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Nonlocal particle approximation for linear and fast diffusion equations

José Antonio Carrillo, Antonio Esposito, Jakub Skrzeczkowski, Jeremy Sheung-Him Wu

TL;DR

This work develops a deterministic, nonlocal particle framework to approximate linear and fast-diffusion PDEs as $2$-Wasserstein gradient flows. By regularising the internal energy with mollified kernels $V_\varepsilon$ (compactly or globally supported) and employing the JKO scheme, it constructs weak solutions to nonlocal heat and fast-diffusion equations and proves convergence to the local equations as $(\varepsilon,\sigma)\to(0,0)$ (or in steps). A key technical contribution is a commutator lemma bounding the noncommutativity of convolution with nonlinearities, enabling the nonlocal-to-local limit for globally supported kernels, together with geodesic convexity results that yield convergence rates and a deterministic particle scheme. The fast-diffusion extension uses a Barenblatt-type kernel and shows analogous nonlocal-to-local limits and particle-approximation results. Overall, the framework provides rigorous, rate-controlled deterministic particle methods for sampling Gibbs-type and heavy-tailed distributions via gradient-flow dynamics.

Abstract

We construct deterministic particle solutions for linear and fast diffusion equations using a nonlocal approximation. We exploit the $2$-Wasserstein gradient flow structure of the equations in order to obtain the nonlocal approximating PDEs by regularising the corresponding internal energy with suitably chosen mollifying kernels, either compactly or globally supported. Weak solutions are obtained by the JKO scheme. From the technical point of view, we improve known commutator estimates, fundamental in the nonlocal-to-local limit, to include globally supported kernels which, in particular cases, allow us to justify the limit without any further perturbation needed. Furthermore, we prove geodesic convexity of the nonlocal energies in order to prove convergence of the particle solutions to the nonlocal equations towards weak solutions of the local equations. We overcome the crucial difficulty of dealing with the singularity of the first variation of the free energies at the origin. As a byproduct, we provide convergence rates expressed as a scaling relationship between the number of particles and the localisation parameter. The analysis we perform leverages the fact that globally supported kernels yield a better convergence rate compared to compactly supported kernels. Our result is relevant in statistics, more precisely in sampling Gibbs and heavy-tailed distributions.

Nonlocal particle approximation for linear and fast diffusion equations

TL;DR

This work develops a deterministic, nonlocal particle framework to approximate linear and fast-diffusion PDEs as -Wasserstein gradient flows. By regularising the internal energy with mollified kernels (compactly or globally supported) and employing the JKO scheme, it constructs weak solutions to nonlocal heat and fast-diffusion equations and proves convergence to the local equations as (or in steps). A key technical contribution is a commutator lemma bounding the noncommutativity of convolution with nonlinearities, enabling the nonlocal-to-local limit for globally supported kernels, together with geodesic convexity results that yield convergence rates and a deterministic particle scheme. The fast-diffusion extension uses a Barenblatt-type kernel and shows analogous nonlocal-to-local limits and particle-approximation results. Overall, the framework provides rigorous, rate-controlled deterministic particle methods for sampling Gibbs-type and heavy-tailed distributions via gradient-flow dynamics.

Abstract

We construct deterministic particle solutions for linear and fast diffusion equations using a nonlocal approximation. We exploit the -Wasserstein gradient flow structure of the equations in order to obtain the nonlocal approximating PDEs by regularising the corresponding internal energy with suitably chosen mollifying kernels, either compactly or globally supported. Weak solutions are obtained by the JKO scheme. From the technical point of view, we improve known commutator estimates, fundamental in the nonlocal-to-local limit, to include globally supported kernels which, in particular cases, allow us to justify the limit without any further perturbation needed. Furthermore, we prove geodesic convexity of the nonlocal energies in order to prove convergence of the particle solutions to the nonlocal equations towards weak solutions of the local equations. We overcome the crucial difficulty of dealing with the singularity of the first variation of the free energies at the origin. As a byproduct, we provide convergence rates expressed as a scaling relationship between the number of particles and the localisation parameter. The analysis we perform leverages the fact that globally supported kernels yield a better convergence rate compared to compactly supported kernels. Our result is relevant in statistics, more precisely in sampling Gibbs and heavy-tailed distributions.
Paper Structure (18 sections, 42 theorems, 293 equations)

This paper contains 18 sections, 42 theorems, 293 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nonlocal approximating equation for the heat equation
  4. Perturbation by vacuum exclusion
  5. Uniform-in-$\sigma$,$\varepsilon$ estimates and compactness
  6. Nonlocal-to-local limit for the heat equation
  7. Commutator estimates
  8. Joint limit $\sigma, \varepsilon \to 0$
  9. First $\sigma \to 0$, then $\varepsilon \to 0$.
  10. Particle approximation for the heat equation
  11. Extension to Fast Diffusion equations
  12. Nonlocal equation for fast diffusion
  13. Nonlocal-to-local limit
  14. Particle approximation
  15. Auxiliary results
  16. ...and 3 more sections

Key Result

Lemma 3.1

Let $\varepsilon>0$, $\sigma \in [0,1)$. There exists an absolutely continuous curve $\rho^{\varepsilon, \sigma}:[0,T]\to {\mathcal{P}_2({\mathbb{R}^{d}})}$ such that, up to a subsequence, we have There is a constant $C>0$ depending only on $T, \,m_2(\rho_0), \, m_2(V_1)$, and $\mathcal{H}_\sigma^\varepsilon[\rho_0]$ where $\rho_0 = \rho_\tau^{\varepsilon, \sigma}(0) = \rho^{\varepsilon, \sigma}(

Theorems & Definitions (92)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.1: Weak measure solution to \ref{['eq:nonlocal_he_gauss']}
  • Remark 2.3
  • Definition 2.2: Weak measure solution to \ref{['eq:nonlocal_he']}
  • Remark 2.4
  • Definition 2.3: Weak solution to \ref{['eq:heat_eq']}
  • Lemma 3.1
  • proof
  • Theorem 3.1
  • ...and 82 more