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Guaranteeing Higher Order Convergence Rates for Accelerated Wasserstein Gradient Flow Schemes

Raymond Chu, Matt Jacobs

TL;DR

This work develops a fully rigorous, higher-order time discretization scheme for Wasserstein gradient flows by marrying Eulerian and Lagrangian viewpoints through a second-order Lagrangian trapezoidal method. Under smooth lifted energies, it achieves $O(\tau^2)$ convergence, while under weaker assumptions it retains $O(\tau)$ convergence via a discrete EVI framework, with robust stability properties. The authors provide concrete functionals satisfying the smoothness requirements and prove exponential decay of gradients and energy when $\lambda>0$ and the energy is $L$-smooth. They also supply a comprehensive set of numerical experiments confirming the predicted rates and stability, including particle-based implementations of blob-type energies. Overall, the paper delivers the first fully rigorous proof of accelerated second-order convergence rates for smooth Wasserstein gradient flows and clarifies the scheme’s relation to the classical JKO method under displacement convexity.

Abstract

In this paper, we study higher-order-accurate-in-time minimizing movements schemes for Wasserstein gradient flows. We introduce a novel accelerated second-order scheme, leveraging the differential structure of the Wasserstein space in both Eulerian and Lagrangian coordinates. For sufficiently smooth energy functionals, we show that our scheme provably achieves an optimal quadratic convergence rate. Under the weaker assumptions of Wasserstein differentiability and $λ$-displacement convexity (for any $λ\in \mathbb{R}$), we show that our scheme still achieves a first-order convergence rate and has strong numerical stability. In particular, we show that the energy is nearly monotone in general, while when the energy is $L$-smooth and $λ$-displacement convex (with $λ>0$), we prove the energy is non-increasing and the norm of the Wasserstein gradient is exponentially decreasing along the iterates. Taken together, our work provides the first fully rigorous proof of accelerated second-order convergence rates for smooth functionals and shows that the scheme performs no worse than the classical scheme JKO scheme for functionals that are $λ$-displacement convex and Wasserstein differentiable.

Guaranteeing Higher Order Convergence Rates for Accelerated Wasserstein Gradient Flow Schemes

TL;DR

This work develops a fully rigorous, higher-order time discretization scheme for Wasserstein gradient flows by marrying Eulerian and Lagrangian viewpoints through a second-order Lagrangian trapezoidal method. Under smooth lifted energies, it achieves convergence, while under weaker assumptions it retains convergence via a discrete EVI framework, with robust stability properties. The authors provide concrete functionals satisfying the smoothness requirements and prove exponential decay of gradients and energy when and the energy is -smooth. They also supply a comprehensive set of numerical experiments confirming the predicted rates and stability, including particle-based implementations of blob-type energies. Overall, the paper delivers the first fully rigorous proof of accelerated second-order convergence rates for smooth Wasserstein gradient flows and clarifies the scheme’s relation to the classical JKO method under displacement convexity.

Abstract

In this paper, we study higher-order-accurate-in-time minimizing movements schemes for Wasserstein gradient flows. We introduce a novel accelerated second-order scheme, leveraging the differential structure of the Wasserstein space in both Eulerian and Lagrangian coordinates. For sufficiently smooth energy functionals, we show that our scheme provably achieves an optimal quadratic convergence rate. Under the weaker assumptions of Wasserstein differentiability and -displacement convexity (for any ), we show that our scheme still achieves a first-order convergence rate and has strong numerical stability. In particular, we show that the energy is nearly monotone in general, while when the energy is -smooth and -displacement convex (with ), we prove the energy is non-increasing and the norm of the Wasserstein gradient is exponentially decreasing along the iterates. Taken together, our work provides the first fully rigorous proof of accelerated second-order convergence rates for smooth functionals and shows that the scheme performs no worse than the classical scheme JKO scheme for functionals that are -displacement convex and Wasserstein differentiable.

Paper Structure

This paper contains 18 sections, 36 theorems, 297 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and previous results
  3. Main results
  4. Preliminaries on the Differential Structure of W2
  5. Notation
  6. Derivatives on Wasserstein Space
  7. Convexity and Differentiability of the Lifted Functional
  8. The Lagrangian Trapezoidal Scheme: Definition and Numerical Stability
  9. O(tau2) Convergence via Differential Methods
  10. Examples of Functionals Satisfying Higher-Order Convergence
  11. O(tau) convergence via Discrete EVI
  12. Discrete EVI
  13. Interpolations and Differential Inequalities
  14. $O(\tau + \eta)$ Convergence of $d_{\tau,\eta}(t;t)$ for $\lambda \geq 0$
  15. $O(\tau + \eta)$ Convergence of $d_{\tau,\eta}(t;t)$ for $\lambda < 0$
  16. ...and 3 more sections

Key Result

Theorem 1.1

Fix a terminal time $T>0$, $\rho_0 \in \mathcal{P}_2(\mathbb{R}^d)$, and assume that the smoothness Assumption smoothness_assumption holds. Define the Lipschitz constants Then, for any time step $\tau \leq 1/L(\phi)$ and integer $n \in \mathbb{N}$ such that $n\tau \leq T$, one has where $\rho^{\tau}_n$ and $X^{\tau}_n$ are defined by trapezoid_variational_form. Here, $\rho_t := (X(t))_{\#} \rho_

Figures (2)

  • Figure 1: Numerical results for the trapezoidal scheme with potentials $(f,V,W) = (0, |x|^2, -\tfrac{1}{4\pi}\log(\varepsilon^2 + |x|^2))$, where $\varepsilon = 10^{-2}$, $N = 64$, and $\tau = 1/25$. (a)–(b) show the particle configurations at the beginning and end of the simulation; (c) displays the energy and Wasserstein gradient norm over time; (d) reports the estimated time–step convergence rate where $\tau_{\text{Ref}} = 1/4096$, while the other $\tau \in [1/1024,1/64]$.
  • Figure 2: Numerical results for the trapezoidal scheme with potentials $(f,V,W) = (\tfrac{1}{2}\log(x+\varepsilon), |x|^2, 0)$, where $\varepsilon = 10^{-2}$, $\sigma = 1/10$, $N = 64$, and $\tau = 1/100$. (a)–(b) show the particle configurations at the beginning and end of the simulation; (c) displays the energy and Wasserstein gradient norm over time; (d) reports the estimated time–step convergence rate where $\tau_{\text{Ref}} = 0.5/4096$, while the other $\tau \in [0.5/1024,0.5/64]$.

Theorems & Definitions (86)

  • Theorem 1.1: $O(\tau^2)$ Convergence: Theorem \ref{['higher_order_convergence_theorem']}, Theorem \ref{['strong_gf_solution']}, Theorem \ref{['limiting_PDE']}
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4: Convexity and Differentiability Assumptions
  • Theorem 1.5: $O(\tau)$ Convergence: Theorem \ref{['linear_convergence_rate']}, Theorem \ref{['refined_convergence']}, Theorem \ref{['strong_gf_solution']}, Theorem \ref{['limiting_PDE']}
  • Theorem 1.6: Numerical Stability: Lemma \ref{['trapezoid_well_defined']}, Lemma \ref{['energy_almost_decreasing']}, Lemma \ref{['refined_decay']}
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 76 more