An Algebraic Substructuring Method for Large-Scale Eigenvalue Calculation
An Algebraic Substructuring Method for Large-Scale Eigenvalue Calculation
复制标题
大规模特征值计算的代数子结构方法
DOI：
10.1137/040613767
复制
发表时间：
2004
期刊：
影响因子：
--
通讯作者：
E. Ng
中科院分区：
文献类型：
--
作者：
Chao Yang;Weiguo Gao;Z. Bai;X. Li;Lie;P. Husbands;E. Ng
我们从纯代数的角度研究了求解大规模广义特征值问题的子结构方法。我们使用术语代数子结构来指代应用矩阵重新排序和划分算法将大型稀疏矩阵划分为较小的子矩阵的过程，从这些子矩阵中提取并组合频谱分量的子集以提供原始问题的近似解决方案。我们感兴趣的问题，其中频谱componentsone应该提取每个子结构，以产生一个近似的解决方案，以所需的精度水平的原始问题。对小的Esteigen对近似的误差估计。估计导致一个简单的启发式从每个子结构中选择频谱分量（模式）。数值例子证明了这种启发式的有效性。我们表明，代数子结构可以有效地用于解决所产生的加速器结构的模拟广义本征值问题。该应用程序的一个有趣的特点是，刚度矩阵产生的分层矢量有限元格式包含一个大的零空间。我们提出了一个有效的计划，紧缩这个零空间的代数子结构的过程。
We examine sub-structuring methods for solving large-scale generalized eigenvalue problems from a purely algebraic point of view. We use the term algebraic sub-structuring to refer to the process of applying matrix reordering and partitioning algorithms to divide a large sparse matrix into smaller submatrices from which a subset of spectral components are extracted and combined to provide approximate solutions to the original problem. We are interested in the question of which spectral componentsone should extract from each sub-structure in order to produce an approximate solution to the original problem with a desired level of accuracy. Error estimate for the approximation to the small esteigen pair is developed. The estimate leads to a simple heuristic for choosing spectral components (modes) from each sub-structure. The effectiveness of such a heuristic is demonstrated with numerical examples. We show that algebraic sub-structuring can be effectively used to solve a generalized eigenvalue problem arising from the simulation of an accelerator structure. One interesting characteristic of this application is that the stiffness matrix produced by a hierarchical vector finite elements scheme contains a null space of large dimension. We present an efficient scheme to deflate this null space in the algebraic sub-structuring process.