2020/02/28 by Maxime Chupin, Mi-Song Dupuy, Chupin, Maxime +5 · 2 citations
Computer Science · Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2002.12850
openalex publication_date 2020/02/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper deals with a general class of algorithms for the solution of\nfixed-point problems that we refer to as \Anderson--Pulay acceleration.\nThis family includes the DIIS technique and its variant sometimes called\ncommutator-DIIS, both introduced by Pulay in the 1980s to accelerate the\nconvergence of self-consistent field procedures in quantum chemistry, as well\nas the related Anderson acceleration which dates back to the 1960s, and the\nwealth of techniques they have inspired. Such methods aim at accelerating the\nconvergence of any fixed-point iteration method by combining several iterates\nin order to generate the next one at each step. This extrapolation process is\ncharacterised by its \depth, i.e. the number of previous iterates stored,\nwhich is a crucial parameter for the efficiency of the method. It is generally\nfixed to an empirical value. In the present work, we consider two\nparameter-driven mechanisms to let the depth vary along the iterations. In the\nfirst one, the depth grows until a certain nondegeneracy condition is no longer\nsatisfied; then the stored iterates (save for the last one) are discarded and\nthe method "restarts". In the second one, we adapt the depth continuously by\neliminating at each step some of the oldest, less relevant, iterates. In an\nabstract and general setting, we prove under natural assumptions the local\nconvergence and acceleration of these two adaptive Anderson--Pulay methods, and\nwe show that one can theoretically achieve a superlinear convergence rate with\neach of them. We then investigate their behaviour in quantum chemistry\ncalculations. These numerical experiments show that both adaptive variants\nexhibit a faster convergence than a standard fixed-depth scheme, and require on\naverage less computational effort per iteration. This study is complemented by\na review of known facts on the DIIS, in particular its link with the Anderson\nacceleration and some multisecant-type quasi-Newton methods.\n