This paper addresses the numerical solution of nonlinear eigenvector problems such as the Gross–Pitaevskii and Kohn–Sham equation arising in computational physics and chemistry. These problems characterize critical points of energy minimization problems on the infinite-dimensional Stiefel manifold. To efficiently compute minimizers, we propose a novel Riemannian gradient descent method induced by an energy-adaptive metric. Quantified convergence of the methods is established under suitable assumptions on the underlying problem. A non-monotone line search and the inexact evaluation of Riemannian gradients substantially improve the overall efficiency of the method. Numerical experiments illustrate the performance of the method and demonstrates its competitiveness with well-established schemes.
Keywords: Riemannian optimization, Stiefel manifold, Kohn–Sham model, Gross–Pitaevskii eigenvalue problem, nonlinear eigenvector problem
@article{M2AN_2022__56_5_1629_0,
author = {Altmann, Robert and Peterseim, Daniel and Stykel, Tatjana},
title = {Energy-adaptive {Riemannian} optimization on the {Stiefel} manifold},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1629--1653},
year = {2022},
publisher = {EDP-Sciences},
volume = {56},
number = {5},
doi = {10.1051/m2an/2022036},
mrnumber = {4454161},
zbl = {1502.65177},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2022036/}
}
TY - JOUR AU - Altmann, Robert AU - Peterseim, Daniel AU - Stykel, Tatjana TI - Energy-adaptive Riemannian optimization on the Stiefel manifold JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2022 SP - 1629 EP - 1653 VL - 56 IS - 5 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2022036/ DO - 10.1051/m2an/2022036 LA - en ID - M2AN_2022__56_5_1629_0 ER -
%0 Journal Article %A Altmann, Robert %A Peterseim, Daniel %A Stykel, Tatjana %T Energy-adaptive Riemannian optimization on the Stiefel manifold %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2022 %P 1629-1653 %V 56 %N 5 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2022036/ %R 10.1051/m2an/2022036 %G en %F M2AN_2022__56_5_1629_0
Altmann, Robert; Peterseim, Daniel; Stykel, Tatjana. Energy-adaptive Riemannian optimization on the Stiefel manifold. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 5, pp. 1629-1653. doi: 10.1051/m2an/2022036
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