Convergence of a particle method for gradient flows on the $L^p$-Wasserstein space

Rong Lei

REVIEW 4 major objections 4 minor 20 references

Convergence of a particle method for gradient flows on the $L^p$-Wasserstein space

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Particle method is proved convergent for L^p-Wasserstein gradient flows.

desk verdict A genuine p>1 extension of the CPSW particle method, with new slope estimates, but the proof as written invokes Serfaty's theorem with squared metric derivatives while verifying only p-th powers, so the main theorem is not logically closed for p≠2. read the letter →

arxiv 2501.03459 v1 pith:S76G5HUG submitted 2025-01-07 math.NA cs.NAmath.OC

Rong Lei This is my paper · ORCID

classification math.NAcs.NAmath.OC MSC 49Q2235K5565M75

keywords L^p-WassersteinspacegradientflowsparticlemethoddoublynonlineardiffusionequationcurveofmaximalslopeGamma-convergenceporousmedium

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that a particle approximation to gradient flows on the L^p-Wasserstein space over the real line converges to the continuum gradient flow. The continuum limit is the doubly nonlinear diffusion equation, of which the porous medium equation and the $p$-Laplacian heat equation are special cases. The proof preserves the steepest-descent structure at the particle level by discretizing the energy through nonoverlapping intervals centered on the particles, and then passes the discrete flows to the continuum limit. On bounded intervals, with an extra growth condition on the energy density $B$, the metric derivatives, the energies, and the slopes of the discrete flows converge to their continuum counterparts. If $p=2$ or the energy is power-law, the whole sequence of particle flows converges rather than only a subsequence.

What carries the argument

The load-bearing object is the discrete energy $\mathcal{E}_N$ built from nonoverlapping balls centered at the $N$ particles: in one dimension the balls are intervals, so $\mathcal{E}_N(\mu_N)=\sum_i |B_i| B(1/(N|B_i|))=(1/N)\sum_i h(N\Delta x_i)$ with $h(x)=xB(1/x)$. This is exactly the continuum energy of the piecewise-constant interval density, and it gives the particle system a gradient-flow structure at the discrete level. To pass to the limit, the proof builds an interpolation $\tilde{\rho}_N$ from the affine interpolation of the dual variables $\sigma_i=-N h'(N\Delta x_i)$ by inverting through $h'$, and shows that the discrete slope controls the continuum Fisher-type information of this interpolation. Hypothesis 2 is used at the last step, to control the effect of the normalization constant in $\tilde{\rho}_N$ when comparing discrete and continuum slopes.

What would settle it

Choose a one-dimensional energy density $B$ that satisfies Hypothesis 1 but violates the dilation inequality of Hypothesis 2, prepare particles with spacing bounds as in Definition 11, and compute the discrete slopes $|\partial\mathcal{E}_N|(\mu_N)$ together with the continuum slope of the interpolation $\tilde{\rho}_N$. If the ratio of these quantities does not tend to a limit at least 1 as $N\to\infty$, or if the normalization constant in $\tilde{\rho}_N$ diverges, condition (C3) fails and the conclusion of Theorem 3 would not hold for that $B$.

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Extended reading notes

Core claim

The central claim, Theorem 3, is that if the energy density $B$ satisfies the convexity and doubling assumptions in Hypothesis 1 and the initial particle configurations are well-prepared for a continuum density $\rho$, then the discrete gradient flows $\mu_N$ converge narrowly, up to a subsequence, to an absolutely continuous curve $\rho$ in $P_p(\Omega)$. When $\Omega=[-\ell,\ell]$ and Hypothesis 2 holds, that is $B''>0$ together with the dilation comparison $B''(\alpha x)\ge \varphi(\alpha)B''(x)$, the limit is a continuum gradient flow of the energy, and the discrete metric derivatives, energies, and local slopes converge to the continuum ones in the sense of (8). For $p=2$ or for power-law energies, uniqueness of the continuum solution upgrades the convergence to the full sequence, and the limit solves the doubly nonlinear diffusion equation in one dimension.

Load-bearing premise

The proof of the slope lower-semicontinuity condition (C3) requires the energy density $B$ to satisfy $B''>0$ with a dilation scaling inequality $B''(\alpha x)\ge \varphi(\alpha)B''(x)$, and it requires the initial particle spacings to be bounded between two constants divided by $N$; if either condition fails, the key comparison between discrete and continuum slopes has no justification.

Editorial extensions

If this is right

The discrete gradient-flow equations for the particle positions are a faithful approximation of the doubly nonlinear diffusion equation in one dimension.
When the theorem applies, discrete energies, metric derivatives, and local slopes pass to the continuum limit, so the approximation preserves the energy-dissipation structure of the flow.
The method extends the variational particle approximation from the $L^2$-Wasserstein setting to every $p>1$, covering $p$-Laplacian heat and porous-medium type flows.
On bounded intervals with $p=2$ or with power-law energies, uniqueness of the continuum solution turns subsequential convergence into convergence of the entire sequence of particle flows.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the proof forces neighboring particle spacings to equalize as $N$ grows, the maximum ratio $|\Delta x_{i+1}/\Delta x_i-1|$ is a natural numerical diagnostic: for well-prepared initial data it should decay to zero, and its decay rate may track the practical accuracy of the method.
Hypothesis 2 is a mild self-similarity condition on $B$; if it is close to being violated, the slope comparison suggests that non-power-law energy densities will converge more slowly, a statement the paper does not make explicitly.
An analogous result in more than one dimension would need a different cell construction, because nonoverlapping balls leave gaps and the one-dimensional ordering of particles is used essentially in the interpolation argument.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

New here is a genuine extension: the particle method of Carrillo–Patacchini–Sternberg–Wolansky, previously for p=2, is pushed to general p>1 and applied to doubly nonlinear diffusion in 1D. The energy discretization, the slope computations, and especially Lemma 6 and the renormalized interpolation in Proposition 4 are real new estimates. The paper also credits [3] honestly and has no fitted parameters or circularity.

The soft spot is not the originality; it is the logic of the convergence proof. Theorem 4, as stated, is Serfaty's theorem with (C1) in squared metric derivatives. The flows here are p-curves of maximal slope (Definition 5), and Proposition 2 verifies only the p-th power version of (C1): liminf ∫|μ'_N|^p ≥ ∫|ρ'|^p. On a fixed time interval, that liminf in L^p does not imply the L^2 liminf when p≠2, and the conclusion (8) is written in L^p. So unless there is a p,q version of Serfaty's theorem stated and proved (with (C1) in p-th powers and (C3) in q-th powers), Theorem 3 is not logically closed for the advertised range p>1. This is independent of the C3/Hypothesis 2 interpolation issue; it is the load-bearing step.

There are also several important lemmas not proved in the text: Lemmas 3–5 and the details of Theorem 5 are dismissed with 'same strategy as in [3]'. That may be acceptable for a follow-up paper, but here p≠2 changes the inequalities, so the reader cannot check them without redoing [3] by hand. Lemma 6 is new and treated; Proposition 4 is sketched at a level that is plausible but has a confusing normalization (ν_N = A_N \tilde ρ_N after \tilde ρ_N was already normalized).

Bottom line: the result is probably true and the paper is worth engaging with, but in its current form the main theorem is not fully proved for p≠2 because of the Serfaty exponent mismatch. The author should be asked to either prove a p,q Gamma-convergence theorem or show how the p-th power liminf reduces to the squared one. Send it to referees, with a request for major revision; readers in numerical analysis and optimal transport will get value once the gap is fixed.

Referee Report

4 major / 4 minor

Summary. The paper adapts the particle method of Carrillo, Patacchini, Sternberg, and Wolansky to gradient flows on the L^p-Wasserstein space over R, for p>1. The continuum energy is discretized by replacing each empirical measure with a piecewise-constant density on nonoverlapping balls centered at the particles, and the discrete gradient flow is defined as a p-curve of maximal slope in a weighted l^p norm. The main theorem (Theorem 3) claims that, for well-prepared initial data in a class G(Omega), the discrete gradient flows converge, up to a subsequence, narrowly to an absolutely continuous curve, and that on a bounded interval, under an additional scaling condition on B'', the limit is a continuum gradient flow in the sense of a p-curve of maximal slope, with convergence of metric derivatives, energies, and slopes. For p=2 or for power-law energies, convergence of the whole sequence to the doubly nonlinear diffusion equation is claimed. The proof follows Serfaty's Gamma-convergence-of-gradient-flows framework and attempts to verify conditions (C1)-(C3).

Significance. If the result is established, it would be a useful extension of the one-dimensional particle method to p-curves of maximal slope and to doubly nonlinear diffusion equations. The discrete energy is a direct, parameter-free discretization of the continuum energy, and the claimed limit statements are concrete and falsifiable. However, as the manuscript stands, the central proof is not closed: the invoked Serfaty theorem is stated with the wrong exponents, and several load-bearing lemmas and the recovery-sequence construction are delegated to [3] without proof. The significance is therefore conditional on a substantial revision.

major comments (4)

[Section 3, Theorem 4 and Proposition 2] Theorem 4, as stated, invokes Serfaty's theorem with condition (C1) involving squared metric derivatives and condition (C3) with no exponent, while the discrete flows of Definition 9 are p-curves of maximal slope. Proposition 2 proves only the p-th-power inequality liminf_N ∫_0^t |μ'_N|^p ds ≥ ∫_0^t |ρ'|^p ds. For p≠2, a lower bound on L^p norms over a bounded interval does not imply the corresponding L^2 lower bound, and the slopes in (C3) are not stated in the correct conjugate exponent. Moreover, the convergence of slopes in (8) is stated in L^p, although for a p-curve of maximal slope the natural integrability of the slope is L^q with 1/p+1/q=1. Since Theorem 4 is the mechanism that converts (C1)-(C3) into the continuum gradient flow, the proof of Theorem 3 is not logically closed for the advertised range p>1. The manuscript should either state and prove a p,q version of Serfaty's theorem with (C1) in p-th powers and (C3) in q-th powers, or restrict the main theorem to p=2.
[Section 3.2, Lemmas 3-5] Lemmas 3, 4, and 5 are load-bearing: Lemma 3 gives the lower bound |η_i| ≥ |σ_i - σ_{i+1}| used in Lemma 6, Lemma 4 fixes the behavior of the boundary particles, and Lemma 5 provides the uniform spacing bounds E_1/N ≤ Δx_i ≤ E_2/N used throughout the proof of Proposition 4 and in the interpolation. The paper states that these are obtained by the same strategy as in [3] and omits all details. Because the discrete gradient flow here is taken with respect to a weighted l^p norm, the subdifferential is different from the p=2 case, and the adaptation is not automatic. These proofs must be supplied.
[Section 3.2, proof of Proposition 4] The proof of (C3) is not self-contained. It asserts without proof that the interpolation ρ̃_N(t) converges narrowly to ρ(t), that the normalization constant A_N converges to 1, and that the limit passage (14)-(15) is valid after using Hypothesis 2. The step relating liminf_N I_p(ρ̂_N) and liminf_N I_p(ρ̃_N) via the scaling inequality B''(αx) ≥ φ(α)B''(x) is only sketched, and the finiteness assumption in Lemma 6 is not justified for every t∈[0,T]. Since (C3) is the last condition needed to apply Theorem 4, this is a major gap.
[Section 4, Theorem 5] Theorem 5 states the Gamma-convergence of the discrete energies, but its proof consists entirely of 'we follow the same strategy as in [3]' and 'we omit details here.' If this theorem is part of the paper's claims, it needs a proof; if it is only meant as a reference to known results, it should be stated as such. Moreover, the construction of recovery sequences is needed to make the well-preparedness assumption in Definition 11 non-vacuous, so this omission affects the interpretation of the main theorem.

minor comments (4)

[Section 3.1, proof of Proposition 2] After extracting a subsequence for the weak convergence of |μ'_N| in L^p, the notation reverts to the full sequence in equation (10); the subsequence should be relabeled consistently throughout the proof.
[Section 3.2, proof of Proposition 4] The displayed formula for the Fisher information of ρ̂_N appears to contain an exponent inconsistency: with the definition in Lemma 1 the denominator should involve ρ^{-1}, not ρ^{1-3p}; this should be checked.
[Section 3, Definition 10] The condition 'minsupp ρ ρ > 0' should be written as min over the support of ρ, for example min_{ρ>0} on supp ρ, to avoid ambiguity.
[Section 3, Remark 2] The uniqueness references for the p-heat equation and the Leibenson equation should be accompanied by precise hypotheses matching the class G(Ω); as written it is unclear whether the cited uniqueness theorems apply to the weak solutions obtained in Theorem 3.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence proof is built on the external Serfaty theorem and direct verification of its conditions, with no fitted parameter or self-citation chain carrying the conclusion.

full rationale

The derivation chain is not circular. The continuum object is the L^p-Wasserstein gradient flow of the energy E (Definition 6), and the discrete object is the p-curve of maximal slope for the discrete energy E_N (Definition 9), where E_N is defined directly as E applied to the piecewise-constant density reconstruction rho_N (Definition 8, equation (6)). No parameter is fitted to the quantities being predicted: the initial data are only assumed to be well-prepared (Definition 11), with E1/N <= Delta x_i <= E2/N and limsup E_N(rho_N(0)) <= E(rho(0)); these are hypotheses on the data, not outputs derived from the conclusion. The central convergence theorem (Theorem 3) is obtained by invoking Serfaty's gamma-convergence-of-gradient-flows theorem [17] (Theorem 4) and then verifying conditions (C1)-(C3): (C1) is proved in Proposition 2, (C2) in Proposition 3, and (C3) in Proposition 4, with the discrete slope computed explicitly in Lemma 2 and controlled via the interpolation (12) and Lemma 6. The Gamma-convergence of E_N to E (Theorem 5) is an independent claim proved in Section 4. The only borrowings from [3] are the discretization construction and several technical lemmas whose proofs are said to follow by the same method; [3] is an external published source, not a self-citation, so this does not make the argument circular. The proof does contain a non-circular rigor gap: Theorem 4 is quoted with condition (C1) on squared metric derivatives and (C3) on slopes, while Proposition 2 proves only the L^p version liminf integral |mu'_N|^p >= integral |rho'|^p, and Proposition 4 proves the pointwise slope inequality without an exponent; for p != 2 this mismatch is not repaired in the paper. That is a correctness or completeness concern about the applicability of the invoked theorem, not a reduction of the conclusion to the hypotheses, so it is not counted as circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the assumed energy hypotheses and a special initial data condition. No fitted parameters or invented entities are introduced; the constants E1, E2 are assumed to exist rather than fitted.

assumptions (4)

domain assumption Hypothesis 1: B is proper, convex, non-negative, superlinear, satisfies a doubling condition, and h(s)=s^p B(s^{-p}) is strictly convex and non-increasing on (0,∞).
This ensures displacement convexity and lower semicontinuity of the energy, used throughout Theorem 3 and Proposition 2.
domain assumption Hypothesis 2: B'' > 0 and there exists a continuous φ with φ(1)=1 such that B''(αx) ≥ φ(α)B''(x) for all x, α > 0.
Used in Proposition 4 to control the normalization constant in the interpolation and obtain condition (C3); not satisfied by all admissible B.
domain assumption Well-prepared initial data (Definition 11): initial empirical measures are recovery sequences with E1/N ≤ Δx_i ≤ E2/N and, when ρ∈G(Ω), boundary particles satisfy x_N = -x_1 = ℓ.
Required for tightness and for the slope estimates; this restricts the class of allowed initial particle configurations.
domain assumption Boundary condition using fictitious particles x_0 and x_{N+1} to keep real particles inside Ω.
Needed for the well-posedness of the discrete gradient flow and borrowed from [3]; the paper assumes the same boundary condition without proof.

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Cite this review

Pith. "Pith review of Convergence of a particle method for gradient flows on the $L^p$-Wasserstein space." pith.science (2026). https://pith.science/paper/S76G5HUG

@misc{pith2026250103459,
  author       = {Pith},
  title        = {Pith review of: Convergence of a particle method for gradient flows on the $L^p$-Wasserstein space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S76G5HUG}},
  note         = {Machine review of arXiv:2501.03459}
}

abstract

We study the particle method to approximate the gradient flow on the $L^p$-Wasserstein space. This method relies on the discretization of the energy introduced by [3] via nonoverlapping balls centered at the particles and preserves the gradient flow structure at the particle level. We prove the convergence of the discrete gradient flow to the continuum gradient flow on the $L^p$-Wasserstein space over $\mathbb R$, specifically to the doubly nonlinear diffusion equation in one dimension.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [3]

A., Patacchini, F

Carrillo, J. A., Patacchini, F. S., Sternberg, P ., Wolansky, G.: Convergence of a particle method for diﬀusive gradient ﬂows in one dimension. SIAM J. Math. An al. 48(6), 3708-3741 (2016)

work page 2016

[2] [1]

Birkh ¨auser Basel, 2005

Ambrosio, L, Gigli, N, Savar ´e, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Birkh ¨auser Basel, 2005. 18 Rong Lei

work page 2005

[3] [2]

H.: Discrete duality ﬁnite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations

Andreianov, B., Bendahmane, M., Karlsen, K. H.: Discrete duality ﬁnite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations. J. Hyperbolic Diﬀer. Equ. 7(01), 1-67 (2010)

work page 2010

[4] [4]

A., Huang, Y ., Patacchini, F

Carrillo, J. A., Huang, Y ., Patacchini, F. S., Wolansky, G .: Numerical study of a particle method for gradient ﬂows. arXiv preprint arXiv:1512.03029 (2015)

arXiv 2015

[5] [5]

Droniou, J., Eymard, R., Gallouet, T., Herbin, R.: Gradie nt schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal el liptic and parabolic equations. Math. Models Methods Appl. Sci. 23(13), 2395-2432 (2013)

work page 2013

[6] [6]

Boun dary value problems for partial diﬀerential equations and applications, 81–98, RMA Res

De Giorgi, E.: New problems on minimizing movements. Boun dary value problems for partial diﬀerential equations and applications, 81–98, RMA Res. No tes Appl. Math., 29, Masson, Paris, 1993

work page 1993

[7] [7]

Att Accad Naz

De Giorgi, E., Marino, A., Tosques, M.: Problems of evolut ion in metric spaces and maximal decreasing curve. Att Accad Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8)68, 180-187 (1980)

work page 1980

[8] [8]

Evje, S., Hvistendahl Karlsen, K.: Discrete approximations of BV solutions to doubly nonlinear degenerate parabolic equations. Numer. Math. 86(3), 377-4 17 (2000)

work page 2000