High Performance Interactive Solvers

这个项目的目标是发展高性能的迭代技术来解决一般稀疏线性方程组。 这里，高性能意味着（1）本质上高效和（2）为高性能平台设计的双重意义。 这项研究的动机主要是相信，并行迭代方法将完全接受的应用世界，只有当他们是强大的，以及有效的解决线性系统所产生的典型应用，如计算流体动力学。 由于这个原因，将强调利用高精度的预处理技术。 这些包括完全因式分解中的高水平填充，例如ILUT和ILU（k）。 最一般的框架，不规则结构的稀疏线性系统将被考虑。 这些矩阵被划分并映射到一个分布式内存多处理器系统，然后用预处理Krylov子空间技术解决系统-使用消息传递库，如MPI处理器之间的通信。 有三个主要问题需要仔细考虑。 首先，预处理技术专门为分布式稀疏矩阵将进行调查。 第二，技术，以减少克雷洛夫子空间加速器的内积的影响将被考虑。 随着处理器数量的增加，这些内积导致并行实现变得低效。 最后，独立的并行性的考虑，强大的迭代求解器将寻求与改善目前的最先进的最终目标。 在预处理器方面，待开发的方法有三种类型：（1）从区域分解观点导出但适用于一般稀疏矩阵，（2）具有多重消除的ILU（ILUM）和相关技术，以及（3）从近似逆技术导出的混合方法。在加速器方面，将研究降低内积计算影响的混合技术以及块Krylov方法。 ***

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. ***