适用于高性能计算的大型线性系统预处理技术的最新进展

IF 0.6 Q3 MATHEMATICS
A. Franceschini, M. Ferronato, C. Janna, V. Magri
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引用次数: 2

摘要

现实世界工程问题的数值模拟创建了具有数百万甚至数十亿自由度的模型。这些模拟大多集中在非线性方程组的解上,这些方程组一旦线性化,就会变成一系列线性系统,其解通常是最耗时的任务。因此,为了增加对大型案例建模的能力,利用高性能计算体系结构的资源是至关重要的。在这个框架下,开发新的算法来加速求解多核体系结构的线性系统是一个非常活跃的研究领域。我们的主要重点是代数预处理,在各种选项中,我们选择开发对称和正定(SPD)线性系统[22]的近似逆,作为AMG技术的独立预处理或平滑。这种选择主要是由这些算法固有的近乎完美的并行性所支持的。选取Janna和Ferronato[18]以自适应形式提出的分解稀疏近似逆(FSAI)作为基本核。最近的研究进展包括:1)基于FSAI预处理的SPD问题鲁棒多级方法,该方法消除了独立于预处理稀疏度[14]的算法故障的机会;2)一种新颖的AMG方法,其特点是自适应FSAI方法具有灵活的平滑性,以及自适应计算扩展算子的新方法。在后一项工作中,还提出了一种构建延拓的新技术。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Recent advancements in preconditioning techniques for large size linear systems suited for high performance computing
The numerical simulations of real-world engineering problems create models with several millions or even billions of degrees of freedom. Most of these simulations are centered on the solution of systems of non-linear equations, that, once linearized, become a sequence of linear systems, whose solution is often the most time-demanding task. Thus, in order to increase the capability of modeling larger cases, it is of paramount importance to exploit the resources of High Performance Computing architectures. In this framework, the development of new algorithms to accelerate the solution of linear systems for many-core architectures is a really active research field. Our main focus is algebraic preconditioning and, among the various options, we elect to develop approximate inverses for symmetric and positive definite (SPD) linear systems [22], both as stand-alone preconditioner or smoother for AMG techniques. This choice is mainly supported by the almost perfect parallelism that intrinsically characterizes these algorithms. As basic kernel, the Factorized Sparse Approximate Inverse (FSAI) developed in its adaptive form by Janna and Ferronato [18] is selected. Recent developments are i) a robust multilevel approach for SPD problems based on FSAI preconditioning, which eliminates the chance of algorithmic breakdowns independently of the preconditioner sparsity [14] and ii) a novel AMG approach featuring the adaptive FSAI method as a flexible smoother as well as new approaches to adaptively compute the prolongation operator. In this latter work, a new technique to build the prolongation is also presented.
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来源期刊
CiteScore
1.70
自引率
7.70%
发文量
0
审稿时长
8 weeks
期刊介绍: Dolomites Research Notes on Approximation is an open access journal that publishes peer-reviewed papers. It also publishes lecture notes and slides of the tutorials presented at the annual Dolomites Research Weeks and Workshops, which have been organized regularly since 2006 by the Padova-Verona Research Group on Constructive Approximation and Applications (CAA) in Alba di Canazei (Trento, Italy). The journal publishes, on invitation, survey papers and summaries of Ph.D. theses on approximation theory, algorithms, and applications.
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