论文标题
与隐式放气的块预处理最陡后的特定层的收敛分析
Convergence analysis of a block preconditioned steepest descent eigensolver with implicit deflation
论文作者
论文摘要
可以通过使用预处理和放气技术来加速解决赫尔米尔特征值问题的梯度型迭代方法。具有隐式通缩（PSD-ID）的预处理最陡的下降迭代就是这样的方法之一。最近，根据萨莫基什对预处理的最陡下降法（PSD）的开创性工作，对PSD-ID的收敛行为进行了研究。由此产生的非反应估计表明，在最初的猜测中，PSD-ID的超线性收敛性。本文在较弱的假设下利用了Neymeyr对PSD进行的替代收敛分析。我们使用PSD-ID的有限公式将Neymeyr的方法嵌入了PSD-ID的分析中。更重要的是，我们将PSD-ID的新收敛分析扩展到PSD-ID或BPSD-ID的实际优选块版本，并显示BPSD-ID的群集鲁棒性。提供数值示例以验证理论估计。
Gradient-type iterative methods for solving Hermitian eigenvalue problems can be accelerated by using preconditioning and deflation techniques. A preconditioned steepest descent iteration with implicit deflation (PSD-id) is one of such methods. The convergence behavior of the PSD-id is recently investigated based on the pioneering work of Samokish on the preconditioned steepest descent method (PSD). The resulting non-asymptotic estimates indicate a superlinear convergence of the PSD-id under strong assumptions on the initial guess. The present paper utilizes an alternative convergence analysis of the PSD by Neymeyr under much weaker assumptions. We embed Neymeyr's approach into the analysis of the PSD-id using a restricted formulation of the PSD-id. More importantly, we extend the new convergence analysis of the PSD-id to a practically preferred block version of the PSD-id, or BPSD-id, and show the cluster robustness of the BPSD-id. Numerical examples are provided to validate the theoretical estimates.
