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-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 an 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.

💡 Research Summary

This paper presents a comprehensive theoretical and algorithmic framework for Preconditioned Riemannian Gradient (P-RG) methods to minimize the Gross-Pitaevskii (GP) energy functional, which is central to modeling rotating Bose-Einstein condensates. The minimization is constrained to the L^2 unit sphere, forming a Riemannian manifold.

The core challenge addressed is the inherent symmetry of the GP functional: it is invariant under global phase shifts (φ → e^{iα}φ) and, for rotationally symmetric potentials, under coordinate rotations (φ(x) → φ(A_β x)). These symmetries cause minimizers to form continuous manifolds rather than isolated points, complicating local convergence analysis.

The authors’ first major contribution is a unified framework for P-RG methods that encompasses existing projected Sobolev gradient approaches. Under mild assumptions on the preconditioning operator P(φ) (uniform boundedness and coercivity), they rigorously prove global convergence of the algorithm to the set of critical points.

The more significant and novel contribution lies in the local convergence analysis. Assuming the GP functional is Morse-Bott (its critical set is a non-degenerate submanifold) near a minimizer, the authors derive a sharp Polyak-Łojasiewicz (PL) inequality on a complement subspace N_φ M. This subspace is orthogonal (in L^2 sense) to the symmetry directions iφ and iL_zφ. The PL inequality holds on N_φ M and relates the function value decrease to the gradient norm, bypassing the need for isolated minimizers.

Leveraging this PL inequality, Theorem 4.2 precisely characterizes the local linear convergence rate of the P-RG method. The rate is shown to be (q = \sqrt{1 - μ/L} + ε), where μ and L are the lower and upper spectral bounds, respectively, of a composite operator involving the preconditioner P(φ_g) and the Hessian E’’(φ_g), restricted to the subspace N_{φ_g} M. This reveals that the convergence speed is governed by the condition number L/μ of this composite operator on that subspace.

This insight leads to the design of an optimal preconditioner. By performing a spectral analysis, the authors construct a preconditioner that minimizes the effective condition number L/μ on N_{φ_g} M. Theorem 4.3 establishes that the corresponding optimal local convergence rate is ((L - μ)/(L + μ) + ε), which is the best possible rate achievable by a Riemannian gradient method for this problem structure. To the authors’ knowledge, this is the first rigorous derivation of an optimal local convergence rate for preconditioned gradient methods applied to GP functionals with two continuous symmetries.

Finally, the theoretical findings are validated through extensive numerical experiments on rapidly rotating BECs in 2D and 3D. The performance of P-RG with different preconditioners (including the common choice P=H_φ, the linear part H_0, and the theoretically optimal one) is compared. The results consistently match the predicted convergence rates, demonstrating the practical effectiveness of the optimal preconditioner and confirming the entire theoretical framework.