Parallel Implementation of Discretization Methods for Nonstructured Meshes on Distributed Shared Memory Architectures

1.有限元方法的代数多重网格预条件算法本课题组于1992年提出并实现的几何多重网格预条件共轭梯度法是求解对流扩散方程最有效、最具伸缩性的算法之一，但在求解各向异性问题时存在收敛速度慢的缺点。针对几何各向异性的影响，提出了一种基于平滑聚合的代数多重网格预条件共轭梯度法AMGCG，并在具有256个CPU的SPARC工作站机群上实现。在250^3个网格点上的实验表明，对于最大网格，该方法的收敛速度是不完全Cholesky分解的共轭梯度法ICCG的3倍以上。此外，在最粗网格上使用并行直接求解器，该方法即使在各向同性问题上也表现出恒定的收敛时间。分布式共享内存上的高性能计算分布式共享内存体系结构是使商用共享内存机上的高性能计算可扩展的关键技术之一。我们已经在两台基于Intel处理器的分布式共享内存计算机上构建了Linux环境，NEC的Azusa和IBM的xSeries440。我们的底层基本线性代数例程和快速傅立叶变换代码的实现表明，我们可以在这种商品化的DSM架构上实现良好的可扩展性，但还需要仔细调整操作系统以获得LAIR内存带宽。

This research has achieved the following results on the two main targets of the project :1. Algebraic Multigrid Preconditioner for Finite Element MethodsMGCG, the geometric multigrid preconditioned conjugate gradient method, which has been proposed and implemented by our group in 1992, is one of the most effective and scalable solvers for convection diffusion equations, but it has a drawback of slow convergence on anisotropic problems. This study proposed AMGCG, the algebraic multigrid preconditioned conjugate gradient method based on smoothed aggregation, which cancels the effect of geometric anisotropy, and implemented on a SPARC workstation cluster with 256 CPUs. Experiments up to 250^3 mesh points showed that the method achieves more than three times as fast convergence as ICCG, the conjugate gradient method with incomplete Cholesky factorization, for the largest size. Furthermore, using parallel direct solver on the coarsest grid, the method shows constant convergence time even on isotropic problems.2. High Performance Computing on Distributed Shared MemoryDistributed shared memory architecture is one of the key technologies to make high performance computing on commodity shared memory machines scalable. We have built Linux environments on two Intel Processor based distributed shared memory computers, NEC's AzusA and IBM's xSeries 440. Our implementation of low level basic linear algebra routines and fast Fourier transform codes shows that we can achieve good scalability on such commodity DSM architectures, but careful tuning of the operation system is also required to get lair memory bandwidth.