Parallel Numerical Processing of Linear Systems with Irregularly Sparse Coefficient Matrix

具有不规则稀疏系数矩阵的线性系统的并行数值处理

• 批准号: 11680341
• 负责人: OYANAGI Yoshio
• 金额: $ 2.11万
• 依托单位: UNIVERSITY OF TOKYO
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11680341/
• 关键词: linear equations; sparse matrix; conjugate gradient method; preconditioning; multigrid; algebraic multigrid; AMG; 代数的マルチグリット; 連立一次方程式; Cholesky分解; Jocobi前処理

Parallel solution of large-scale linear systems which arise from the discretization of unstructured grid systems with irregular structures is studied. Although preconditioned conjugate gradient (PCG) methods are applicable to positive symmetric equations, the effectiveness of the PCG critically depends on the acceleration of convergence by the preconditioning and the parallelizability of the precontitioning. In this study, an algorithms to generated automatically generate the miltigrid using geometrical structures and to solve the systems of equations given by irregular finite element method for elliptic partial differential equations proposed. We found that although this method is effective for homogeneous problems, the convergence is not fast enough for problems with inhomogeneity. We proposed a kind of automatica semi-coursening method using an algebraic information at the same time for inhomogeneous problems. We have shown that our method gives almost the same performance as the ICCG (Incomplete Cholesky Conjugate Gradient) method. Since the ICCG cannot be parallelized, we believe our method can be applied to practical problems. For problems with the inhomogeneity of 100 or more, our method is inferior to the ICCG.We would like to continue our research toward the AMG, algebraic multi-grid method, and its application as a preconditioning.

研究了由不规则结构的非结构化网格系统离散化而产生的大型线性系统的并行解。虽然预条件共轭梯度（PCG）方法适用于正对称方程，但PCG方法的有效性主要取决于预条件的收敛速度和预条件的并行性。本文提出了一种利用几何结构自动生成微网格和求解椭圆型偏微分方程的不规则有限元法方程组的算法。我们发现该方法对齐次问题是有效的，但对非齐次问题的收敛速度不够快。针对非齐次问题，提出了一种同时使用代数信息的自动半训练方法。我们已经证明，我们的方法给出了几乎相同的性能ICCG（不完全Cholesky共轭梯度）方法。由于ICCG不能并行化，我们相信我们的方法可以应用于实际问题。对于100或更多的非均匀性问题，我们的方法不如ICCG。我们将继续研究AMG，代数多网格方法及其作为预处理的应用。

项目成果

西田晃: "Least Squares Iterative Arnoldi and its Convergence."Proc.of Riken Symposium. 164-171 (1999)

Akira Nishida：“最小二乘迭代阿诺尔迪及其收敛性”。理研研讨会论文集 164-171 (1999)。

塙与志夫,小柳義夫: "A Dented Tridiagonal Parallel Conditioner for CGM"Proc.of Riken Symposium. 35-41 (1999)

Yoshio Hanawa、Yoshio Koyanagi：“CGM 的凹状三对角平行调节器”Proc.of Riken Symposium 35-41 (1999)。

西田晃,小柳義夫: "大規模固有値問題のためのJawbi-Diwidson法とその特性について"情報処理学会論文誌(HPS). 41-8. 101-106 (2000)

Akira Nishida、Yoshio Koyanagi：“关于大规模特征值问题的 Jawbi-Diwidson 方法及其特征”，日本信息处理协会汇刊 (HPS) 41-8 (2000)。

西田 晃: "Least Squares Iterative Arnoldi and its Convergence"Ptoc.of RIKEN Symaporuim. 164-171 (1999)

Akira Nishida：“最小二乘迭代 Arnoldi 及其收敛性”Ptoc.of RIKEN Symaporuim 164-171 (1999)。

Ito,S.,Zhang,S.L.,Oyanagi,Y. and Natori,M.: "Spectral Properties by Using Splitting Correction Preconditioner fir Kubear Systems that arise from Periodic B.P."Proceedings of International Conference on Information Society in the 21st Century. 425-431 (200

伊藤S.，张S.L.，小柳Y.