arXiv subjects

Zixu Feng

Publications and source records attributed to Zixu Feng.

4 recordsLinked to original sources

arXiv2026

Structure and symmetry of the Gross-Pitaevskii ground-state manifold

The structure and degeneracy of ground states of the Gross-Pitaevskii energy functional play a central role in both analysis and computation, yet a precise characterization of the ground-state manifold in the presence of symmetries remains a fundamental challenge. In this paper, we establish sharp theoretical 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 intrinsic symmetries of the energy functional, namely phase shifts and spatial rotations. This yields a structural characterization of the ground-state manifold in terms of these symmetries. Building on this insight, we characterize the local convergence behavior of general preconditioned Riemannian gradient methods (P-RG). Under the Morse-Bott condition, we derive sharp local $Q$-linear convergence estimates 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 P-RG exhibits local linear convergence nearby. When the condition fails, we establish a local sublinear convergence rate. Taken together, these results show that the Morse-Bott condition is the exact threshold separating linear from sublinear convergence, while determining the symmetry-induced structure of the ground-state manifold, connecting geometry, symmetry, and convergence behavior in a unified framework.

math.NA↗

arXiv2026

Optimal local linear convergence of Nesterov's accelerated gradient method for $C^2$ functions under the Polyak--{\L}ojasiewicz inequality

In this work, we establish that Nesterov's accelerated gradient method, applied to $C^2$ functions satisfying the Polyak--{\L}ojasiewicz inequality around local minimizers, achieves the optimal local linear convergence rate $\rho=\frac{\sqrt{3L+\mu}-2\sqrt{\mu}}{\sqrt{3L+\mu}}+\varepsilon$, 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 $\sqrt{\mu}$ for the continuous-time Heavy Ball dynamics. Finally, a representative numerical experiment corroborates our theoretical findings.

math.OC↗

arXiv2025

Energy dissipation and global convergence of a discrete normalized gradient flow for computing ground states of two-component Bose-Einstein condensates

The gradient flow with semi-implicit discretization (GFSI) is the most widely used algorithm for computing the ground state of Gross-Pitaevskii energy functional. Numerous numerical experiments have shown that the energy dissipation holds when calculating the ground states of multicomponent Bose-Einstein condensates (MBECs) with GFSI, while rigorous proof remains an open challenge. By introducing a Lagrange multiplier, we reformulate the GFSI into an equivalent form and thereby prove the energy dissipation for GFSI in two-component scenario with Josephson junction and rotating term, which is one of the most important and topical model in MBECs. Based on this, we further establish the global convergence to stationary states. Also, the numerical results of energy dissipation in practical experiments corroborate our rigorous mathematical proof, and we numerically verified the upper bound of time step that guarantees energy dissipation is indeed related to the strength of particle interactions.

math.NA↗

arXiv2025

On preconditioned Riemannian gradient methods for minimizing the Gross-Pitaevskii energy functional: algorithms, global convergence and optimal local convergence rate

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 a quasi-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.

math.NA↗