论文标题

代数杂化的平行能量最小化延长

Parallel Energy-Minimization Prolongation for Algebraic Multigrid

论文作者

Janna, Carlo, Franceschini, Andrea, Schroder, Jacob B., Olson, Luke

论文摘要

代数多机(AMG)是由离散偏微分方程引起的线性方程系统最广泛使用的解决方案技术之一。 AMG的普及源于其在几乎线性时间内求解线性系统的潜力,即具有O(n)的复杂性,其中N是问题大小。目前,这种能力至关重要,大量HPC平台的可用性越来越多地推动了解决非常大问题的解决方案。快速收敛的AMG方法的关键是平滑和粗网格校正之间的良好相互作用,进而需要使用有效的延长。从理论上的角度来看,延长必须准确地表示核分量附近,同时又在能量规范中限制。但是,对于具有挑战性的问题,确保这两个要求都不容易,这正是这项工作的目标。我们提出了一个有限的最小化程序,旨在减少延长能量,同时在插值范围内保留近核成分。所提出的算法是基于先前的能量最小化方法,利用预处理的限制共轭梯度方法,但具有新功能,并且针对并行性能和实现具有特定的重点。结果表明,当用于各种应用程序领域的大型现实世界问题时,所得的求解器具有出色的收敛速率和可伸缩性,并且至少胜过一些更传统的AMG方法。

Algebraic multigrid (AMG) is one of the most widely used solution techniques for linear systems of equations arising from discretized partial differential equations. The popularity of AMG stems from its potential to solve linear systems in almost linear time, that is with an O(n) complexity, where n is the problem size. This capability is crucial at the present, where the increasing availability of massive HPC platforms pushes for the solution of very large problems. The key for a rapidly converging AMG method is a good interplay between the smoother and the coarse-grid correction, which in turn requires the use of an effective prolongation. From a theoretical viewpoint, the prolongation must accurately represent near kernel components and, at the same time, be bounded in the energy norm. For challenging problems, however, ensuring both these requirements is not easy and is exactly the goal of this work. We propose a constrained minimization procedure aimed at reducing prolongation energy while preserving the near kernel components in the span of interpolation. The proposed algorithm is based on previous energy minimization approaches utilizing a preconditioned restricted conjugate gradients method, but has new features and a specific focus on parallel performance and implementation. It is shown that the resulting solver, when used for large real-world problems from various application fields, exhibits excellent convergence rates and scalability and outperforms at least some more traditional AMG approaches.

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