Performance and Scalability of Hierarchical Hybrid Multigrid Solvers for Stokes Systems
Performance and Scalability of Hierarchical Hybrid Multigrid Solvers for Stokes Systems
复制标题
Stokes 系统分层混合多重网格求解器的性能和可扩展性
DOI：
10.1137/130941353
复制
发表时间：
2015
期刊：
影响因子：
--
通讯作者：
B. Wohlmuth
中科院分区：
文献类型：
--
作者：
B. Gmeiner;U. Rüde;H. Stengel;C. Waluga;B. Wohlmuth
在许多涉及不可压缩流体流动的应用中，斯托克斯系统发挥着重要作用。复杂的流动问题可能需要极其精细的分辨率，很容易导致具有超过一万亿 ($10^{12}$) 未知数的鞍点问题。即使在最先进的超级计算机上，快速求解此类方程组也是一项非常重要且具有挑战性的任务。在这项工作中，我们考虑实现迭代鞍点求解器，该求解器在数学上基于压力的 Schur 补公式，在算法上基于分层混合网格的抽象概念。我们的快速多重网格求解器的设计以计算内核的创新性能分析与通信开销的量化相结合为指导。现代超级计算机的不同特征类型都展示了出色的节点性能和近百万个并行线程的良好可扩展性。
In many applications involving incompressible fluid flow, the Stokes system plays an important role. Complex flow problems may require extremely fine resolutions, easily resulting in saddle-point problems with more than a trillion ($10^{12}$) unknowns. Even on the most advanced supercomputers, the fast solution of such systems of equations is a highly nontrivial and challenging task. In this work we consider a realization of an iterative saddle-point solver which is based mathematically on the Schur-complement formulation of the pressure and algorithmically on the abstract concept of hierarchical hybrid grids. The design of our fast multigrid solver is guided by an innovative performance analysis for the computational kernels in combination with a quantification of the communication overhead. Excellent node performance and good scalability to almost a million parallel threads are demonstrated on different characteristic types of modern supercomputers.