Zixu Feng

Zixu Feng, Qinglin Tang

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-Łojasiewicz (PL) inequality near minimizers. This allows precise characterization of the local convergence rate via the condition number $μ/L$, where $μ$ 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 a quasi-optimal preconditioner achieving the best possible local convergence rate: $(L-μ)/(L+μ)+\varepsilon$ ($\varepsilon>0$ 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.

Zixu Feng, Patrick Henning, Qinglin Tang

The structure and degeneracy of ground states of the Gross-Pitaevskii energy functional play a central role in both analysis and computation, yet a characterization of the ground-state manifold in the presence of symmetries remains a fundamental challenge. In this paper, we establish sharp results describing the geometric structure of local minimizers and its implications for optimization algorithms. We show that when local minimizers are non-unique, the Morse-Bott condition provides a natural and sufficient criterion under which the ground-state set partitions into finitely many embedded submanifolds, each coinciding with an orbit generated by the intrinsic symmetries of the energy functional, namely phase shifts and spatial rotations. This yields a structural characterization of the ground-state manifold purely in terms of these natural symmetries. Building on this geometric insight, we analyze the local convergence behavior of the preconditioned Riemannian gradient method (P-RG). Under the Morse-Bott condition, we derive the optimal local $Q$-linear convergence rate and prove that the condition holds if and only if the energy sequence generated by P-RG converges locally $Q$-linearly. In particular, on the ground-state set, the Morse-Bott condition is satisfied if and only if the minimizers decompose into finitely many symmetry orbits and the P-RG exhibits local linear convergence in a neighborhood of this set. When the condition fails, we establish a local sublinear convergence rate. Taken together, these results provide a precise picture: for the Gross-Pitaevskii minimization problem, the Morse-Bott condition acts as the exact threshold separating linear from sublinear convergence, while simultaneously determining the symmetry-induced structure of the ground-state manifold. Our analysis thus connects geometric structure, symmetry, and algorithmic performance in a unified framework.

Zixu Feng, Hao Yuan

In this work, we establish that Nesterov's accelerated gradient method, applied to $C^2$ functions satisfying the Polyak--Łojasiewicz inequality around local minimizers, achieves the optimal local linear convergence rate $ρ=\frac{\sqrt{3L+μ}-2\sqrtμ}{\sqrt{3L+μ}}+\varepsilon$, where $\varepsilon$ is an arbitrarily small constant. Our analysis requires neither higher-order smoothness beyond $C^2$ of the objective function nor any additional geometric regularity of the submanifold of local minimizers. The key novelty lies in a two-stage argument: we first establish a coarse yet valid local linear convergence rate and then, building upon this a priori convergence guarantee, obtain a refined characterization of the linearized iteration operator, which yields the optimal rate. As a result, we only need to slightly strengthen the standard $C^{1,1}$ assumption, which is commonly required in theoretical analyses of linear convergence for first-order methods, to $C^2$ smoothness. Moreover, the same analytical framework allows us to recover, under identical conditions, the optimal local exponential convergence rate $\sqrtμ$ for the continuous-time Heavy Ball dynamics. Finally, a representative numerical experiment corroborates our theoretical findings.