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IMA Journal of Numerical Analysis

Riemannian optimization methods for ground states of multicomponent Bose–Einstein condensates

Robert Altmann, Martin Hermann, Daniel Peterseim, Tatjana Stykel

Published Apr 17, 2025• 1 citation
Publisher

Abstract

Abstract This paper addresses the computation of ground states of multicomponent Bose–Einstein condensates, defined as the global minimizer 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 characterize 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 discretization. Numerical experiments highlight the computational efficiency of both the Riemannian gradient descent and Newton methods.

DOI
10.1093/imanum/draf046

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Abstract This paper addresses the computation of ground states of multicomponent Bose–Einstein condensates, defined as the global minimizer 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 characterize 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 discretization. Numerical experiments highlight the computational efficiency of both the Riemannian gradient descent and Newton methods.
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We develop and analyze Riemannian optimization methods for computing ground states of rotating multicomponent Bose-Einstein condensates, defined as minimizers of the Gross-Pitaevskii energy functional. To resolve the non-uniqueness of ground states induced by phase invariance, we work on a quotient manifold endowed with a general Riemannian metric. By introducing an auxiliary phase-aligned iteration and employing fixed-point convergence theory, we establish a unified local convergence framework for Riemannian gradient descent methods and derive explicit convergence rates. Specializing this framework to two metrics tailored to the energy landscape, we study the energy-adaptive and Lagrangian-based Riemannian gradient descent methods. While monotone energy decay and global convergence are established only for the former, a quantified local convergence analysis is provided for both methods. Numerical experiments confirm the theoretical results and demonstrate that the Lagrangian-based method, which incorporates second-order information on the energy functional and mass constraints, achieves faster local convergence than the energy-adaptive scheme.
Abstract. In this paper, we propose a preconditioned Riemannian conjugate gradient method for computing ground states of spin-1 Bose-Einstein condensates, which can be reformulated as an optimization problem with the mass and magnetization constraints. The energy functional and constraints can be discretized using Fourier pseudospectral schemes, thereby transforming the problem into an optimization task on a manifold. We derive vector transports by differentiating three existing retractions and propose an initial step size selection strategy based on the second-order approximation of the energy function for the proposed Riemannian optimization algorithm. In addition, a preconditioner is derived and used. To further accelerate the convergence, we combine the proposed algorithm with a multigrid method. Numerical experiments demonstrate the efficiency and accuracy of the proposed method while confirming the usefulness of our preconditioning technique and the effectiveness of the proposed strategy for initial step size selection.
Abstract. The computation of the ground state of special multicomponent Bose–Einstein condensates (BECs) can be formulated as an energy functional minimization problem with spherical constraints. It leads to a nonconvex quartic-quadratic optimization problem after suitable discretizations. First, we generalize the Newton-based methods for single-component BECs to the alternating minimization scheme for multicomponent BECs. Second, the global convergent alternating Newton-Noda iteration (ANNI) is proposed. In particular, we prove the positivity preserving property of ANNI under mild conditions. Finally, our analysis is applied to a class of more general “multiblock” optimization problems with spherical constraints. Numerical experiments are performed to evaluate the performance of proposed methods for different multicomponent BECs, including pseudo spin-1/2, antiferromagnetic spin-1 and spin-2 BECs. These results support our theory and demonstrate the efficiency of our algorithms.
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ABSTRACT In this paper, we present a highly accurate discretization model for directly computing the ground states of Bose–Einstein condensates (BECs). The energy functional and constraint are discretized by introducing the fourth‐order compact finite difference and the composite Simpson quadrature, and thus the original infinite‐dimensional constrained minimization problem is approximated by a finite‐dimensional minimization problem with a spherical constraint uniformly. We develop a projected Riemannian gradient (PRG) algorithm to solve the optimization problem and establish the corresponding convergence results. Extensive numerical tests of non‐rotating BECs as well as rotating BECs under different potentials and parameter regimes are provided to verify the feasibility and high accuracy of our method. In addition, we apply it to investigate the ground states of multi‐component BECs and the numerical results again validate the power of this method.