Massively Parallel Preconditioners for Krylov Subspace Methods

这项研究的目的是研究求解一般大型稀疏线性方程组的高度并行预条件技术。特别令人感兴趣的是由非结构网格上的偏微分方程组离散化而产生的系统。研究的重点是高效的预处理器，当处理器数量很大时，预条件器具有很好的伸缩性。在这种情况下，标准的不完全分解预条件子并不能令人满意。该项目调查了一些非标准备选方案，这些方案强调基于“高精度”的近似值。这些类似于具有高填充级别的ILU分解。许多并行预处理器存在一个缺点，即需要比标准预处理器进行更多的迭代才能实现收敛。这往往超过了更高的并行度带来的潜在收益。然而，最近的经验表明，增加预处理器的数量是通过减少迭代次数来实现更快执行的好策略。对多种着色技术的使用也进行了研究。

The goal of this research is to investigate highly parallel preconditioning techniques for solving general large sparse linear systems of equations. Of particular interest are systems that arise from the discretization of partial differential equations on unstructured grids. The focuses on efficient preconditioners which scale well when number of processors is very large. In this situation, the standard incomplete factorization preconditioners are not satisfactory. The project investigates a number of non-standard alternatives which emphasize approximations based on ``high level of accuracy.'' These are similar to ILU factorizations with high level of fill-in. Many of the parallel preconditioners, suffer from the disadvantage that entail a larger number of iterations than standard preconditioners to achieve convergence. This often outweighs potential gains arising from the higher degree of parallelism. However, recent experience suggests that increasing the of the preconditioner is a good strategy for achieving faster execution by reducing the number of iterations. The use of multi-coloring techniques is also investigated.