Convergence of a Riemannian gradient method for the Gross-Pitaevskii energy functional in a rotating frame

TL;DR

The paper develops and analyzes an energy-adaptive Riemannian gradient method for computing ground states of rotating Bose-Einstein condensates by minimizing the Gross-Pitaevskii energy $E$ on the $L^2$-unit sphere. It leverages Sobolev gradients with an adaptively changing metric so that $\nabla_X E(u)=u$, and proves global energy dissipation and convergence of densities to critical points, even when minimizers are not isolated due to gauge symmetry. Locally, the convergence rates are characterized through a weighted eigenvalue problem tied to the first spectral gap of $E''(u)|_{T_u\mathbb{S}}$, and an auxiliary tangent-space iteration enables a precise rate analysis via Ostrowski-type theorems. Numerical experiments confirm both global convergence and the predicted local rates, and demonstrate the practical efficiency of the energy-adaptive metric over standard Sobolev-gradients in achieving faster convergence to ground states in a rotating frame.

Abstract

This paper investigates the numerical approximation of ground states of rotating Bose-Einstein condensates. This problem requires the minimization of the Gross-Pitaevskii energy $E$ on a Hilbert manifold $\mathbb{S}$. To find a corresponding minimizer $u$, we use a generalized Riemannian gradient method that is based on the concept of Sobolev gradients in combination with an adaptively changing metric on the manifold. By a suitable choice of the metric, global energy dissipation for the arising gradient method can be proved. The energy dissipation property in turn implies global convergence to the density $|u|^2$ of a critical point $u$ of $E$ on $\mathbb{S}$. Furthermore, we present a precise characterization of the local convergence rates in a neighborhood of each ground state $u$ and how these rates depend on the first spectral gap of $E^{\prime\prime}(u)$ restricted to the $L^2$-orthogonal complement of $u$. With this we establish the first convergence results for a Riemannian gradient method to minimize the Gross-Pitaevskii energy functional in a rotating frame. At the same, we refine previous results obtained in the case without rotation. The major complication in our new analysis is the missing isolation of minimizers, which are at most unique up to complex phase shifts. For that, we introduce an auxiliary iteration in the tangent space $T_{\mathrm{i} u} \mathbb{S}$ and apply the Ostrowski theorem to characterize the asymptotic convergence rates through a weighted eigenvalue problem. Afterwards, we link the auxiliary iteration to the original Riemannian gradient method and bound the spectrum of the weighted eigenvalue problem to obtain quantitative convergence rates. Our findings are validated in numerical experiments.

Figures (3)

• Figure 1: Left: The ground state density $|u|^2$. Right: Number of iterations versus the energy error $E(u^n)-E(u)$ for the Riemannian gradient method with adaptive $\tau$ and fixed $\tau=1$.
• Figure 2: Number of iterations versus the energy error $E(u^n)-E(u)$ for the Riemannian gradient method with $a_u$-adaptive metric and fixed $H^1$-metric.
• Figure 3: Left: Comparison of contraction rates for the inverse iteration (i.e. iteration \ref{['method']} with $\tau_n=1$), with $|\mu_1|$ and the first spectral gap of $E^{\prime\prime}(u)\vert_{T_{u}\mathbb{S}}$. The results are consistent with \ref{['estimate-for-inverse-iteration']}. Right: Five largest eigenvalues in magnitude of the weighted eigenvalue problem \ref{['weighted-evp1']}.

Theorems & Definitions (38)

• Proposition 2.1: Existence of ground states
• Definition 2.2: Locally quasi-unique ground state
• Lemma 2.3
• proof
• Definition 3.1: Energy-adaptive Riemannian gradient method
• Remark 3.2: Optimal step length
• Remark 3.3: Continuous gradient flow
• Theorem 3.4: Global energy-dissipation
• Lemma 3.5
• proof