GPA:基于多面体网格几何并行性的矩阵特征多项式异步因式分解器--纪念我国现代计算数学的开拓者之一周毓麟先生诞辰100周年  

Construction of intrinsic schemes for eigen-computation based on the polyhedron grid matrix in 3-D

作　　者：孙家昶[1] Jiachang Sun

机构地区：[1]中国科学院软件研究所并行软件与计算科学实验室,北京100080

出　　处：《中国科学：数学》2023年第6期859-894,共36页Scientia Sinica：Mathematica

基　　金：中国科学院软件研究所高性能计算基础研究计划资助项目

摘　　要：满足高次方程G^(m)=I的几何网格矩阵G,可在复数范围内进行因式分解,并且G与偏微分方程(partial differential equation,PDE)离散后的刚度矩阵A和质量矩阵B之间的乘法存在互易性:AG=GA,BG=GB,从而利用几何不变性可以将A正交分解为m-块对角块矩阵(m<N=dim(A)).本文在作者前期工作的基础上,继续深入研究求解数学物理方程离散特征值问题的几何网格异步因式分解算法(geometry pre-processing asynchronous algorithm,GPA),针对非规则的二维单元和典型三维单元(如六面体、四面体和十二面体单元等),提出计算PDE离散特征值问题的高效异步并行预处理降阶算法,给出相关的理论证明及数值计算实例.通过研究得到“三维几何网格预变换的并行度主要与多面体的面数成正比”的结论,并进一步揭示“几何网格矩阵与刚度矩阵的互易性对于特征值并行计算降阶算法的特殊重要性”。A geometric asynchronous parallel algorithm for solving large-scale discrete mathematical-physical systems in the 2-D case was recently presented by the author in 2022.Different from the traditional precondition-ing,we applied the intrinsic geometric invariance to get the grid matrix G,and required the discrete PDE(partial differential equation)stiff matrix A,and the mass matrices B and G satisfy the reciprocal relations AG=GA and BG=GB,where G satisfies Gm=I,m<N=dim(G),i.e.,large scale system solvers can be transformed to a smaller block-solver as a pre-treatment in real or complex domain.In this paper,we expand our geometry pre-processing asynchronous algorithm(GPA)to the 2-D irregular mesh and the 3-D mathematical-physical dis-crete eigenvalue problems over more wide polyhedron grids such as hexahedrons,tetrahedrons and dodecahedrons.We give the theory and numerical examples of an efficient asynchronous parallel order reduction algorithm.We obtain the conclusion that“the parallelism of 3-D geometric mesh pre-transformation is mainly proportional to the number of the faces of the polyhedron”,and further find that“the reciprocity of the grid mesh matrix and the stiff matrix is an important basis for the feasibility and reliability of the GPA algorithm”.

关 键 词：三维数理方程离散特征值 互易算子 几何块预处理子 特征值问题因式分解 异步并行算法 

分 类 号：O241[理学—计算数学] TP338.6[理学—数学]