SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page C193-C220, January 2020.
The multigroup neutron transport equation is crucial for studying the motion of neutrons and their interaction with materials. Numerical simulation of the multigroup neutron transport equation is computationally challenging because the equation is defined on a high-dimensional phase space, the computational spatial domain is complex, and the materials are heterogeneous. A scalable parallel solver is required to address such a challenge. In this paper, we study a highly parallel Newton--Krylov--Schwarz (NKS) method consisting of a Newton-based eigenvalue solver, a Krylov subspace method, and a novel multilevel Schwarz preconditioner. The multilevel method is one of the most popular preconditioners for accelerating neutron transport calculations, but the construction of coarse spaces can be expensive and often unscalable when a large number of processors is used. We propose a novel matrix coarsening algorithm in which a multilevel hierarchy is constructed using a single-component matrix instead of the full matrix of the neutron transport equation. This new coarsening algorithm is referred to as “subspace-based coarsening.” Above 8,000 processors, we show a 13x enhancement in multilevel preconditioner setup time when using the subspace-based coarsening method. A partition-based balancing strategy is studied to enhance the parallel efficiency of the NKS algorithm by equalizing the work for each processor. A hierarchical mesh partitioning algorithm is employed to generate a large number of submeshes while minimizing off-node communication. We demonstrate that the proposed algorithm is scalable with more than 10,000 processors for a realistic application on three-dimensional unstructured meshes with a few billion degrees of freedom. Neutron transport calculations using the improved NKS algorithm are twice as fast as those based on the unmodified NKS solver when over 8,000 processors are employed.

中文翻译：

---

三维非结构网格上多组中子输运方程的高度并行多层牛顿-Krylov-Schwarz方法，具有基于子空间的粗化和基于分区的平衡

SIAM科学计算杂志，第42卷，第5期，第C193-C220页，2020年1月。
多组中子输运方程对于研究中子的运动及其与材料的相互作用至关重要。多组中子输运方程的数值模拟在计算上具有挑战性，因为该方程是在高维相空间上定义的，计算空间域是复杂的，并且材料是异质的。需要一个可扩展的并行求解器来解决这一难题。在本文中，我们研究了高度并行的牛顿-克雷洛夫-舒瓦兹（NKS）方法，该方法由基于牛顿的特征值求解器，克雷洛夫子空间方法和新型多级Schwarz前置条件组成。多级方法是加速中子输运计算最流行的前提条件之一，但是当使用大量处理器时，粗糙空间的构建可能很昂贵，并且通常无法扩展。我们提出了一种新颖的矩阵粗化算法，其中使用单分量矩阵而不是中子输运方程的完整矩阵构造了多级层次结构。这种新的粗化算法称为“基于子空间的粗化”。在8,000个以上的处理器上，使用基于子空间的粗化方法时，多层预处理器的设置时间提高了13倍。研究了基于分区的平衡策略，以通过均衡每个处理器的工作来提高NKS算法的并行效率。采用分层网格划分算法来生成大量的子网格，同时使节点外通信最小化。我们证明了所提出的算法可扩展到超过10,000个处理器，用于具有数十亿自由度的三维非结构化网格的实际应用。当使用8,000多个处理器时，使用改进的NKS算法的中子输运计算速度是基于未经修改的NKS求解器的中子输运速度的两倍。