[论文解读] Convergence analysis of adaptive DIIS algorithms with application to electronic ground state calculations

Maxime Chupin, Mi-Song Dupuy+2 more|arXiv (Cornell University)|Feb 28, 2020

Numerical methods for differential equations参考文献 67被引用 22

一句话总结

本文提出两种自适应深度策略用于Anderson–Pulay加速方法——基于重启的自适应与连续自适应策略——通过动态调整存储的迭代次数，以提升自洽场计算中的收敛性能。作者证明了在无需假设外推系数有界的条件下，局部超线性收敛性；数值结果表明，两种变体均优于固定深度方案，且平均计算成本更低。

ABSTRACT

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.

研究动机与目标

• 为解决固定深度DIIS与Anderson加速在电子结构计算中性能不佳的问题。
• 开发自适应机制，在迭代过程中动态调整深度（存储迭代次数），以提升收敛速度。
• 在弱于先前工作的假设下，对自适应Anderson–Pulay方法提供严格的理论收敛性分析。
• 通过数值实验表明，自适应深度可实现更快收敛速度与更低的平均计算成本，优于固定深度方案。
• 在统一框架下阐明DIIS、CDIIS与Anderson加速之间的理论联系。

提出的方法

• 提出一种基于重启的自适应深度机制：当非退化条件不成立时，丢弃较旧的迭代数据，从而有效重启历史记录。
• 引入一种新颖的连续自适应深度策略：在每一步中，根据新准则移除最旧且最不相关的迭代数据。
• 在抽象不动点设定下分析两种方法，证明在弱化假设下的局部收敛性与加速性。
• 通过将自适应机制直接嵌入算法中，建立外推系数的先验界，避免依赖于事后有界性假设。
• 将方法应用于量子化学中的自洽场（SCF）计算，以分子体系作为测试案例。
• 采用广义残差最小化框架，通过最小二乘法推导外推系数，并将深度控制整合到系数计算过程中。

实验结果

研究问题

• RQ1 Anderson–Pulay加速方法中的自适应深度控制是否能在电子结构计算中实现比固定深度方案更快的收敛速度？
• RQ2 所提出的自适应机制是否能确保理论收敛性，并避免因最小二乘问题中线性相关性导致的数值不稳定性？
• RQ3 在收敛速度与计算成本方面，自适应方法相较于标准DIIS与Anderson加速方法的性能如何？
• RQ4 与传统‘经验法则’取值相比，自适应深度机制对平均存储迭代次数有何影响？
• RQ5 理论收敛保证是否可扩展至DIIS与CDIIS方法，且在更弱的非退化性与有界性假设下依然成立？

主要发现

• 自适应深度机制可在无需外推系数先验有界性的条件下，确保局部超线性收敛。
• 两种自适应变体——基于重启与连续自适应——在电子基态计算中均比其固定深度对应方法收敛更快。
• 自适应深度变体表现尤为突出，表明其在实际应用中可能最具效率。
• 数值实验表明，存储迭代的平均深度显著低于标准实现中常用的固定值。
• 随着自适应参数（τ 和 δ）减小，收敛速率持续提升，但超过某一临界点后收益递减，最优性能出现在 10−4 附近。
• 即使参数取值远大于理论估计值，方法仍保持有效，表明其在实际应用中具有强鲁棒性。