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Riemannian Hessian analysis of Sobolev gradient flows for the Gross--Pitaevskii ground state problem

Shixin Zheng, Xiangxiong Zhang

arXiv:2610.12833v1math.NAmath.OCphysics.comp-ph

TL;DR

The Gross–Pitaevskii ground state computation becomes harder with finer meshes, larger domains, and stronger interactions. This paper analyzes Sobolev gradient flows using Riemannian Hessian conditioning and introduces new metrics. In tests, the Thomas–Fermi flow uses the fewest iterations, while separable metrics are fastest in computational time.

  • Problem

    Computing the discrete ground state becomes harder as the mesh is refined, the box enlarged, or the interaction strength increased.

    Source: §1.1 · p. 2

  • Method

    The paper bounds the Riemannian Hessian's Rayleigh quotient at the ground state to compare metric-induced Sobolev gradient flows and their convergence.

    Source: §1.3 · p. 4

  • Results

    The Thomas–Fermi flow needs the fewest iterations, while the shifted Laplacian and separable-potential metric are fastest in computational time because their iterations are cheapest.

    Source: §6 · p. 28

  • Takeaways & Limitations

    In three-dimensional tests with up to 10^9 unknowns, the Thomas–Fermi flow used 8 iterations versus 35 for au, while separable modified H1 and modified a0 operators were fastest in wall-clock time.

    Source: §1.3 · p. 4

    The Thomas–Fermi metric is smooth near the ground state only when no node satisfies Vi = λ1(Aβ); at equality it is Lipschitz but not differentiable there.

    Source: §3.1 · p. 9

Abstract

from arXiv · show

Sobolev gradient flows for the Gross--Pitaevskii ground state problem are Riemannian gradient descent methods on the sphere under a metric induced by a discrete elliptic operator. We analyze their convergence through the condition number of the Riemannian Hessian at the ground state. Bounds of the Rayleigh quotient of the Riemannian Hessian give a uniform analysis of these methods for different metrics. The uniform analysis applies to the classical metrics, such as the shifted Laplacian and the energy-adaptive metrics. Inspired by this analysis, we also consider three new metrics. One of them is a Thomas--Fermi metric, whose condition number is bounded uniformly in the interaction strength for the one-dimensional harmonic trap and whose iteration count does not grow in the Thomas--Fermi regime in the numerical tests. The performance suggested by our analysis is consistent with three-dimensional numerical tests with up to one billion unknowns, for both Riemannian gradient descent and Riemannian conjugate gradient methods, whose practical performance is also related to the Hessian condition number. The Thomas--Fermi flow needs the fewest iterations, while the metrics inverted exactly by tensor-product solvers are the fastest in computational time on one GPU card.

1. Introduction

The Gross–Pitaevskii ground state problem becomes computationally harder as the mesh is refined, the box enlarged, or the interaction strengthened. The paper analyzes Sobolev gradient flows through the Riemannian Hessian, deriving metric comparisons and introducing variants for improved conditioning and practical performance.

1.1. The Gross–Pitaevskii ground state

  • The fully discretized problem depends on mesh size h, box size L, and coupling β; computation becomes harder as h decreases, L increases, or β increases.The box truncation approximates the whole-space problem, and large β is the physical regime.Source: §1.1 · pp. 1–2

1.2. Sobolev gradient flows as Riemannian gradient descent

  • Sobolev gradient descent projects a metric-induced Sobolev gradient onto the sphere's tangent space and renormalizes; the au metric makes the gradient equal to u, avoiding a solve.Other metrics require elliptic solves for the gradient and projection; the paper's three new metrics are variants inspired by its Hessian analysis.Source: §1.2 · pp. 2–3

1.3. Main results and related work

  • The Riemannian Hessian has the same quadratic form across metrics, enabling Rayleigh-quotient bounds, condition-number comparisons, and local convergence-rate estimates.The paper also proves convergence of the whole iterate sequence to a critical point from any initial value.Source: §1.3 · pp. 3–4
  • The Thomas–Fermi metric has a condition number bounded uniformly in β for the one-dimensional harmonic trap, while the Thomas–Fermi flow needs 8 iterations versus 35 for the au flow in three-dimensional tests.Its iteration count also does not grow over a factor of 64 in β; modified H1 and modified a0 are fastest in wall-clock time because their operators are separable.Source: §1.3 · p. 4

1.4. Organization of the paper

  • The paper proceeds from the discrete problem and metrics through Hessian analysis and convergence, then reports three-dimensional experiments and concludes.Appendix A compares bounds with exact spectra; Appendix B covers the connection, arbitrary-point Hessian, and vector transport.Source: §1.4 · p. 5

2. Riemannian gradient descent

The method discretizes the energy and minimizes it on a sphere in the resulting finite-dimensional space. It equips that sphere with metrics induced by positive elliptic operators and applies projected, normalized Riemannian gradient descent.

2.1. Discretization

  • The discretization produces a finite-dimensional energy on R^N, with the discrete problem constrained to the unit sphere; monotone schemes yield a unique positive ground state up to sign.The monotonicity assumption includes second-order finite differences in any dimension and mass-lumped P1 elements under the stated mesh condition.Source: §2.1 · pp. 5–6

2.2. Riemannian metrics

  • Each metric is specified by a positive self-adjoint operator B(u); six choices are elliptic operators whose inverses are applied through sparse linear solves.The modified H1 and modified a0 operators have separable diagonals and admit exact tensor-product direct solvers on rectangular Qk meshes.Source: §2.2 · pp. 6–7

2.3. The manifold and the iteration

  • The unit sphere has tangent space {ξ : ⟨ξ,u⟩h = 0}; descent projects the ambient gradient onto it, takes a step, and renormalizes with a retraction.The retraction replaces the exponential map, which has no closed form for the u-dependent metrics.Source: §2.3 · p. 7

3. Riemannian Hessian analysis

This section analyzes ground-state convergence using Riemannian Hessian Rayleigh-quotient bounds, deriving bounds for multiple metrics and introducing modified and Thomas–Fermi metrics. It establishes parameter conditions for a uniform-in-β condition-number bound for the Thomas–Fermi metric, notes that it is not uniformly equivalent to A_β, and tests predicted orders against exact Hessian spectra in one and three dimensions.

3.1. The Hessian at a critical point

  • At the ground state, the second variation is metric-independent; when the Thomas–Fermi metric is nonsmooth, the Hessian is defined by a self-adjoint operator satisfying the same quadratic-form identity.The section’s results use this identity for every metric.Source: §3.1 · pp. 8–9, §3 · p. 8

3.2. Two-sided bounds at the ground state

  • Two-sided Rayleigh-quotient bounds apply to every metric at the ground state, yielding condition-number bounds; for the classical au metric, the stated bound is below 2 under Assumption 2.1.For the harmonic trap and β ≥ 10^2, the lower bound is within 9% of the exact minimum in one dimension and within 21% in two and three dimensions; the spectral-gap-only bound can be smaller by up to 3×10^4.Source: §3.2 · pp. 10–11

3.3. The modified a0 metric

  • The modified a0 metric uses a separable operator as its metric, retaining a0-like conditioning up to bounded factors while remaining robust to domain size.Its iteration count does not grow with domain size in the reported tests, provided the coercivity constant is uniformly bounded below for the stated uniform condition-number result.Source: §3.3 · pp. 11–12

3.4. The modified au metric

  • The modified au metric uses A3β(u), which stays positive definite away from the ground state and gives an explicit Hessian spectrum; related metrics with c in [2,3] share its stated condition-number bound under Assumption 2.1.Only c = 3 has a scalar Zc and the explicit spectrum, and its bound holds without a scheme assumption.Source: §3.4 · pp. 12–13

3.5. The Thomas–Fermi metric

  • The new Thomas–Fermi metric replaces the approximation error with a positive shift while retaining the Laplacian, and its Hessian condition number is bounded uniformly in β for every ℓ > 0.The section also derives sharp Thomas–Fermi asymptotics for the au metric’s coercivity constant and compares condition-number behavior among metric families in the one-dimensional harmonic problem.Source: §3.5 · pp. 14–15
  • The Thomas–Fermi metric’s condition number is bounded uniformly in β for every ℓ > 0, since both hypotheses of (29) hold for large β.The proof’s two eigenvalues enclose the condition number.Source: §3.5 · p. 15
  • Numerically, C0 ≈ 0.43, so CTF can be chosen in (C0, 1/2); for ℓ = 1/2, the first factor in (29) equals one.The passage defines C0 as a finite supremum and gives its numerical value as ν0(0)^2 ≈ 0.43.Source: §3.5 · p. 16
  • Unlike the au metric and its family, the Thomas–Fermi metric is not uniformly equivalent to A_β.Appendix A tests the predicted orders, including the Thomas–Fermi metric’s uniform bound, against exact Riemannian Hessian spectra in one and three dimensions.Source: §3.5 · p. 16

4. Convergence

The section gives a uniform convergence analysis of the Sobolev gradient flows and relates local convergence to the Riemannian Hessian spectrum. It also establishes convergence of the full iteration sequence to a critical point.

4. Convergence

  • The convergence results are treated uniformly for all six Sobolev gradient flows under the same hypotheses.The section notes that extending results for a specific flow is not automatic for the Thomas–Fermi metric, whose operator is continuous but not smooth in u.Source: §4 · p. 16

4.1. Local convergence via the Riemannian Hessian

  • For the Thomas–Fermi metric, Theorem 4.2 requires that no node satisfy Vi = λ1(Aβ) so the metric is smooth near u*; Proposition 4.3 does not require this condition.Source: §4.1 · p. 16
  • For 0 < τ < 2/ρN−1, the fixed-step iteration converges locally linearly; its optimal step gives rate (κ(u*)−1)/(κ(u*)+1).The local rate is controlled by the Hessian eigenvalues, and the optimal step is τ* = 2/(ρ1 + ρN−1).Source: §4.1 · p. 17
  • For the au metric, the undamped inverse iteration is the Sobolev gradient-flow iteration with τ = 1, and its local rate is set by the weighted eigenproblem spectrum.Under Assumption 2.1, the theorem applies at τ = 1.Source: §4.1 · p. 18
  • On the specified sublevel set S0, the proof obtains energy contraction by the factor 1−ρ−/(4ρ+) per gradient step.Here ρ− and ρ+ are the smallest and largest Riemannian Hessian eigenvalues at the ground state.Source: §4.1 · pp. 18–19

4.2. Global convergence

  • With step τk = 1/CE, the full sequence converges to a single critical point, its gradients tend to zero, and its total path length is finite.The result also holds, with adjusted constants, for step rules satisfying the stated decrease and step-length conditions.Source: §4.2 · pp. 19–20

5. Numerical experiments

Three-dimensional tests compare six metrics with Riemannian gradient descent and conjugate gradients across interaction strengths and box sizes. Their observed performance broadly follows the Riemannian Hessian conditioning analysis, with iteration counts and computational time favoring different metrics.

5.1. Setup of the three-dimensional comparisons

  • At β=1600, α=20 gives the fewest modified H1 iterations: 169 for gradient descent and 31 for conjugate gradients.The corresponding counts at α=1 are 582 and 62.Source: §5.1 · p. 21

5.2. Comparison of the metrics

  • Iteration counts are essentially mesh independent for the five metrics tested on both grids, differing by at most one on the 993 grid.The 993 grid has 125 times fewer unknowns than the 4993 grid.Source: §5.2 · p. 22
  • The condition numbers at β=1600 range from 1.30 for Thomas–Fermi to 39.9 for modified H1, and their ordering accounts for much of the iteration behavior.The four metrics with condition numbers from 4.73 to 24.3 have iteration counts differing by a factor of five, so condition number alone does not fully predict counts.Source: §5.2 · pp. 22–23
  • The Thomas–Fermi flow needs the fewest iterations, while the modified a0 and modified H1 conjugate-gradient variants are fastest on the 4993 grid.The Thomas–Fermi flow has the fewest iterations among gradient flows; the two conjugate-gradient variants are the fastest methods.Source: §5.2 · p. 22, Figure 1 · p. 23
  • On the Q20 grid with about 10^9 unknowns, iteration ordering carries over, but Thomas–Fermi is slower than au in wall-clock time.The two metrics with separable operators remain cheapest per iteration; the wall-time difference arises from the inner solve.Source: Figure 2 · p. 24, §5.2 · p. 24

5.3. Dependence on the interaction strength

  • The interaction-strength sweep tests β values from 1600 to 102400 on a larger box because the Thomas–Fermi radius leaves the smaller box near β=2×10^4.The tested β range spans a factor of 64.Source: §5.3 · p. 25
  • Across the tested range, the Thomas–Fermi metric’s iteration count does not grow; the au flow count grows by 2.74 versus predicted λ^(2/3) growth, while modified a0 reaches the iteration cap at the largest β.Source: §5.3 · p. 25

5.4. Dependence on the domain size

  • Across L=8, 16, and 32, the five potential-based metrics have nearly L-independent error curves, while modified H1 iterations grow and miss tolerance at L=32.The conjugate-gradient modified a0 method is fastest among the six on the two larger boxes.Source: §5.4 · p. 26

6. Conclusions

The paper analyzes discrete Sobolev gradient flows through the ground-state Riemannian Hessian and compares metrics using its conditioning. The analysis and three-dimensional tests support distinct trade-offs: Thomas–Fermi minimizes iterations, while exactly tensor-solvable metrics minimize computational time.

6. Conclusions

  • Rayleigh-quotient bounds for the metric-independent constrained Hessian yield condition-number bounds and local convergence rates for different metrics.The fixed-step iteration rate is determined by the Hessian spectrum, and iterates converge to a single critical point from any initial value.Source: §6 · p. 27
  • The Thomas–Fermi flow needs the fewest iterations, while the shifted Laplacian and separable-potential metrics are fastest in computational time.The latter metrics are fastest because their iterations are cheapest.Source: §6 · pp. 27–28

Funding

The acknowledgments identify funding, AI-tool use, and code availability.

Funding

  • Xiangxiong Zhang was supported in part by NSF DMS-2208518.Source: Funding · p. 28

Use of AI-assisted technologies

  • The authors report using ChatGPT and Claude for mathematical discussions, implementation, and drafting, and state they reviewed and take responsibility for the manuscript.Source: Use of AI-assisted technologies · p. 29

Data Availability Statements

A. Exact spectra of the Riemannian Hessian

The appendix tests Riemannian-Hessian spectral bounds and condition-number predictions in harmonic traps, alongside mesh checks and iteration comparisons. The Thomas–Fermi metric remains well-conditioned across the reported three-dimensional interaction strengths, while the modified au metric has a mesh-dependent upper spectral edge.

A. Exact spectra of the Riemannian Hessian

  • The spectral computations use dense diagonalization in one dimension, shift-invert Lanczos in two, and locally optimal block preconditioned conjugate gradients in three dimensions.The harmonic-trap tests use second-order finite differences, and the reported three-dimensional mesh checks include β = 10^5 on a 3993 grid and β = 3 × 10^5 on a 2993 grid.Source: A. Exact spectra of the Riemannian… · p. 31
  • For β ≥ 10^2, the au lower bound is within 9% of the exact minimum in one dimension and 21% in two and three dimensions; its condition-number bound is within 2% for β ≥ 10^4 in one dimension.For β ≥ 10^4, the spectral-gap-only lower bound undershoots the exact minimum by factors of 31 to 3 × 10^4.Source: A. Exact spectra of the Riemannian… · p. 32, Table A.1 · p. 32
  • Across β ∈ [10^4, 3 × 10^5] in one dimension, the slopes against λ1 are −0.66 for minρu∗, 0.333 for λ1(A3β)−λ1, and −1.00 for the spectral gap.These slopes confirm the reported asymptotics of Corollary 3.11.Source: A. Exact spectra of the Riemannian… · p. 32
  • At β = 10^2, 10^3, and 10^4, the modified au metric takes 21, 47, and 114 iterations, against 32, 84, and 217.The reported comparison uses one-dimensional harmonic-trap tests with exact line search, N = 2000, a Thomas–Fermi-profile start, and energy error below 10^-12.Source: A. Exact spectra of the Riemannian… · p. 32
  • In three dimensions, κTF(u∗) falls from 1.89 to 1.17 as β ranges from 1 to 3 × 10^5, while κa0(u∗) grows by a factor 330.The measured exponents against λ1 are 0.62, 1.62, and −0.12 for the au, a0, and Thomas–Fermi metrics, consistent with the predicted 2/3 and 5/3 slopes and (29).Source: A. Exact spectra of the Riemannian… · p. 33
  • At β = 10^4, the Thomas–Fermi bound κTF(u∗) ≤ 1.38 compares with the measured 1.16, a factor 39 smaller than κau(u∗).With the nonseparable bump, κTF(u∗) stays between 1.16 and 1.18.Source: A. Exact spectra of the Riemannian… · p. 33
  • The modified au upper spectral edge depends on mesh: at β = 10^4, refinement from N = 2000 to N = 4000 changes κeau(u∗) from 21.5 to 22.5 in one dimension.The three-dimensional mesh checks show changes of at most 4%.Source: A. Exact spectra of the Riemannian… · p. 33

B. Riemannian connection, Hessian and vector transport

This appendix gives connection, Hessian, and transport formulas for the sphere equipped with metrics induced by discrete elliptic operators. It specifies the smoothness conditions for their use and distinguishes critical-point Hessian identities from properties that need not hold away from critical points.

B. Riemannian connection, Hessian and vector transport

  • The five metrics other than the Thomas–Fermi metric have the form (B.2), with c = 0 for the three Euclidean metrics, c = 1 for au, and c = 3 for modified au; c affects the metric, not the energy.The analysis assumes an open set where B(u) is smooth and MB(u) is symmetric positive definite.Source: B. Riemannian connection, Hessian and… · p. 33
  • The ambient Levi–Civita connection is the directional derivative of the vector field plus Γu, and the sphere’s induced connection is its gu-orthogonal projection onto the tangent space.For the metric family (B.2), the derivative of B contributes through 2cβdiag(u ⊙ ξ).Source: B. Riemannian connection, Hessian and… · p. 34
  • At a critical point, the Riemannian Hessian quadratic form is the second variation of the energy along the normalization retraction and is the same for every smooth metric.Away from critical points, λB(u) cannot in general be replaced by λ(u).Source: B. Riemannian connection, Hessian and… · pp. 34–35
  • The Thomas–Fermi metric depends on u through λ(u) and is smooth on the stated open set where u ≠ 0 and λ(u) differs from every nodal potential Vi.The general connection and Hessian formulas apply there.Source: B. Riemannian connection, Hessian and… · p. 35
  • Projection transport maps tangent vectors to the new tangent space and is linear, smoothly dependent on its inputs, and equal to the identity for zero step.It is generally not parallel transport; when the metric depends on u, its norm need not contract relative to the starting-point metric.Source: B. Riemannian connection, Hessian and… · p. 35
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