Energy-adaptive Riemannian optimization on the Stiefel manifold

TL;DR

This paper introduces an energy-adaptive Riemannian gradient descent method for solving nonlinear eigenvector problems on the Stiefel manifold, improving efficiency and convergence in computational physics and chemistry applications.

Contribution

It proposes a novel energy-adaptive metric-based Riemannian gradient descent method with convergence analysis and practical enhancements like non-monotone line search and inexact gradients.

Findings

Method demonstrates competitive performance in numerical experiments.

Energy-adaptive metric improves convergence efficiency.

Non-monotone line search enhances overall method robustness.

Abstract

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.