时间相关偏微分方程隐式离散后,通常需要求解一个稀疏线性代数方程组序列.利用序列中相邻方程组性质的差异性与相似性,自适应地选取预条件子,提升方程组序列的并行求解效率,从而缩短总体求解时间,是一个值得研究的问题.本文针对科学与工程计算中广泛使用的代数多重网格（AMG）预条件子,设计了方程组序列相关的自适应预条件策略.通过惯性约束聚变（ICF）的辐射流体力学数值模拟典型应用,验证了该策略的有效性.测试结果表明,在某高性能计算机的3125个CPU核上,自适应预条件策略可将并行效率从47%提升到61%,将模拟总时间从19.7 h降为14.5 h.
A series of sparse linear systems must be solved in applications that are based on the implicit solution of time-dependent partial differential equations（PDEs）.Preconditioned iterative methods are usually employed to solve such sparse linear systems.AMG is one of the most popular preconditioners in real applications.However,it results in poor parallel scalability,owing to its setup phase.In this paper,by utilizing the differences and similarities in property among the systems in series,an adaptive AMG preconditioning strategy is presented to improve the parallel scalability.The results obtained for a radiation hydrodynamics computation within an ICF simulation demonstrate the efficiency and improvement of the adaptive strategy.For a typical model,the new strategy improves the parallel efficiency from 47% to 61%,and reduces the CPU time from 19.7 h to 14.5 h.
Scientia Sinica Informationis
通信作者．E-mail：xwxu@iapcm．ac．cnXiaowen XU was born in 1978. He received his B.S degree in computational mathematics from Xiangtan University （2002）, and tlis Ph.D. degree in computational mathematics from the Graduate School of CAEP （2007）, both in China. Currently, he is a professor of IAPCM. His research interests include parallel algorithms, scalable linear solvers, and large-scale simulations for real applications.
Zeyao MO was born in 1971. He received his B.S in applied mathematics （1992） and Ph.D. in computer science （1997） both from National Defense University of Technology in China. Currently, he is a professor of IAPCM. His research interests include parallel computing, and parallel programming frameworks for large-scale numerical simulations.