Volume 55, Number 6, November-December 2021
|Page(s)||2785 - 2825|
|Published online||25 November 2021|
Convergence analysis of adaptive DIIS algorithms with application to electronic ground state calculations
CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, Université PSL, Place du Maréchal de Lattre de Tassigny, 75775 Paris, Cedex 16, France
2 Laboratoire Jacques-Louis Lions, UMR CNRS 7598, Sorbonne Université, boîte courrier 187, 75252 Paris, Cedex 05, France
* Corresponding author: firstname.lastname@example.org
Accepted: 20 October 2021
This paper deals with a general class of algorithms for the solution of fixed-point problems that we refer to as 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 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.
Mathematics Subject Classification: 65B99 / 65H10 / 65Z05 / 81-08 / 90C53
Key words: Fixed-point iteration / Anderson acceleration / DIIS / nonlinear GMRES / quantum chemistry
© The authors. Published by EDP Sciences, SMAI 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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