求解超大规模稀疏线性方程组的可扩展分层并行算法
Scalable hierarchical parallel algorithm for the solution of super large-scale sparse linear equations
摘要：
能够有效使用1000多个处理器的并行线性方程组求解器成为大规模隐式工程仿真的瓶颈。本文提出了一种新的在分布式计算机集群中求解超大规模稀疏线性方程组的分层并行主从结构迭代算法。通过交替进行全局平衡计算和局部松弛，我们提出的算法将在几个迭代步骤达到特定的精度要求。此外，每个集合/从处理器主要与其最近的邻居通信，并且集合/从处理器与主处理器之间的数据传输总是远低于集合-邻居通信。基于MPI库实现了相应的隐式有限元分析算法，并在清华大学“Exploration 100”机群上对一个超10亿自由度、2001个处理器的超大型二维正方形三角网格桁架结构体系进行了随机静载荷下的数值模拟。数值实验表明，该算法具有良好的并行效率和可扩展性，在其他隐式模拟中具有广泛的应用前景。
英文摘要：
The parallel linear equations solver capable of effectively using 1000+ processors becomes the bottleneck of large-scale implicit engineering simulations. In this paper, we present a new hierarchical parallel master-slave-structural iterative algorithm for the solution of super large-scale sparse linear equations in distributed memory computer cluster. Through alternatively performing global equilibrium computation and local relaxation, our proposed algorithm will reach the specific accuracy requirement in a few of iterative steps. Moreover, each set/slave-processor majorly communicate with its nearest neighbors, and the transferring data between sets/slave-processors and master is always far below the set-neighbor communication. The corresponding algorithm for implicit finite element analysis has been implemented based on MPI library, and a super large 2-dimension square system of triangle-lattice truss structure under random static loads is simulated with over one billion degrees of freedom and up to 2001 processors on "Exploration 100" cluster in Tsinghua University. The numerical experiments demonstrate that this algorithm has excellent parallel efficiency and high scalability, and it may have broad application in other implicit simulations.
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