Pipelined Parallel JFNK Method and its Application in Neutron k Eigenvalue Problem
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摘要: JFNK（Jacobian-free Newton-Krylov）方法是求解中子k本征值和反应堆多物理场耦合等非线性问题的高效加速方法，其中的Krylov迭代常用广义极小残差法（简称GMRES）。并行JFNK方法是实现更大规模问题求解的必要手段，其核心是解决GMRES中Gram-Schmidt （简称GS）正交化过程集合通信多、并行效率低的问题。本文以三维中子k本征值问题为研究对象，开发了基于消息传递接口并行编程模型和空间区域分解技术的并行JFNK方法。针对GS正交化过程并行可扩展性差的问题，分析讨论了流水线方法，以提高并行JFNK的并行效率，并对比了采用经典GS正交化的并行JFNK、采用修正GS正交化的并行JFNK和采用流水线方法的并行JFNK的计算时间和并行效率。最后选用三维扩散基准题IAEA-3D进行了数值测试，测试结果表明采用流水线方法的并行JFNK并行效率显著高于使用经典或修正GS正交化的并行JFNK，且收敛性未受影响。Abstract: JFNK (Jacobian-free Newton-Krylov) is an efficient acceleration method for solving nonlinear problems such as neutron k eigenvalue and coupling of multiple physical fields in the reactor, and the generalized minimum residual (GMRES) algorithm is commonly used in Krylov iteration. The parallel JFNK method is a necessary means for solving large-scale problems, and its main problem lies in the low parallel efficiency of Gram-Schmidt (GS) orthogonalization procedure in GMRES, which causes massive collective communications. In this paper, the parallel JFNK method based on the parallel programming model of message passing interface and spatial domain decomposition technology is developed for the three-dimensional neutron k eigenvalue problem. Aiming at the poor parallel scalability of GS orthogonalization procedure, the pipelined method is studied to improve the parallel efficiency of parallel JFNK. Then, the computation time and parallel efficiency of parallel JFNK using classical GS orthogonalization, modified GS orthogonalization and pipelined method are compared. The IAEA-3D three-dimensional diffusion benchmark problem was used for numerical test. The results show that the parallel efficiency of pipelined parallel JFNK is significantly superior to that using classical or modified GS orthogonalization, and the convergence of pipelined parallel JFNK is not affected.
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图 3 并行JFNK方法中的计算和通信示意图
Figure 3. Computing and Communication Schematic in Parallel JFNK Method
图 5 流水线方法的并行通信示意图
Figure 5. Schematic Diagram of Parallel Communication of Pipeline Method
图 6 使用MPI非阻塞通信计算残差函数
Figure 6. Computing Residual Using MPI Non-Blocking Communication
图 7 IAEA-3D基准题归一化组件功率计算结果
Figure 7. Normalized Assembly Power Results of IAEA-3D Benchmark
图 8 使用不同正交化过程的并行JFNK收敛对比
Figure 8. Convergence Comparison of Parallel JFNK Using Different Orthogonalization Procedures
图 9 使用不同正交化过程的并行JFNK的各种运算用时对比
Figure 9. Time Comparison of Various Operations of Parallel JFNK Using Different Orthogonalization Procedures
图 10 使用不同正交化过程的并行JFNK的计算和通信用时对比
Figure 10. Time Comparison of Computing and Communication of Parallel JFNK Using Different Orthogonalization Procedures
图 11 使用不同正交化过程的并行JFNK的并行加速比和并行效率
Figure 11. Speedup Ratio and Parallel Efficiency of Parallel JFNK Using Different Orthogonalization Procedures
表 1 3种正交化过程的通信和计算次数对比
Table 1. Comparison of Communication and Computation Among Three Orthogonalization Procedures
正交化过程 CGS MGS 流水线 集合通信次数 2m m(m+3)/2 m+1 矩阵向量积次数 m m m+1 AXPY运算次数 m m 2m 
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表 2 不同计算节点数下的JFNK的计算用时
Table 2. Total Computing Time of JFNK under Different Number of Nodes
计算节点数 计算时间/s JFNK-CGS JFNK-MGS JFNK-流水线 1 177.13 222.86 234.72 4 41.66 45.08 53.20 8 22.10 26.41 25.84 12 14.81 22.55 17.70 16 11.59 19.98 13.64 24 9.19 20.29 9.12 32 8.47 22.89 7.37 40 8.06 31.95 6.17
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