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Parallel ADI solver based on processor scheduling

机译：基于处理器调度的并行ADI求解器

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Gaussian elimination is used for the direct solution of banded linear systems that typically appear in implicit numerical methods for PDEs. Gaussian elimination for narrow-banded systems (also known as the Thomas algorithm (TA)) includes forward and backward recurrences along lines of a numerical grid. Multi-domain decomposition, essential for parallelization of implicit solvers, spans the recurrences across processors in one or more directions. Processor idle time and inter-processor communication time are two interdependent reasons for the poor parallelization efficiency of TAs. In this research an efficient parallel algorithm for 3D directionally split problems is developed. The proposed solver is based on the static scheduling of processors where local and non-local, data-dependent and data-independent computations are scheduled while processors are idle. The proposed algorithm uses a reformulated version of the pipelined Thomas algorithm that starts the backward step computations immediately after the completion of the forward step computations for the first portion of lines. This algorithm has data available for other computational tasks while processors are idle from the TA. A theoretical model of parallelization efficiency is used to define optimal parameters of the algorithm, to show an asymptotic parallelization penalty and to obtain an optimal cover of a global domain with subdomains. It is shown by computational experiments and by the theoretical model that the proposed algorithm considerably reduces the communication cost and processor idle time over the basic algorithm for the range of the number of processors (subdomains) considered and the number of grid nodes per subdomain. (C) 2002 Elsevier Science Inc. All rights reserved. [References: 24]
机译：高斯消除法用于带状线性系统的直接求解，该线性系统通常出现在PDE的隐式数值方法中。窄带系统的高斯消除（也称为Thomas算法（TA））包括沿数字网格线的正向和反向递归。多域分解对于隐式求解器的并行化必不可少，它跨一个或多个方向跨处理器重复执行。处理器空闲时间和处理器间通信时间是TA并行效率差的两个相互依存的原因。在这项研究中，针对3D定向分裂问题开发了一种有效的并行算法。提出的求解器基于处理器的静态调度，其中处理器空闲时调度本地和非本地，数据相关和数据独立的计算。所提出的算法使用流水线Thomas算法的重新编写的版本，该算法在行的第一部分的前向步计算完成后立即开始后向步计算。当处理器从TA空闲时，该算法具有可用于其他计算任务的数据。并行化效率的理论模型用于定义算法的最佳参数，以显示渐近并行化代价，并获得带有子域的全局域的最优覆盖。通过计算实验和理论模型表明，在所考虑的处理器（子域）数量范围和每个子域的网格节点数量范围内，与基本算法相比，该算法大大降低了通信成本和处理器空闲时间。 （C）2002 Elsevier Science Inc.保留所有权利。 [参考：24]

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