arXiv:2608.07329v1

Sampling and Optimization meet Enhanced Flows

Yuan Gao, Siming He, Eitan Tadmor

math.OCmath.AP35B4035Q8435Q3537A2565C05

Paper metadata

Original submission: August 7, 2026
Claimed MSC: 35B40, 35Q84, 35Q35, 37A25, 65C05
arXiv categories: math.OC, math.AP
Paper license: CC BY 4.0 ↗

Abstract

It is well known that the computational realization of Gibbs probability measures, $e^{-\mathbb{U}(\mathbf{x})}/Z$ , plays a central role in sampling and optimization. In this paper, we introduce two types of dynamics that exhibit rapid convergence towards these Gibbs measures. The mechanism driving this rapid convergence is the enhanced dissipation associated with these transport-diffusion dynamics. Motivated by these enhanced dynamics, we design numerical algorithms for sampling from the target Gibbs measure. Finally, we provide the corresponding particle systems that may yield other effective numerical samplers.

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Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsCorrect

The enhanced convergence results for the first- and second-order sampling dynamics, the alternating-shear enhanced-dissipation theorem, and the mass-searching theorem are correct as stated.

Theorem 1.1Correct

The first-order dynamics converge at the enhanced rate

Pages 3 and 10–15 · Theorem 1.1 and Section 2.2 · arXiv:2608.07329v1

Dividing by the target density and removing the scalar reaction factor reduces the nonlinear equation to a drift-diffusion equation. Mass conservation controls the time integral of the reaction coefficient. On intervals of length $\delta^{-1}\nu^{-1/2}$ , the alternating-shear estimate contracts the mean-zero component, while the potential term differs from the passive-scalar evolution by $O(\nu^{1/4})$ . Choosing $\nu$ below the displayed thresholds closes the bootstrap and yields the asserted $e^{-\delta\nu^{1/2}t}$ decay.

Theorem 1.2Correct

The kinetic dynamics have the stated sampling, approximation, and mixing bounds

Pages 5 and 15–22 · Theorem 1.2 and Section 3 · arXiv:2608.07329v1

The $x$ -fluctuation of the renormalized kinetic density decays by the sine-shear hypocoercive estimate, with the potential contribution absorbed for small $\nu$ and $0<\kappa\leq\nu$ . Mass conservation converts this into $L^1$ convergence of the hydrodynamic density. An $x$ -Sobolev estimate grows only like $e^{C\kappa t}$ , so interpolation with the $e^{-c\nu^{1/2}t}$ bound gives the claimed $L^\infty$ rate. The short-time dual estimate follows from the proved viscous mixing bound.

Theorems 1.3 and 1.4Correct

Alternating shears provide enhanced dissipation and recover the target mass

Pages 7–10 and 27–34 · Theorems 1.3, 1.4 and Appendices D–E · arXiv:2608.07329v1

The modewise hypocoercive functional gives a dimension-independent $e^{-\delta_0\nu^{1/2}t}$ contraction for every nonzero streamwise mode. Alternating the two simultaneous coordinate shears contracts first the $x$ -fluctuation and then the remaining $y$ -average, giving the all-time estimate after a cycle decomposition. For the mass-searching equation, $v=\omega e^{-W}$ is exactly a passive scalar with conserved spatial average $M=\int e^{-W}$ , so Theorem 1.3 yields Theorem 1.4 directly.

02Proofs3 reported findingsCorrect

The central proofs are correct and complete. The normalization, bootstrap, hypocoercive, interpolation, alternating-cycle, and passive-scalar arguments establish their stated conclusions with the required parameter dependencies.

Sections 2 and 3Correct and complete

The nonlinear bootstrap arguments close

Pages 10–22 · Sections 2–3 · arXiv:2608.07329v1

The absolute bounds on the accumulated reaction factors follow from the exact scalar variation-of-constants formulas and positivity. The comparison solutions start from the correct fluctuations, the energy errors over one enhanced-dissipation interval are $O(\nu^{1/4})$ , and the selected smallness conditions make each interval contractive. The omitted repetitions in the kinetic case are the same verified estimates with $\kappa\leq\nu$ and do not conceal a new case.

Appendix DCorrect and complete

The hypocoercive and alternating-shear estimates are complete

Pages 27–34 · Appendix D · arXiv:2608.07329v1

The spectral inequality controls the selected nonzero Fourier direction, the time-weighted functional is coercive, and the parameter choice makes its derivative negative at the claimed universal rate. The two shear stages use orthogonal averaging: after the first stage the average in the second shear direction is controlled by the contracted first fluctuation. Iterating full cycles and using ordinary $L^2$ dissipation between cycle endpoints proves the estimate for arbitrary starting and elapsed times.

Theorem 1.2, Steps 1–3Correct and complete

The hydrodynamic estimates preserve all normalizations and constants

Pages 18–22 · proof of Theorem 1.2 · arXiv:2608.07329v1

The normalization $|\mathbb T|^{-d}\lVert h\rVert_1\to1$ follows from the exact mass identity and the decaying $x$ -fluctuation. Positivity supplies the denominator lower bounds used in both $L^1$ decompositions. The higher-regularity estimate involves only $x$ derivatives and grows at rate $O(\kappa)$ , which is dominated uniformly by the $O(\nu^{1/2})$ decay after decreasing the stated threshold. The mixing proof applies its dual estimate only on the announced initial time interval.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2608.07329v1
Authors listed: Yuan Gao, Siming He, Eitan Tadmor
Audit date: August 18, 2026

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Sampling and Optimization meet Enhanced Flows

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The first-order dynamics converge at the enhanced rate

The kinetic dynamics have the stated sampling, approximation, and mixing bounds

Alternating shears provide enhanced dissipation and recover the target mass

The nonlinear bootstrap arguments close

The hypocoercive and alternating-shear estimates are complete

The hydrodynamic estimates preserve all normalizations and constants

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