相关特征值序列问题的块迭代特征解
Block Iterative Eigensolvers for Sequences of Correlated Eigenvalue Problems
摘要：
在基于LAPW方法的密度泛函理论模拟中，每个自洽场周期包括数十个大的密集广义本征问题。与实空间方法相反，在不同的周期上求解问题的特征对被认为是独立的，或者至多是非常松散的连接。在最近的一项研究[7]中，有人证明，与信念相反，序列中的连续特征值问题彼此之间有很强的相关性。特别是，通过监测连续特征值问题的特征向量之间的子空间角度，结果表明，这些角度显着减少后的前几次迭代，变得接近共线。这最后一个结果表明，我们可以操纵的特征向量，解决一个特定的特征问题的序列，作为一个近似的解决方案，为以下特征问题。在这项工作中，我们提出的结果是符合这种直觉。我们提供了数值示例，其中适当选择的块迭代特征解算器通过实现大幅加速而受益于特征向量的重用。提出的结果最终将开辟一条道路，广泛使用的块迭代本征解的从头计算电子结构代码的基础上的LAPW方法。
英文摘要：
In Density Functional Theory simulations based on the LAPW method, each self-consistent field cycle comprises dozens of large dense generalized eigenproblems. In contrast to real-space methods, eigenpairs solving for problems at distinct cycles have either been believed to be independent or at most very loosely connected. In a recent study [7], it was demonstrated that, contrary to belief, successive eigenproblems in a sequence are strongly correlated with one another. In particular, by monitoring the subspace angles between eigenvectors of successive eigenproblems, it was shown that these angles decrease noticeably after the first few iterations and become close to collinear. This last result suggests that we can manipulate the eigenvectors, solving for a specific eigenproblem in a sequence, as an approximate solution for the following eigenproblem. In this work we present results that are in line with this intuition. We provide numerical examples where opportunely selected block iterative eigensolvers benefit from the reuse of eigenvectors by achieving a substantial speed-up. The results presented will eventually open the way to a widespread use of block iterative eigensolvers in ab initio electronic structure codes based on the LAPW approach.
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