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.
翻译:本文涉及一个用于解决固定点问题的通用算法类别, 我们称之为 \ emph{ Anderson- Pulay 加速 } 。 此组包括 DIIS 技术及其变式, 有时称为 computator- DIIS, 由Pulay 于1980年代推出, 目的是加速量化学领域自相一致的实地程序的趋同, 以及相关的 Anderson 加速, 以及它们所启发的技术的丰富。 这种方法的目的是加速任何固定点递归方法的趋同。 此类方法的目的是通过合并多个迭代方法加速任何固定点递归趋同方法的趋同, 以便在每个步骤产生下一个步骤。 DII 的外推法过程以 \ emph{ 深度为特征, 也就是说, 先前的推算数是这个方法的关键参数, 这是计算方法的效率。 在目前的工作中, 我们考虑两个参数驱动机制, 使 的深度随迭代变化而变化而变化。 在第一个阶段, 深度, 直至某个非变性状态不再满足; 然后, 存储的精确的递化速度, 在一次变变变变变变, 。