Riemannian optimisation methods for ground states of multicomponent Bose-Einstein condensates

TL;DR

This paper develops Riemannian optimization algorithms to compute ground states of multicomponent Bose-Einstein condensates, demonstrating their theoretical properties and computational efficiency through numerical experiments.

Contribution

It introduces Riemannian gradient descent and Newton methods tailored for the energy minimization problem on a manifold, with convergence analysis and practical performance evaluation.

Findings

Methods are globally convergent and robust.

Energy-adaptive metrics improve convergence speed.

Numerical results show high computational efficiency.

Abstract

This paper addresses the computation of ground states of multicomponent Bose-Einstein condensates, defined as the global minimiser of an energy functional on an infinite-dimensional generalised oblique manifold. We establish the existence of the ground state, prove its uniqueness up to scaling, and characterise it as the solution to a coupled nonlinear eigenvector problem. By equipping the manifold with several Riemannian metrics, we introduce a suite of Riemannian gradient descent and Riemannian Newton methods. Metrics that incorporate first- or second-order information about the energy are particularly advantageous, effectively preconditioning the resulting methods. For a Riemannian gradient descent method with an energy-adaptive metric, we provide a qualitative global and quantitative local convergence analysis, confirming its reliability and robustness with respect to the choice of the spatial discretisation. Numerical experiments highlight the computational efficiency of both the Riemannian gradient descent and Newton methods.