double sense of (1) intrinsically efficient and (2) designed for high- performance platforms. This research is motivated mainly by the belief that parallel iterative methods will be fully accepted in the applications world only if they are robust as well as efficient for solving linear systems arising from typical applications, such as Computational Fluid Dynamics. For this reason preconditioning techniques that utilize a high order of accuracy will be emphasized. These include the high level of fill in complete factorizations such as ILUT and ILU(k). The most general framework of irregularly structured sparse linear systems will be considered. These matrices are partitioned and mapped into a distributed memory multiprocessor system and the systems are then solved with a preconditioned Krylov subspace technique -- using a message passing library such as MPI for communicating between processors. There are three major issues to be considered with care. First, preconditioning techniques developed specifically for distributed sparse matrices will be investigated. Second, techniques to reduce the impact of inner products in Krylov subspace accelerators will be considered. These inner products cause the parallel implementations to become inefficient as the number of processors increases. Finally, independent of consideration of parallelism, robust iterative solvers will be sought with the ultimate goal of improving the current state-of-the-art. On the preconditioners side, the methods to be developed are of three types: (1) derived from a Domain Decomposition viewpoint but adapted to general sparse matrices, (2) ILU with multi- elimination (ILUM) and related techniques, and (3) Hybrid methods derived from approximate inverse techniques. On the accelerators side, hybrid techniques that lowe r the impact of inner-product computations, as well as block Krylov methods will be investigated. ***