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A positive-conservative Fourier optimization method for ground states of Bose--Einstein condensates with higher-order interactions

Bo Lin, Xinran Ruan

arXiv:2609.38965v1math.NAcs.NA

TL;DR

Near zero density, singular kinetic terms and the potential slope complicate high-accuracy computation in the convex density formulation. The paper combines convex regularization, a mass-conservative positive Fourier discretization, and FISTA followed by Newton refinement. The regularized discretization shows spectral-type convergence, and smaller potential regularization reduces error at the cost of finer resolution.

  • Problem

    At zero density, the kinetic term is singular and the potential slope can activate positivity constraints and reduce Fourier-relevant regularity.

    Source: §2.1 · p. 4

  • Method

    The method regularizes the convex density energy, discretizes it with a Fourier pseudospectral method preserving mass and nodal positivity, then applies potential-proximal FISTA and Newton refinement.

    Source: §1 · p. 2, §3.2 · p. 8, §4.1 · p. 9

  • Results

    Fixed-regularization Fourier discretization shows spectral-type convergence; smaller potential regularization reduces error but requires finer resolution, while Newton refinement reduces stationarity residual after FISTA.

    Source: §6 · p. 18

  • Takeaways & Limitations

    On the computational domain, the regularized energies Γ-converge to the original density energy as the regularization parameters vanish.

    Source: §6 · p. 18

    A joint analysis of domain truncation, Fourier discretization, and regularization remains for future work.

    Source: §6 · p. 18

Abstract

from arXiv · show

We develop a positive-conservative Fourier optimization (PCFO) method for computing ground states of Bose--Einstein condensates with higher-order interactions. The ground-state problem admits a convex density formulation, but the singular behavior near zero density poses difficulties for high-accuracy computation. We introduce convex regularizations of the density energy and establish their $\Gamma$-convergence on the computational domain as the regularization parameters vanish. The regularized problem is discretized by a Fourier pseudospectral method with mass conservation and nodal positivity. For the resulting constrained optimization problem, we use an accelerated first-order method for the main energy reduction, followed by a Newton refinement to reduce the remaining optimality residual. Numerical experiments show spectral-type convergence for regularized problems, quantify the effects of the regularization parameters on accuracy and spatial resolution, and demonstrate the effectiveness of the two-stage optimization method.

1 Introduction

Higher-order interactions extend the Gross–Pitaevskii model, and their ground-state problem can be written as a convex density optimization despite a kinetic singularity at zero density. This work advances that framework with Fourier discretization, positivity and mass preservation, and a two-stage solver.

1 Introduction

  • Higher-order interactions arise as a correction beyond the contact-potential approximation and may matter under strong confinement or near narrow Feshbach resonances.Their competition with contact interactions can affect ground-state structures and asymptotic regimes.Source: §1 · pp. 1–2
  • The density formulation is strictly convex for β ≥ 0, but its kinetic term remains singular at zero density.A ground state may be chosen real and nonnegative, motivating ρ = |φ|^2.Source: §1 · p. 2
  • For the higher-order-interaction model, Bao and Ruan used convex density minimization with shifted kinetic regularization, finite differences, and accelerated projected gradient.The present work replaces the finite-difference discretization and improves the optimization solver.Source: §1 · p. 2
  • The method advances density-based ground-state computation with Fourier pseudospectral discretization, positivity and mass preservation, and a potential-proximal first-order method followed by Newton refinement.The regularizations preserve convexity and Γ-converge on the computational domain as their parameters vanish.Source: §1 · p. 2

2 Density formulation and convex regularization

The density energy is strictly convex, while its kinetic singularity and potential behavior near zero density motivate convex regularization. On the periodic computational domain, the joint regularizations are equicoercive and Γ-converge as their parameters vanish, with convergence of minimizers.

2.1 Convex regularization of the kinetic and potential terms

  • Admissible denominator regularizations extend the kinetic term to vacuum regions while keeping its integrand jointly convex.The admissibility conditions order the denominators and make them converge to s as ε vanishes.Source: §2.1 · p. 3
  • The original density energy is strictly convex for β ≥ 0 and δ > 0, so it has at most one minimizer.The kinetic integrand is treated as a lower-semicontinuous convex extension at zero density.Source: §2.1 · p. 3
  • Near zero density, the linear potential's nonzero slope can activate positivity constraints and reduce Fourier-relevant regularity, motivating replacement of ρ by pσ(ρ).The potential scale σ is independent of the kinetic regularization scale, and pσ(s) approaches s as σ vanishes.Source: §2.1 · p. 4
  • The regularized energy retains strict convexity on the admissible set and is strongly convex in L2 when β > 0.The regularization preserves the convex structure of the original density problem.Source: §2.1 · p. 4

2.2 Γ-convergence of the regularized energies

  • On the bounded periodic domain, the joint regularizations are equicoercive and Γ-converge as ε vanishes, provided σ(ε) also vanishes.The theorem assumes V is continuous and nonnegative, β ≥ 0, δ > 0, and an admissible denominator family.Source: §2.2 · p. 5
  • The potential smoothing is a uniformly vanishing perturbation and therefore does not change the Γ-limit.This allows the main convergence argument to be carried out for the kinetic regularization.Source: §2.2 · pp. 5–6
  • The Γ-convergence result yields convergence of minimum values and minimizers to those of the unregularized energy on the periodic domain.Uniqueness of the limiting minimizer gives convergence of the whole family, not only a subsequence.Source: §2.2 · p. 6

3 Fourier pseudospectral discretization

The regularized density problem is discretized on a periodic domain with a Fourier pseudospectral method. Its constrained optimization uses a nodal projection that preserves positivity and mass.

3 Fourier pseudospectral discretization

  • The periodic Fourier pseudospectral discretization uses a one-dimensional formulation, with multidimensional cases obtained by tensor-product extension.The first-derivative operator is evaluated using FFTs.Source: §3 · p. 7, §3.1 · p. 7

3.1 Discrete energy, gradient and Hessian structure

  • The discrete energy is convex on the feasible set, and for β > 0 it is strongly convex with a unique minimizer.Its Hessian is positive semidefinite, and positive definite when β > 0.Source: §3.1 · pp. 7–8

3.2 Positive mass-conservative projection

  • The constrained optimization uses a nodal projection that preserves nonnegativity and the unit-mass constraint.The projection's scalar multiplier is determined by the mass constraint.Source: §3.2 · pp. 8–9

4 Two-stage optimization method

The discrete convex minimization uses an accelerated proximal method to reduce energy, followed by Newton refinement to reduce the stationarity residual. The potential regularization is included in the proximal step alongside positivity and mass constraints.

4 Two-stage optimization method

  • The two-stage strategy first reduces energy with an accelerated proximal method, then refines the state through the KKT system to reduce stationarity residual.Because the objective and feasible set are convex, every KKT point is a global minimizer.Source: §4 · p. 9

4.1 Potential-proximal FISTA

  • Potential-proximal FISTA places the regularized potential and positivity and mass constraints in the proximal step.The potential term is included there because its curvature near zero can become large as σ decreases.Source: §4.1 · pp. 9–10

4.2 Newton refinement

  • Newton refinement starts from the FISTA output and uses damped updates from a linearized KKT system, with positivity-preserving damping and a residual line search.A Schur complement reduces each interior step to two linear systems with the same symmetric positive definite Hessian, solved by PCG.Source: §4.2 · pp. 11–12

5 Numerical experiments

The experiments assess Fourier spatial accuracy, regularization effects, solver performance, and two-dimensional ground states. They show spectral-type convergence under fixed regularization, trade-offs between regularization and resolution, and faster residual reduction with Newton refinement.

5.1 Spatial accuracy

  • With ε = σ = 10^-2, state and energy errors decrease rapidly with increasing empirical orders until reaching a numerical floor, consistent with spectral-type spatial convergence.The comparison state uses Nref = 8192.Source: §5.1 · p. 13, Table 5.1 · p. 13
  • The C∞ continuation removes the high-resolution algebraic error floor of the direct harmonic potential, whose observed order approaches two on the periodic domain.The two calculations are nearly indistinguishable on coarse and moderate meshes.Source: §5.1 · p. 13, Figure 5.1 · p. 13

5.2 Regularization effects

  • The potential-regularization transition sharpens as σ decreases, while both energy and state perturbations decrease approximately linearly with σ over the tested range.For small σ, the minimum density exhibits the same scaling, so the positive low-density background vanishes with σ.Source: §5.2 · p. 14, Table 5.2 · p. 14, Figure 5.2 · p. 14
  • Decreasing σ reduces regularization error but requires finer Fourier meshes for the same spatial accuracy; σ = 0 showed no spectral-type decay over tested resolutions.With σ = 0, the nonzero slope of the linear potential at zero density can activate positivity and reduce density regularity.Source: §5.2 · p. 15, Figure 5.3 · p. 15
  • The three kinetic regularizations have essentially the same asymptotic behavior: common-functional energy error approaches second order and state error is approximately first order.The piecewise C1 denominator modestly reduces error at larger ε, but neither piecewise option shows a clear asymptotic rate advantage over the shifted denominator.Source: §5.2 · p. 16, Figure 5.4 · p. 16

5.3 Efficiency tests

  • Newton refinement reduces the projected-gradient residual much faster than continued FISTA from the same switching state, after FISTA has rapidly reduced energy.The comparison uses the same switching state for both branches.Source: §5.3 · p. 17, Figure 5.5 · p. 17
  • Decreasing ε at fixed σ substantially increases FISTA iterations and total time, whereas decreasing σ at fixed ε changes computational cost only modestly over the tested range.The smallest ε run reaches the prescribed FISTA iteration limit and has a somewhat larger final residual.Source: Table 5.3 · p. 17, §5.3 · p. 18

5.4 Two-dimensional examples

  • In the anisotropic harmonic trap, stronger y-direction confinement yields a narrower density profile in that direction.Adding a square optical lattice creates multiple local wells and corresponding density modulation.Source: §5.4 · p. 18

6 Conclusion

The PCFO method combines convex regularization with Fourier discretization and two-stage optimization; numerical results show spectral-type convergence under fixed regularization and a tradeoff between regularization error and spatial resolution. A joint analysis of domain truncation, Fourier discretization, and regularization remains for future work.

6 Conclusion

  • The method regularizes kinetic and potential terms while preserving convexity, and its regularized energies Γ-converge to the original density energy on the computational domain as parameters vanish.Source: §6 · p. 18
  • With fixed regularization parameters, Fourier discretization shows spectral-type convergence, while lower potential regularization reduces regularization error but requires finer Fourier resolution.FISTA provides the main energy reduction, and Newton refinement further reduces the stationarity residual to high accuracy.Source: §6 · p. 18
  • A joint analysis of domain truncation, Fourier discretization, and regularization remains for future work.Source: §6 · p. 18
  • Two-dimensional density examples use a 512 × 512 grid on [−16, 16)², with an anisotropic harmonic trap and, in one case, a square optical lattice.Both computations use ε = 10⁻², σ = 10⁻⁴, and β = δ = 10.Source: Figure 5.6 · p. 19

A Implementation of the Newton refinement

The appendix specifies Hessian actions for the Newton refinement and a PCG preconditioner. It states that the preconditioner is symmetric positive definite in the strongly convex regime.

A Implementation of the Newton refinement

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