High Performance Interactive Solvers

这个项目的目标是开发高性能的迭代技术来求解一般的稀疏线性方程组。在这里，高性能意味着(1)内在高效和(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只有在鲁棒性和求解典型应用（如计算流体动力学）中产生的线性系统的效率高时，才会在应用世界中被完全接受。出于这个原因，将强调利用高阶精度的预处理技术。这些包括高水平的填充完整分解，如ILUT和ILU(k)。将考虑不规则结构稀疏线性系统的最一般框架。这些矩阵被划分并映射到一个分布式内存多处理器系统中，然后用预处理的Krylov子空间技术求解这些系统——使用消息传递库（如MPI）在处理器之间进行通信。有三个主要问题需要仔细考虑。首先，将研究专门为分布式稀疏矩阵开发的预处理技术。其次，将考虑降低Krylov子空间加速器内积影响的技术。随着处理器数量的增加，这些内部乘积导致并行实现变得低效。最后，在不考虑并行性的情况下，寻求鲁棒迭代解，最终目标是提高当前的技术水平。在预条件方面，需要开发的方法有三种类型：(1)从域分解的观点出发，但适用于一般稀疏矩阵；(2)具有多消去（ILUM）及其相关技术的ILU；(3)从近似逆技术出发的混合方法。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. ***