On preconditioned Riemannian gradient methods for minimizing the Gross-Pitaevskii energy functional: algorithms, global convergence and optimal local convergence rate
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| Format: | Preprint |
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2025
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| author | Feng, Zixu Tang, Qinglin |
| author_facet | Feng, Zixu Tang, Qinglin |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_13516 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On preconditioned Riemannian gradient methods for minimizing the Gross-Pitaevskii energy functional: algorithms, global convergence and optimal local convergence rate Feng, Zixu Tang, Qinglin Numerical Analysis 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. |
| title | On preconditioned Riemannian gradient methods for minimizing the Gross-Pitaevskii energy functional: algorithms, global convergence and optimal local convergence rate |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2510.13516 |