基于Distance-2算法的并行Jacobian矩阵计算及其在耦合问题中的应用  

Parallel Jacobian Computation Based on Distance-2 Algorithmand Its Application in Coupling Problems

作　　者：刘礼勋 张汉 彭心茹 窦沁榕 邬颖杰 郭炯[1] 李富[1] LIU Lixun;ZHANG Han;PENG Xinru;DOU Qinrong;WU Yingjie;GUO Jiong;LI Fu(Institute of Nuclear and New Energy Technology,Key Laboratory of Advanced Reactor Engineering and Safety of Ministry of Education,Tsinghua University,Beijing 100084,China)

机构地区：[1]清华大学核能与新能源技术研究院,先进反应堆工程与安全教育部重点实验室,北京100084

出　　处：《原子能科学技术》2024年第6期1201-1209,共9页Atomic Energy Science and Technology

基　　金：国家自然科学基金面上项目(12275150);国家重点研发计划(2022YFB1903000);北京市自然科学基金(1212012)。

摘　　要：并行Newton-Krylov方法是求解大规模多物理耦合问题的有效方法,如何高效自动计算Jacobian矩阵是一大难点。利用有限差分方法,可避免推导Jacobian矩阵的表达式,实现矩阵的自动计算。现有工作表明,在串行环境下利用矩阵的稀疏性和图着色算法,Jacobian矩阵的计算效率可提高至少1个量级。但在并行环境下,串行着色算法失效,需采用相应的并行着色算法。本研究将图论领域的Distance-2算法应用于Jacobian矩阵的并行着色。通过求解一个简化多物理耦合问题检验了该并行算法的正确性和计算效率。测试结果表明,该并行算法得到的Jacobian矩阵完全正确;着色数随着并行核数的增加略微有所增加,100个进程下并行效率为56%;基于该算法求解多物理耦合问题,其计算时间和Krylov迭代次数较JFNK减少了约1/2。The fission nuclear reactor is a typical complicated multi-physics coupling system because of the nonlinear coupled terms among different physical fields.The Newton-Krylov method is an effective method for solving the multi-physics coupling problem,featuring strong stability and a high-order convergence rate.Recently,the Newton-Krylov method with an explicit Jacobian matrix has become popular.Compared with the JFNK(Jacobian-free Newton-Krylov)method which doesn’t form the Jacobian matrix explicitly,it has a better preconditioner matrix(the Jacobian matrix itself)and can achieve a more stable and fast convergence.How to calculate the Jacobian matrix efficiently and automatically is a major challenge.The finite difference method is an effective way to compute the Jacobian matrix automatically and can avoid the derivation of matrix element expressions.Besides,the serial graph coloring algorithm has been utilized to achieve an efficient computation of Jacobian.By exploiting the sparsity of the Jacobian matrix,all structurally orthogonal columns can be computed simultaneously through one function evaluation.Thus,the Jacobian computational cost can be reduced by one order of magnitude.However,when solving the larger scale problem under a distributed-memory parallel environment,the parallel graph coloring algorithm is required because the Jacobian is distributed among all processors.In this study,a Distance-2 graph coloring algorithm,arising from the field of graph theory,is applied to color the Jacobian matrix in parallel.This algorithm is performed iteratively,with each iteration consisting of four stages.Local coloring:Each processor tentatively colors its diagonal submatrix using greedy coloring algorithms;Color transfer:The colors of the diagonal submatrix are transferred to other processors to update the off-diagonal submatrix.Conflict detection:Each processor concurrently checks whether the color of the off-diagonal submatrix conflicts with the diagonal submatrix.The conflicts are gathered back to the correspon

关 键 词：Newton-Krylov方法 稀疏Jacobian矩阵 图着色 有限差分 分布式并行计算 

分 类 号：TL329.2[核科学技术—核技术及应用]