| Issue |
ESAIM: M2AN
Volume 56, Number 5, September-October 2022
|
|
|---|---|---|
| Page(s) | 1629 - 1653 | |
| DOI | https://doi.org/10.1051/m2an/2022036 | |
| Published online | 20 July 2022 | |
Energy-adaptive Riemannian optimization on the Stiefel manifold
1
Institute of Mathematics, University of Augsburg, Universitätsstr. 12a, 86159 Augsburg, Germany
2
Institute of Mathematics & Centre for Advanced Analytics and Predictive Sciences (CAAPS), University of Augsburg, Universitätsstr. 12a, 86159 Augsburg, Germany
* Corresponding author: robert.altmann@math.uni-augsburg.de
Received:
23
August
2021
Accepted:
8
April
2022
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.
Mathematics Subject Classification: 65N25 / 81Q10
Key words: Riemannian optimization / Stiefel manifold / Kohn–Sham model / Gross–Pitaevskii eigenvalue problem / nonlinear eigenvector problem
© The authors. Published by EDP Sciences, SMAI 2022
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