用于分析大规模并行异步数值算法的切换动力学系统框架
A Switched Dynamical System Framework for Analysis of Massively Parallel Asynchronous Numerical Algorithms
AI总结：
本文提出用于分析大规模并行异步数值算法的切换动力学系统框架，通过将每个处理元素的计算视为动力学系统，并将异步性建模为随机切换，以解决大规模并行系统中同步带来的计算瓶颈问题。
摘要：
在未来，大规模并行计算系统将有必要解决计算密集型应用。大规模并行数值算法实现中的关键瓶颈是每次迭代后处理元素（PEs）之间的数据同步，这导致了显著的空闲时间。因此，有一个趋势是放松同步并采用异步计算模型以减少空闲时间。然而，这种放松对数值算法的稳定性和准确性的影响尚不清楚。本文提出了一种新的框架来分析此类算法。我们将每个PE的计算视为一个动力学系统，并将异步性建模为随机切换。然后将整个系统分析为切换动力学系统。然而，将大规模并行数值算法建模为切换动力学系统会导致大量的模式，这使得当前可用的分析工具在计算上不可行。我们开发了新的技术来克服这一可扩展性问题。该框架以一维热方程作为案例研究，并通过在nVIDIA Tesla TM GPU机器上求解偏微分方程（PDE）来验证所提出的分析框架，其中核心之间有异步通信。
英文摘要：
In the near future, massively parallel computing systems will be necessary to solve computation intensive applications. The key bottleneck in massively parallel implementation of numerical algorithms is the synchronization of data across processing elements (PEs) after each iteration, which results in significant idle time. Thus, there is a trend towards relaxing the synchronization and adopting an asynchronous model of computation to reduce idle time. However, it is not clear what is the effect of this relaxation on the stability and accuracy of the numerical algorithm. In this paper we present a new framework to analyze such algorithms. We treat the computation in each PE as a dynamical system and model the asynchrony as stochastic switching. The overall system is then analyzed as a switched dynamical system. However, modeling of massively parallel numerical algorithms as switched dynamical systems results in a very large number of modes, which makes current analysis tools available for such systems computationally intractable. We develop new techniques that circumvent this scalability issue. The framework is presented on a one-dimensional heat equation as a case study and the proposed analysis framework is verified by solving the partial differential equation (PDE) in a $\mathtt{nVIDIA\: Tesla^{\scriptsize{TM}}}$ GPU machine, with asynchronous communication between cores.
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