Efficient spectral computation of the stationary states ofrotating Bose-Einstein condensates by preconditioned nonlinear conjugate gradient methods

JOURNAL OF COMPUTATIONAL PHYSICS

卷:343

Article

Antoine, Xavier¹  Levitt, Antoine^2,3  Tang, Qinglin^1,4 

[1] Univ Lorraine, Inst Elie Cartan Lorraine, UMR 7502, Inria Nancy Grand Est,SPHINX Team, F-54506 Vandoeuvre Les Nancy, France

[2] Inria Paris, F-75589 Paris 12, France

[3] Univ Paris Est, CERMICS ENPC, F-77455 Marne La Vallee, France

[4] Beijing Computat Sci Res Ctr, 10 East Xibeiwang Rd, Beijing 100193, Peoples R China

关键词: Bose-Einstein condensation;    Rotating Gross-Pitaevskii equation;    Stationary states;    Fourier spectral method;    Steepest descent;    Conjugate gradient;    Optimization algorithms on Riemannian manifolds;    Preconditioner;   

DOI  :  10.1016/j.jcp.2017.04.040

来源: Elsevier

【 摘 要 】

We propose a preconditioned nonlinear conjugate gradient method coupled with a spectral spatial discretization scheme for computing the ground states (GS) of rotating Bose-Einstein condensates (BEC), modeled by the Gross-Pitaevskii Equation (GPE). We first start by reviewing the classical gradient flow (also known as imaginary time(IMT)) method which considers the problem from the PDE standpoint, leading to numerically solve a dissipative equation. Based on this IMT equation, we analyze the forward Euler (FE), Crank-Nicolson (CN) and the classical backward Euler (BE) schemes for linear problems and recognize classical power iterations, allowing us to derive convergence rates. By considering the alternative point of view of minimization problems, we propose the preconditioned steepest descent (PSD) and conjugate gradient (PCG) methods for the GS computation of the GPE. We investigate the choice of the preconditioner, which plays a key role in the acceleration of the convergence process. The performance of the new algorithms is tested in 1D, 2D and 3D. We conclude that the PCG method outperforms all the previous methods, most particularly for 2D and 3D fast rotating BECs, while being simple to implement. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

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