The computational kernel in solving the $S_N$ transport equations is the parallel sweep, which corresponds to directly inverting a block lower triangular linear system that arises in discretizations of the linear transport equation. Existing parallel sweep algorithms are fairly efficient on structured grids, but still have polynomial scaling, $P^{1/d}$ for $d$ dimensions and $P$ processors. Moreover, an efficient scalable parallel sweep algorithm for use on general unstructured meshes remains elusive. Recently, a classical algebraic multigrid (AMG) method based on approximate ideal restriction (AIR) was developed for nonsymmetric matrices and shown to be an effective solver for linear transport. Motivated by the superior scalability of AMG methods (logarithmic in $P$) as well as the simplicity with which AMG methods can be used in most situations, including on arbitrary unstructured meshes, this paper investigates the use of parallel AIR (pAIR) for solving the $S_N$ transport equations with source iteration in place of parallel sweeps. Results presented in this paper show that pAIR is a robust and scalable solver. Although sweeps are still shown to be much faster than pAIR on a structured mesh of a unit cube, pAIR is shown to perform similarly on both a structured and unstructured mesh, and offers a new, simple, black box alternative to parallel transport sweeps.

中文翻译：

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求解SN输运方程的并行近似理想限制多重网格

求解$S_N$ 输运方程的计算核心是并行扫描，它对应于直接反转线性输运方程离散化中出现的块下三角线​​性系统。现有的并行扫描算法在结构化网格上相当有效，但仍然具有多项式缩放，$P^{1/d}$ 用于 $d$ 维和 $P$ 处理器。此外，用于一般非结构化网格的高效可扩展并行扫描算法仍然难以捉摸。最近，针对非对称矩阵开发了一种基于近似理想约束 (AIR) 的经典代数多重网格 (AMG) 方法，并证明它是线性传输的有效求解器。受到 AMG 方法优越的可扩展性（以 $P$ 为单位的对数）以及 AMG 方法可在大多数情况下使用的简单性的启发，包括在任意非结构化网格上，本文研究了使用并行 AIR (pAIR) 求解 $S_N$ 传输方程，其中源迭代代替并行扫描。本文中给出的结果表明 pAIR 是一个强大且可扩展的求解器。尽管在单位立方体的结构化网格上扫描仍然显示出比 pAIR 快得多，但 pAIR 在结构化和非结构化网格上的表现相似，并且提供了一种新的、简单的、黑盒替代并行传输扫描。