In this article, we propose a unified framework for preconditioned Riemannian gradient (P-RG) methods to minimize Gross-Pitaevskii (GP) energy functionals with rotation on a Riemannian manifold. This framework enables comprehensive analysis of existing projected Sobolev gradient methods and facilitates the construction of highly efficient P-RG algorithms. Under mild assumptions on the preconditioner, we prove energy dissipation and global convergence. Local convergence is more challenging due to phase and rotational invariances. Assuming the GP functional is Morse-Bott, we derive a sharp Polyak-\L ojasiewicz (PL) inequality near minimizers. This allows precise characterization of the local convergence rate via the condition number $\mu/L$, where $\mu$ and $L$ are the lower and upper bounds of the spectrum of a combined operator (preconditioner and Hessian) on a closed subspace. By combining spectral analysis with the PL inequality, we identify an optimal preconditioner achieving the best possible local convergence rate: $(L-\mu)/(L+\mu)+\varepsilon$ ($\varepsilon>0$ small). To our knowledge, this is the first rigorous derivation of the local convergence rate for P-RG methods applied to GP functionals with two symmetry structures. Numerical experiments on rapidly rotating Bose-Einstein condensates validate the theoretical results and compare the performance of different preconditioners.

翻译：本文针对黎曼流形上含旋转的Gross-Pitaevskii（GP）能量泛函最小化问题，提出了预条件黎曼梯度（P-RG）方法的统一框架。该框架能够系统分析现有的投影Sobolev梯度方法，并为构建高效P-RG算法提供基础。在预条件子满足温和假设的前提下，我们证明了能量耗散特性与全局收敛性。由于相位不变性与旋转不变性的存在，局部收敛性分析更具挑战性。假设GP泛函满足Morse-Bott性质，我们在极小值点附近推导出尖锐的Polyak-Łojasiewicz（PL）不等式。这使我们能够通过条件数$\mu/L$精确刻画局部收敛速率，其中$\mu$和$L$是闭子空间上组合算子（预条件子与Hessian算子）谱的下界与上界。通过将谱分析与PL不等式相结合，我们确定了能实现最优局部收敛速率$(L-\mu)/(L+\mu)+\varepsilon$（$\varepsilon>0$为小量）的预条件子。据我们所知，这是首次对具有双重对称结构的GP泛函P-RG方法给出严格的局部收敛速率推导。针对快速旋转玻色-爱因斯坦凝聚体的数值实验验证了理论结果，并比较了不同预条件子的性能。