This work considers the numerical computation of ground states of rotating Bose-Einstein condensates (BECs) which can exhibit a multiscale lattice of quantized vortices. This problem involves the minimization of an energy functional on a Riemannian manifold. For this we apply the framework of nonlinear conjugate gradient methods in combination with the paradigm of Sobolev gradients to investigate different metrics. Here we build on previous work that proposed to enhance the convergence of regular Riemannian gradients methods by an adaptively changing metric that is based on the current energy. In this work, we extend this approach to the branch of Riemannian conjugate gradient (CG) methods and investigate the arising schemes numerically. Special attention is given to the selection of the momentum parameter in search direction and how this affects the performance of the resulting schemes. As known from similar applications, we find that the choice of the momentum parameter plays a critical role, with certain parameters reducing the number of iterations required to achieve a specified tolerance by a significant factor. Besides the influence of the momentum parameters, we also investigate how the methods with adaptive metric compare to the corresponding realizations with a standard $H^1_0$-metric. As one of our main findings, the results of the numerical experiments show that the Riemannian CG method with the proposed adaptive metric along with a Polak-Ribi\'ere or Hestenes-Stiefel-type momentum parameter show the best performance and highest robustness compared to the other CG methods that were part of our numerical study.

翻译：本文研究旋转玻色-爱因斯坦凝聚体（BECs）基态的数值计算问题，此类体系可呈现具有多尺度结构的量子化涡旋晶格。该问题涉及在黎曼流形上对能量泛函进行最小化。为此，我们采用非线性共轭梯度方法的框架，结合索博列夫梯度范式来研究不同的度量。本研究基于先前工作，该工作提出通过基于当前能量的自适应变化度量来增强常规黎曼梯度方法的收敛性。本文将此方法推广至黎曼共轭梯度（CG）方法分支，并对所提出的方案进行数值研究。特别关注搜索方向中动量参数的选择及其对算法性能的影响。正如类似应用中所知，我们发现动量参数的选择至关重要，某些参数可将达到指定容差所需的迭代次数显著减少。除动量参数的影响外，我们还研究了采用自适应度量的方法与标准 $H^1_0$ 度量下相应实现方式的比较。作为主要发现之一，数值实验结果表明：相较于本数值研究中涉及的其他共轭梯度方法，采用所提自适应度量并结合Polak-Ribiére或Hestenes-Stiefel型动量参数的黎曼共轭梯度方法表现出最佳性能和最高鲁棒性。