论文标题
使用两阶段多幅度求解线性系统的高度可扩展方法
A highly scalable approach to solving linear systems using two-stage multisplitting
论文作者
论文摘要
用于求解大型线性方程式稀疏系统的迭代方法在许多HPC应用中广泛使用。但是，这些方法的极端缩放可能很困难，因为在每次迭代中通常都需要进行全球形成点产品的全球通信。 为了克服这一限制，我们提出了一种混合方法，其中矩阵被划分为块。在每个块中，我们使用一个高度优化的（并行）常规求解器，但是然后使用块jacobi或其他一些可以以同步或异步方式实现的块将块共同搭配在一起。这使我们能够将块大小限制在常规迭代方法不再扩展的点，并避免在所有过程中进行全局通信（并可能同步）。 我们的块框架是为使用PETSC（用于求解稀疏线性系统的流行科学套件，作为同步块内求解器，我们在高达32768个Cray XE6系统的核心上展示结果。在这个规模上，传统的求解器仍然更有效，尽管趋势表明混合方法在较高的核心计数下可能是有益的。
Iterative methods for solving large sparse systems of linear equations are widely used in many HPC applications. Extreme scaling of these methods can be difficult, however, since global communication to form dot products is typically required at every iteration. To try to overcome this limitation we propose a hybrid approach, where the matrix is partitioned into blocks. Within each block, we use a highly optimised (parallel) conventional solver, but we then couple the blocks together using block Jacobi or some other multisplitting technique that can be implemented in either a synchronous or an asynchronous fashion. This allows us to limit the block size to the point where the conventional iterative methods no longer scale, and to avoid global communication (and possibly synchronisation) across all processes. Our block framework has been built to use PETSc, a popular scientific suite for solving sparse linear systems, as the synchronous intra-block solver, and we demonstrate results on up to 32768 cores of a Cray XE6 system. At this scale, the conventional solvers are still more efficient, though trends suggest that the hybrid approach may be beneficial at higher core counts.
