自适应DIIS算法的收敛性分析及其在电子基态计算中的应用
Convergence analysis of adaptive DIIS algorithms with application to electronic ground state calculations
摘要：
本文讨论一类求解不动点问题的一般算法，我们称之为\emph{Anderson--Pulay acceleration}。这个家族包括DIIS技术和它的变体，有时被称为加速器-DIIS，两者都是由Pulay在20世纪80年代引入的，以加速量子化学中自洽场程序的收敛，以及相关的安德森加速，可以追溯到20世纪60年代，以及他们启发的丰富技术。这种方法旨在通过组合几个迭代以在每一步生成下一个迭代来加速任何定点迭代法的收敛。该外推过程的特征在于其\emph{depth}，即存储的先前迭代的数量，其是该方法的效率的关键参数。它通常固定为经验值。在目前的工作中，我们考虑两个参数驱动的机制，让深度沿迭代变化。在第一种方法中，深度不断增长，直到不再满足某个非退化条件;然后丢弃存储的迭代（保存最后一个），方法“重新启动”。在第二种方法中，我们通过在每一步消除一些最老的、不太相关的迭代来不断调整深度。在一个抽象的和一般的设置，我们证明在自然的假设下，这两个自适应安德森-Pulay方法的局部收敛和加速，我们表明，理论上可以实现超线性收敛速率与他们每个人。然后，我们研究它们在量子化学计算中的行为。这些数值实验表明，这两个自适应变量表现出更快的收敛速度比标准的固定深度的计划，平均每次迭代所需的计算量更少。这项研究的补充是回顾已知的事实上的DIIS，特别是它的链接与安德森加速和一些多割线型拟牛顿法。
英文摘要：
This paper deals with a general class of algorithms for the solution of fixed-point problems that we refer to as \emph{Anderson--Pulay acceleration}. This family includes the DIIS technique and its variant sometimes called commutator-DIIS, both introduced by Pulay in the 1980s to accelerate the convergence of self-consistent field procedures in quantum chemistry, as well as the related Anderson acceleration which dates back to the 1960s, and the wealth of techniques they have inspired. Such methods aim at accelerating the convergence of any fixed-point iteration method by combining several iterates in order to generate the next one at each step. This extrapolation process is characterised by its \emph{depth}, i.e. the number of previous iterates stored, which is a crucial parameter for the efficiency of the method. It is generally fixed to an empirical value. In the present work, we consider two parameter-driven mechanisms to let the depth vary along the iterations. In the first one, the depth grows until a certain nondegeneracy condition is no longer satisfied; then the stored iterates (save for the last one) are discarded and the method "restarts". In the second one, we adapt the depth continuously by eliminating at each step some of the oldest, less relevant, iterates. In an abstract and general setting, we prove under natural assumptions the local convergence and acceleration of these two adaptive Anderson--Pulay methods, and we show that one can theoretically achieve a superlinear convergence rate with each of them. We then investigate their behaviour in quantum chemistry calculations. These numerical experiments show that both adaptive variants exhibit a faster convergence than a standard fixed-depth scheme, and require on average less computational effort per iteration. This study is complemented by a review of known facts on the DIIS, in particular its link with the Anderson acceleration and some multisecant-type quasi-Newton methods.
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