Computation of Smallest Eigenvalues using Spectral Schur Complements
Computation of Smallest Eigenvalues using Spectral Schur Complements
复制标题
使用谱 Schur 补数计算最小特征值
DOI：
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复制
发表时间：
2005
影响因子：
3.1
通讯作者：
Y. Saad
中科院分区：
文献类型：
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作者：
C. Bekas;Y. Saad
自动多层子结构方法(AMLS)是最近提出的一种结构工程中计算大型矩阵特征值的方法的替代方法。该技术基于通过区域分解和投影方法进行高水平的降维。本文采用纯代数的方法，并解释了它可以看作是三个部分的组合：(A)逼近原特征问题在子域界面上的限制的非线性特征值问题的一阶展开式，(B)对应于子域内部的部分特征基上的明智投影，(C)递归性。这一观点使我们探索了使用Krylov子空间而不是特征基来构造逼近式子空间的方法的变体。非线性本征值问题的观点产生了一个二阶近似，作为AMLS固有的一阶技术的增强。数值实验结果验证了所提方法的有效性。
The automated multilevel substructuring method (AMLS) was recently presented as an alternative to well-established methods for computing eigenvalues of large matrices in the context of structural engineering. This technique is based on exploiting a high level of dimensional reduction via domain decomposition and projection methods. This paper takes a purely algebraic look at the method and explains that it can be viewed as a combination of three ingredients: (a) A first order expansion to a nonlinear eigenvalue problem that approximates the restriction of the original eigenproblem on the interface between the subdomains, (b) judicious projections on partial eigenbases that correspond to the interior of the subdomains, (c) recursivity. This viewpoint leads us to explore variants of the method which use Krylov subspaces instead of eigenbases to construct subspaces of approximants. The nonlinear eigenvalue problem viewpoint yields a second order approximation as an enhancement to the first order technique inherent to AMLS. Numerical experiments are reported to validate the approaches presented.