Convergence Estimates for the Generalized Davidson Method for Symmetric Eigenvalue Problems I: The Preconditioning Aspect
Convergence Estimates for the Generalized Davidson Method for Symmetric Eigenvalue Problems I: The Preconditioning Aspect
复制标题
对称特征值问题的广义戴维森方法的收敛性估计 I：预处理方面
DOI：
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发表时间：
2003
影响因子：
2.9
通讯作者：
E. Ovtchinnikov
中科院分区：
文献类型：
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作者：
E. Ovtchinnikov
广义Davidson(GD)方法可以看作是求解对称特征值问题的预条件最速下降(PSD)方法的推广。这种概括有两个方面。最明显的是，在GD方法中，新的近似是在一个更大的子空间中寻找的，即除了当前特征向量及其预条件残差之外，跨越所有先前近似特征向量的子空间。另一个方面与预适应有关。与线性系统的情况一样，PSD方法的大多数可用结果都与对预条件的看法相同。因此，他们没有检测到某些“理想”预条件的超线性收敛，例如对应于“精确”版本的Jacobi-Davidson方法的预条件，这是GD方法最熟悉的实例之一。本文从预条件的角度出发，提出了一种度量特征值问题预条件质量的方法，并给出了GD方法和Jacobi-Davidson方法的相应的非渐近收敛估计，特别是正确检测超线性收敛的已知情形。
The generalized Davidson (GD) method can be viewed as a generalization of the preconditioned steepest descent (PSD) method for solving symmetric eigenvalue problems. There are two aspects of this generalization. The most obvious one is that in the GD method the new approximation is sought in a larger subspace, namely the one that spans all the previous approximate eigenvectors, in addition to the current one and the preconditioned residual thereof. Another aspect relates to the preconditioning. Most of the available results for the PSD method are associated with the same view on preconditioning as in the case of linear systems. Consequently, they fail to detect the superlinear convergence for certain "ideal" preconditioners, such as the one corresponding to the "exact" version of the Jacobi--Davidson method---one of the most familiar instances of the GD method. Focusing on the preconditioning aspect, this paper advocates an alternative approach to measuring the quality of preconditioning for eigenvalue problems and presents corresponding non-asymptotic convergence estimates for the GD method in general and Jacobi--Davidson method in particular that correctly detect known cases of the superlinear convergence.