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Convergence rates of particle approximation of forward-backward splitting algorithm for granular medium equations

Matej Benko, Iwona Chlebicka, Jørgen Endal, Błażej Miasojedow

TL;DR

The paper studies the spatially homogeneous granular medium equation with confinement and nonlocal interactions, and develops forward-backward splitting particle algorithms to approximate solutions. It establishes sharp Wasserstein convergence rates for both the local (W=0) and nonlocal (W≠0) problems, supported by moment bounds and propagation of chaos results. The analysis integrates gradient-flow and JKO perspectives, and handles non-globally Lipschitz potentials with provable stability under approximate proximal steps. Numerical simulations in one and two dimensions corroborate the theoretical rates and illustrate the practical performance of the proposed methods.

Abstract

We study the spatially homogeneous granular medium equation \[\partial_tμ=\rm{div}(μ\nabla V)+\rm{div}(μ(\nabla W \ast μ))+Δμ\,,\] within a large and natural class of the confinement potentials $V$ and interaction potentials $W$. The considered problem do not need to assume that $\nabla V$ or $\nabla W$ are globally Lipschitz. With the aim of providing particle approximation of solutions, we design efficient forward-backward splitting algorithms. Sharp convergence rates in terms of the Wasserstein distance are provided.

Convergence rates of particle approximation of forward-backward splitting algorithm for granular medium equations

TL;DR

The paper studies the spatially homogeneous granular medium equation with confinement and nonlocal interactions, and develops forward-backward splitting particle algorithms to approximate solutions. It establishes sharp Wasserstein convergence rates for both the local (W=0) and nonlocal (W≠0) problems, supported by moment bounds and propagation of chaos results. The analysis integrates gradient-flow and JKO perspectives, and handles non-globally Lipschitz potentials with provable stability under approximate proximal steps. Numerical simulations in one and two dimensions corroborate the theoretical rates and illustrate the practical performance of the proposed methods.

Abstract

We study the spatially homogeneous granular medium equation within a large and natural class of the confinement potentials and interaction potentials . The considered problem do not need to assume that or are globally Lipschitz. With the aim of providing particle approximation of solutions, we design efficient forward-backward splitting algorithms. Sharp convergence rates in terms of the Wasserstein distance are provided.
Paper Structure (11 sections, 21 theorems, 168 equations, 5 figures, 4 algorithms)

This paper contains 11 sections, 21 theorems, 168 equations, 5 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and basics
  4. On the existence and uniqueness of solutions
  5. Propagation of chaos to handle nonlocal interactions
  6. Numerical approximation of the local problem
  7. Approximated Proximal Split Discretized Gradient Flow
  8. Comments on the possible extensions and open problems
  9. Simulations
  10. One-dimensional models
  11. Two-dimensional models

Key Result

Theorem 1

Let $\overline{\mu}_\infty$ be the unique minimizer of ${\mathcal{F}}$ given by cF (with $W\equiv0$) for $V$ that satisfies V with $\lambda_V > 0$ and $q_V\geq 1$. Assume further that $\mu$ is a gradient flow solution to eq:model with $W\equiv 0$ and initial datum $\mu_0\in{\mathscr{P}}_2({\mathbb{R

Figures (5)

  • Figure 1: Summary of our approach positioning the content of Theorem \ref{['thm:alg_bound-intro']} provided for Algorithm \ref{['alg:dgf']}, together with Theorems \ref{['thm:alg_bound-nonlocal']} and \ref{['theo:chaos']} for Algorithm \ref{['alg:particle']}.
  • Figure 2: Evolution of density in one dimensional example with mixture of Gaussian as initial distribution, cf. \ref{['eq:initial_1d']}. We show $10\,000$ trajectories for $\tau=10^{-3}$. We run Algorithm \ref{['alg:dgf']} for Model 1, Algorithm \ref{['alg:particle']} for Model 2, and Algorithm \ref{['alg:particle']} with a perturbed proximal step (as in Algorithm \ref{['alg:dgf-pert']}) for Models 3 and 4.
  • Figure 3: Comparison of theoretical convergence rate due (orange) and empirical (blue) for Model E ($W\equiv 0$ and $V_2$) with respect to time step $\tau$, according to Theorem \ref{['thm:alg_bound-intro']}, given as average from $15$ replications, for $N=100\,000$.
  • Figure 4: Comparison of empirical convergence rate (blue) and theoretical one (orange) for Model F (attractive) with respect to the number of particles $N$, according to Theorem \ref{['thm:alg_bound-nonlocal']}, given as average from $15$ replications, with $\tau=10^{-6}$.
  • Figure 5: Numerical solution given by Algorithm \ref{['alg:particle']} with a perturbed proximal step as in Algorithm \ref{['alg:dgf-pert']} for Model G (repulsive) and Model H (attractive). Simulation is done with $N=1\, 000$ particles and time step $\tau=10^{-3}$.

Theorems & Definitions (52)

  • Theorem 1
  • Theorem 2
  • Proposition 2.1: EVI, Ambrosio-Gigli-Savare
  • Theorem 3: Moments bound
  • Remark 3.1
  • proof
  • Theorem 4: Propagation of chaos
  • proof
  • Remark 3.2: Relaxing $\lambda$-convexity
  • Lemma 4.1
  • ...and 42 more