Algebraic multigrid (AMG) is an optimal algorithm for solving large-scale sparse linear systems. However, its complexity makes it challenging to achieve ideal parallel scalability and identify parallel scalability bottlenecks. In this paper, we analyze the performance skeletons and communication patterns of the AMG algorithm to identify three categories of scalability bottlenecks. Additionally, we introduce the concept of the sparse matrix communication domain to characterize the influence of sparse patterns on parallel communication performance. We examine six typical examples with varying sparse pattern features in practical applications such as radiation fluid dynamics, structural mechanics, and aero-engines. Through our analysis, we identify and analyze multi-granularity parallel scalability bottlenecks and provide insights into future directions for improving AMG parallel performance.

In this paper, we study in detail the specific convergence property of the physical-variable-based coarsening two-level iterative method (PCTL) algorithm based on the theory of algebraic multigrid method (AMG), and give a reasonable upper bound on the convergence factor, which provides a theoretical guarantee for the PCTL algorithm. Moreover, we also analyze the algebraic features that affect the convergence of the PCTL algorithm, such as diagonal dominance and coupling strength, hoping to provide theoretical guidance for the applications and algorithm optimization of the PCTL algorithm.

JPSOL(J Parallel Solver Library for Numerical Algebra Problems) is introduced, including the software architecture, matrix vector data structure, three kinds of algorithm libraries (linear, nonlinear and eigenvalue) and domain specific solvers. Then, the high parallel scalability of JPSOL are demonstrated by the testing results of basic iterative methods. Finally, the effect and robustness of JPSOL are demonstrated by several typical practical applications.

Firstly, a review is given by classifying the existing fast algorithms for solving large-scale discrete linear systems arising from the Multi-Group Radiation Diffusion (MGRD) equations. Secondly, based on our recent work on parallel algebraic multigrid (AMG), two preconditioning algorithms and related theoretical frameworks are developed on a higher level. One is the approximate Schur complement type based on physical quantities and the other is the combined type based on physical and algebraic features, and the relevant components of these works are portrayed within these frameworks. Based on the above framework, a approximate Schur complement preconditioner with fundamental approximation property and low computational complexity is designed, and the corresponding spectral equivalence theory is established. Numerical experiments show that the new preconditioner has better robustness and computational efficiency. Finally, several issues that need to be further addressed are presented.

To address the high computational complexity of sparse linear solvers caused by complex physical characteristics in practical applications, this paper presents a unified framework for feature-modified preconditioning algorithms. By refining the algebraic features affecting the efficiency from physical characteristics and combining multilevel feature analysis, we construct feature-modified components. The effectiveness of this framework is demonstrated through several typical feature-modified preconditioning algorithms and their application results.

2022年12月12日, 第八届高性能计算中间件技术研讨会(HPCMid22)成功召开。HPCMid (会议网址: http://www.caep-scns.ac.cn/HPCMid.php)每年举办一次, 面向科学与工程计算数值模拟应用在当前及下一代超级计算机上面临的挑战, 围绕高性能计算中间件关键技术, 邀请相关学者报告最新研究进展并探讨未来发展趋势。第八届研讨会以"适配新型体系结构的性能优化技术"为主题, 聚焦后摩尔时代新型体系结构为科学与工程计算带来的机遇与挑战, 探讨新型体系结构下可移植性能优化技术的发展趋势。本届研讨会的专家座谈(Panel Session)环节由莫则尧研究员和徐小文研究员共同主持, 邀请了王龙、刘杰、谭光明、刘伟峰、喻之斌5位来自高校、科研院所、企业的专家围绕"性能优化: 个性vs共性"这一主题开展了深入的讨论与交流, 翟季冬、杨海龙等多位专家也参与了讨论。专家们针对性能优化技术的研究现状与发展趋势、面临的问题与挑战以及人才培养等方面发表了许多有启发性的观点。《计算物理》编辑部特将本次讨论整理后发表, 以飨读者。限于篇幅, 略有删节。

An algebraic multigrid (AMG) algorithm based on hybrid coarsening is proposed for the linear systems of the discrete pressure Poisson which is derived from the SIMPLE algorithm for the Navier-Stokes equations in complex flows. This algorithm combines a class of non-smoothed aggregation coarsening with classical C/F coarsening to construct grid hierarchy, hoping to reduce the cost in the setup phase of the AMG algorithm without affecting convergence. The high performance of the proposed algorithm is verified by numerical simulation of complex flow in the combustion chamber of aero-engine. The results show that the proposed algorithm can achieve 78% acceleration compared with the classical AMG algorithm.

Targeting at SN algorithm for the neutron transport equation in the two-dimensional spherical coordinate system, we propose a directed graph model based on a (cell, direction) two-tuple, and design a multi-level parallel SN algorithm with controllable granularity on the basis of the existing parallel pipeline algorithm based on directed graph. Among them, a combination of domain decomposition and parallel pipeline is used to mine parallelism in the space-angle direction, and an energy group pipeline parallel method is proposed. Furthermore, by setting appropriate pipeline granularity, the overhead of scheduling, communication and idle waiting are well balanced. Experimental results show that the algorithm can effectively solve the neutron transport equation in the two-dimensional spherical coordinate system. For a typical neutron transport problem with 960 000 grids, 60 directions, 24 energy groups, and billions of degrees of freedom, the parallel program achieved 71% parallel efficiency on 1920 cores of a domestic parallel machine.