Massively Parallel Preconditioners for Krylov Subspace Methods

本研究的目的是探讨求解一般大型稀疏线性方程组的高度并行预处理技术。特别令人感兴趣的是由非结构化网格上的偏微分方程离散化产生的系统。重点是在处理器数量非常大的情况下，高效的预调节器可以很好地扩展。在这种情况下，标准的不完全分解预调节器不能令人满意。该项目研究了一些非标准的替代方案，这些替代方案强调基于“高精确度”的近似值。“这些类似于具有高填充水平的ILU分解。许多并行前置条件的缺点是需要比标准前置条件更多的迭代才能实现收敛。这通常超过了更高并行度所带来的潜在收益。然而，最近的经验表明，增加前置条件是通过减少迭代次数来实现更快执行的好策略。对多色技术的应用也进行了探讨。

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.