Convergence Analysis of a Locally Accelerated Preconditioned Steepest Descent Method for Hermitian-Definite Generalized Eigenvalue Problems
Convergence Analysis of a Locally Accelerated Preconditioned Steepest Descent Method for Hermitian-Definite Generalized Eigenvalue Problems
复制标题
厄米定广义特征值问题的局部加速预条件最速下降法的收敛性分析
DOI：
10.4208/jcm.1703-m2016-0580
复制
发表时间：
2016
影响因子：
0.9
通讯作者：
N. Sukumar
中科院分区：
文献类型：
--
作者：
Yunfeng Cai;Z. Bai;J. Pask;N. Sukumar
通过推广Samokish，Faddeev和Faddeeva，Longsine和McCormick等人的经典分析技巧，证明了求解Hermite定广义特征值问题的预条件隐收缩最速下降法（PSD-id）的收敛性.此外，我们得到的\PSDID方法的收敛速度的非渐近估计。我们表明，与适当的选择的移位，不确定的移位和反转预条件是一个局部加速的预条件，是渐近最优的，导致超线性收敛。数值算例验证了PSDID方法求解电子结构计算中的病态Hermitian定广义本征值问题的收敛性.虽然严格的和全面的收敛证明预条件块最速下降法在实际应用中仍然在很大程度上逃避我们，我们相信本文提出的理论结果揭示了这些块方法的收敛行为的一个更好的理解。
By extending the classical analysis techniques due to Samokish, Faddeev and Faddeeva, and Longsine and McCormick among others, we prove the convergence of preconditioned steepest descent with implicit deflation (PSD-id) method for solving Hermitian-definite generalized eigenvalue problems. Furthermore, we derive a nonasymptotic estimate of the rate of convergence of the \psdid method. We show that with the proper choice of the shift, the indefinite shift-and-invert preconditioner is a locally accelerated preconditioner, and is asymptotically optimal that leads to superlinear convergence. Numerical examples are presented to verify the theoretical results on the convergence behavior of the \psdid method for solving ill-conditioned Hermitian-definite generalized eigenvalue problems arising from electronic structure calculations. While rigorous and full-scale convergence proofs of preconditioned block steepest descent methods in practical use still largely eludes us, we believe the theoretical results presented in this paper sheds light on an improved understanding of the convergence behavior of these block methods.