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首页> 外文期刊>IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences >A Performance Comparison of the Parallel Preconditioners for Iterative Methods for Large Sparse Linear Systems Arising from Partial Differential Equations on Structured Grids
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A Performance Comparison of the Parallel Preconditioners for Iterative Methods for Large Sparse Linear Systems Arising from Partial Differential Equations on Structured Grids

机译：结构网格上由偏微分方程引起的大型稀疏线性系统迭代方法并行预处理器的性能比较

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In this paper we compare various parallel preconditioners such as Point-SSOR (Symmetric Successive OverRelaxation), ILU(0) (Incomplete LU) in the Wavefront ordering, ILU(0) in the Multi-color ordering, Multi-Color Block SOR (Successive OverRelaxation), SPAI (SParse Approximate Inverse) and pARMS (Parallel Algebraic Recursive Multilevel Solver) for solving large sparse linear systems arising from two-dimensional PDE (Partial Differential Equation)s on structured grids. Point-SSOR is well-known, and ILU(0) is one of the most popular pre-conditioner, but it is inherently serial. ILU(0) in the Wavefront ordering maximizes the parallelism in the natural order, but the lengths of the wave-fronts are often nonuniform. ILU(0) in the Multi-color ordering is a simple way of achieving a parallelism of the order N, where N is the order of the matrix, but its convergence rate often deteriorates as compared to that of natural ordering. We have chosen the Multi-Color Block SOR precondi-tioner combined with direct sparse matrix solver, since for the Laplacian matrix the SOR method is known to have a nondeteriorating rate of con-vergence when used with the Multi-Color ordering. By using block version we expect to minimize the interprocessor communications. SPAI computes the sparse approximate inverse directly by least squares method. Finally, ARMS is a preconditioner recursively exploiting the concept of independent sets and pARMS is the parallel version of ARMS. Experiments were conducted for the Finite Difference and Finite Element discretizations of five two-dimensional PDEs with large meshsizes up to a million on an IBM p595 machine with distributed memory. Our matrices are real positive, i.e., their real parts of the eigenvalues are positive. We have used GMRES(m) as our outer iterative method, so that the convergence of GMRES(m) for our test matrices are mathematically guaranteed. Interprocessor communications were done using MPI (Message Passing Interface) primitives. The results show that in general ILU(0) in the Multi-Color ordering and ILU(0) in the Wavefront ordering outperform the other methods but for symmetric and nearly symmetric 5-point matrices Multi-Color Block SOR gives the best performance, except for a few cases with a small number of processors.
机译：在本文中，我们比较了各种并行预处理器，例如Point-SSOR（对称连续过度松弛），波前排序中的ILU（0）（不完全LU），多色排序中的ILU（0），多色块SOR（连续过松弛），SPAI（稀疏近似逆）和pARMS（并行代数递归多级求解器），用于求解结构化网格上二维PDE（偏微分方程）引起的大型稀疏线性系统。 Point-SSOR是众所周知的，并且ILU（0）是最受欢迎的预处理器之一，但它本质上是串行的。波前排序中的ILU（0）使自然顺序中的并行度最大化，但波前的长度通常不均匀。多色排序中的ILU（0）是获得N阶并行度的简单方法，其中N是矩阵的阶数，但与自然排序相比，其收敛速度通常会降低。我们选择了与直接稀疏矩阵求解器组合的Multi-Color Block SOR预处理器，因为对于Laplacian矩阵，SOR方法与Multi-Color排序一起使用时收敛速度不会降低。通过使用块版本，我们希望最大限度地减少处理器间的通信。 SPAI通过最小二乘法直接计算稀疏近似逆。最后，ARMS是递归地利用独立集的概念的前提条件，而pARMS是ARMS的并行版本。在具有分布式内存的IBM p595机器上，进行了五个网格尺寸高达一百万的二维PDE的有限差分和有限元素离散化的实验。我们的矩阵是实数正数，即它们的特征值的实数部分是正数。我们已经使用GMRES（m）作为我们的外部迭代方法，因此在数学上保证了测试矩阵GMRES（m）的收敛性。处理器间通信是使用MPI（消息传递接口）原语完成的。结果表明，通常在多色排序中的ILU（0）和在波前排序中的ILU（0）优于其他方法，但是对于对称和接近对称的5点矩阵，多色块SOR的性能最佳。在少数情况下使用少量处理器。

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