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arxiv: 1907.08305 · v1 · pith:DTCRJIM5new · submitted 2019-07-18 · 🧮 math.NA · cs.NA

A variational finite volume scheme for Wasserstein gradient flows

Cl\'ement Canc\`es , Thomas O. Gallou\"et , Gabriele Todeschi This is my paper

Pith reviewed 2026-05-24 19:20 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Wasserstein gradient flowsfinite volume schemevariational discretizationFokker-Planck equationBenamou-Brenier formulaupstream mobilityconvex energiesnumerical convergence
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The pith

A variational finite volume scheme for Wasserstein gradient flows admits a unique solution for any convex energy and converges for the linear Fokker-Planck equation with positive initial density.

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

The paper introduces a discretization method that turns continuous Wasserstein gradient flows into a sequence of discrete optimization problems. Time steps use an implicit linearization of the distance based on the Benamou-Brenier formula, while space uses upstream mobility finite volumes. This first-discretize-then-optimize strategy keeps the variational structure intact, so the discrete solutions stay non-negative and the energy decreases. Uniqueness holds for every convex energy, and convergence is established when the continuous problem is the linear Fokker-Planck equation started from a strictly positive density.

Core claim

The scheme, obtained by first discretizing the Benamou-Brenier formulation of the Wasserstein distance in time and applying upstream-mobility two-point flux finite volumes in space, then optimizing the resulting discrete energy, admits a unique solution for any convex energy and converges to the solution of the linear Fokker-Planck equation whenever the initial density is strictly positive.

What carries the argument

First-discretize-then-optimize variational finite volume scheme that linearizes the Wasserstein distance implicitly via the Benamou-Brenier formula and uses upstream mobility two-point flux approximations.

If this is right

  • The scheme applies to a wide range of convex energies.
  • Discrete solutions remain non-negative for all time steps.
  • The discrete energy decreases at every step.
  • The scheme is first-order accurate in both time and space.
  • The method remains robust across different energies and initial profiles.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same variational structure could be used to design schemes that handle densities that touch zero, though this is not shown.
  • The approach may transfer to other optimal-transport-based evolution equations beyond Fokker-Planck.
  • Because the scheme is built variationally, it could serve as a template for higher-order or adaptive-mesh extensions.

Load-bearing premise

Convergence is proved only when the initial density stays strictly positive in the linear Fokker-Planck case.

What would settle it

A counter-example or numerical run in which the scheme fails to converge or loses uniqueness for the linear Fokker-Planck equation started from a density that reaches zero at some point.

Figures

Figures reproduced from arXiv: 1907.08305 by Cl\'ement Canc\`es, Gabriele Todeschi, Thomas O. Gallou\"et.

Figure 1
Figure 1. Figure 1: Sequence of regular triangular meshes. 4.2. Fokker-Planck equation. We first tackle the gradient flow of the Fokker-Planck energy, namely eq. (3). In section 3 we showed the L 1 convergence of the scheme. Consider the specific potential ρV (x) = −ρgx: for this case it is possible to design an analytical solution and test the convergence of the scheme. Consider the domain Ω = [0, 1]2 , the time interval [0,… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the dissipation of the system computed with the two numerical schemes and in the real case. Semi-logarithmic plot [PITH_FULL_IMAGE:figures/full_fig_p029_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Evolution of an initial density close to a dirac. In each picture the scaling is different for the sake of the representation. 4.4. Salinity intrusion problem. We want to show now that scheme (36) can be used for the solutions of systems of equations of the type of eq. (1). We consider the problem of salinity intrusion in an unconfined aquifer. Under the assumption that the two fluids, the fresh and the sa… view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of the two interfaces of salt (red) and fresh (blue) water. and initial conditions f(t = 0) = f0, g(t = 0) = g0, with f0, g0 ∈ L∞(Ω), f0, g0 ≥ 0. The quantities f, g, and b represent respectively the thickness of the fresh layer of water, the thickness of the salty water layer and the height of the bedrock. Therefore the quantity b + g represents the height of the sharp interface separating the t… view at source ↗
read the original abstract

We propose a variational finite volume scheme to approximate the solutions to Wasserstein gradient flows. The time discretization is based on an implicit linearization of the Wasserstein distance expressed thanks to Benamou-Brenier formula, whereas space discretization relies on upstream mobility two-point flux approximation finite volumes. Our scheme is based on a first discretize then optimize approach in order to preserve the variational structure of the continuous model at the discrete level. Our scheme can be applied to a wide range of energies, guarantees non-negativity of the discrete solutions as well as decay of the energy. We show that our scheme admits a unique solution whatever the convex energy involved in the continuous problem, and we prove its convergence in the case of the linear Fokker-Planck equation with positive initial density. Numerical illustrations show that it is first order accurate in both time and space, and robust with respect to both the energy and the initial profile.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper proposes a variational finite volume scheme for Wasserstein gradient flows based on an implicit linearization of the Wasserstein distance (via Benamou-Brenier) and upstream-mobility two-point flux finite volumes. The scheme is constructed to preserve the variational structure, guarantees non-negativity and energy decay, admits a unique solution for arbitrary convex energies, and is shown to converge for the linear Fokker-Planck equation when the initial density is strictly positive. Numerical tests indicate first-order accuracy in time and space and robustness across energies and initial profiles.

Significance. If the uniqueness result for arbitrary convex energies holds and the convergence proof is complete for the stated linear case, the work supplies a structure-preserving discretization that respects the underlying variational formulation. The explicit separation of the uniqueness claim (general) from the convergence claim (linear FP, positive density) is a strength; the first-order accuracy and robustness in tests further support utility for this class of problems.

major comments (1)
  1. [Abstract and convergence theorem] Abstract and convergence section: the uniqueness result is stated for arbitrary convex energies, yet convergence is established only for the linear Fokker-Planck equation under the strictly positive initial-density assumption. This restriction is load-bearing for the claim of applicability to a 'wide range of energies,' because the analysis does not address vanishing densities (common in Wasserstein flows) or nonlinear energies; a concrete extension or counter-example discussion would be needed to support the broader positioning.
minor comments (2)
  1. [Section 2] Notation for the discrete Wasserstein distance and the upstream mobility fluxes should be introduced with explicit reference to the continuous Benamou-Brenier formula to improve readability.
  2. [Numerical section] The numerical experiments section would benefit from an additional table reporting the observed convergence rates under the positive-density assumption to make the first-order claim quantitative.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract and convergence theorem] Abstract and convergence section: the uniqueness result is stated for arbitrary convex energies, yet convergence is established only for the linear Fokker-Planck equation under the strictly positive initial-density assumption. This restriction is load-bearing for the claim of applicability to a 'wide range of energies,' because the analysis does not address vanishing densities (common in Wasserstein flows) or nonlinear energies; a concrete extension or counter-example discussion would be needed to support the broader positioning.

    Authors: The manuscript already distinguishes these results explicitly: uniqueness holds for arbitrary convex energies, while convergence is proven only for the linear Fokker-Planck equation with strictly positive initial density. The statement that the scheme 'can be applied to a wide range of energies' refers to its variational construction, which guarantees non-negativity and energy decay for general convex energies (supported by the uniqueness theorem and numerical tests), not to a claim of convergence in all cases. The abstract and convergence section state the proven results without overclaiming. We will nevertheless revise the abstract slightly to reiterate the precise scope of the convergence theorem and add one sentence in the conclusions noting that extension of the convergence analysis to nonlinear energies or vanishing densities remains open. revision: partial

Circularity Check

0 steps flagged

No circularity; discretization follows directly from continuous variational structure

full rationale

The paper derives its finite-volume scheme by first discretizing the Benamou-Brenier formulation of the Wasserstein distance and then optimizing, preserving the variational structure at the discrete level. Uniqueness holds for arbitrary convex energies by direct analysis of the resulting minimization problem, and convergence is proven only for the linear Fokker-Planck case with strictly positive initial data. No step reduces a claimed result to a fitted parameter, self-citation chain, or definitional tautology; the stated limitations on convergence are explicit rather than hidden circularities. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The scheme rests on standard domain assumptions from optimal transport and numerical analysis for convex energies and positive densities; no free parameters are fitted to data and no new entities are postulated.

axioms (2)
  • domain assumption The energy functional is convex.
    Invoked to guarantee existence and uniqueness of the discrete solution for any such energy.
  • domain assumption Initial density is strictly positive for the convergence statement.
    Explicitly required in the abstract for the linear Fokker-Planck convergence proof.

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