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Wasserstein convergence rates for stochastic particle approximation of Boltzmann models

Giacomo Borghi, Lorenzo Pareschi

TL;DR

This work develops a rigorous framework to quantify Wasserstein-1 convergence rates for stochastic particle approximations of Boltzmann-type equations via Nanbu-type Monte Carlo schemes. By coupling Nanbu particles to forward Euler dynamics and employing stability in the $W_1$ metric, the authors derive explicit error bounds that capture the nonlinear interaction structure and moment requirements, and extend the theory to Time Relaxed Monte Carlo methods with asymptotic-preserving properties. A key by-product is the existence and uniqueness of weak measure solutions for a broad class of Boltzmann-type equations. The results bridge probabilistic particle methods and deterministic numerical analysis, providing a unified convergence theory for Monte Carlo solvers in kinetic models and offering practical guidance for high-dimensional simulations.

Abstract

We establish quantitative convergence rates for stochastic particle approximation based on Nanbu-type Monte Carlo schemes applied to a broad class of collisional kinetic models. Using coupling techniques and stability estimates in the Wasserstein-1 (Kantorovich-Rubinstein) metric, we derive sharp error bounds that reflect the nonlinear interaction structure of the models. Our framework includes classical Nanbu Monte Carlo method and more recent developments as Time Relaxed Monte Carlo methods. The results bridge the gap between probabilistic particle approximations and deterministic numerical error analysis, and provide a unified perspective for the convergence theory of Monte Carlo methods for Boltzmann-type equations. As a by-product, we also obtain existence and uniqueness of solutions to a large class of Boltzmann-type equations.

Wasserstein convergence rates for stochastic particle approximation of Boltzmann models

TL;DR

This work develops a rigorous framework to quantify Wasserstein-1 convergence rates for stochastic particle approximations of Boltzmann-type equations via Nanbu-type Monte Carlo schemes. By coupling Nanbu particles to forward Euler dynamics and employing stability in the metric, the authors derive explicit error bounds that capture the nonlinear interaction structure and moment requirements, and extend the theory to Time Relaxed Monte Carlo methods with asymptotic-preserving properties. A key by-product is the existence and uniqueness of weak measure solutions for a broad class of Boltzmann-type equations. The results bridge probabilistic particle methods and deterministic numerical analysis, providing a unified convergence theory for Monte Carlo solvers in kinetic models and offering practical guidance for high-dimensional simulations.

Abstract

We establish quantitative convergence rates for stochastic particle approximation based on Nanbu-type Monte Carlo schemes applied to a broad class of collisional kinetic models. Using coupling techniques and stability estimates in the Wasserstein-1 (Kantorovich-Rubinstein) metric, we derive sharp error bounds that reflect the nonlinear interaction structure of the models. Our framework includes classical Nanbu Monte Carlo method and more recent developments as Time Relaxed Monte Carlo methods. The results bridge the gap between probabilistic particle approximations and deterministic numerical error analysis, and provide a unified perspective for the convergence theory of Monte Carlo methods for Boltzmann-type equations. As a by-product, we also obtain existence and uniqueness of solutions to a large class of Boltzmann-type equations.

Paper Structure

This paper contains 18 sections, 9 theorems, 77 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Collisional kinetic equations and Nanbu method
  3. Contribution of the paper
  4. Literature review
  5. Outline of the paper
  6. Main results
  7. Assumptions and examples
  8. Convergence of Monte Carlo approximations
  9. Convergence of forward Euler and full error analysis
  10. Extension to Time Relaxed Monte Carlo methods
  11. Preliminaries and notation
  12. Error analysis of Nanbu Monte Carlo method
  13. The Nanbu particle system
  14. The nonlinear particle system
  15. Coupling and proof of Theorem \ref{['t:nanbu']}
  16. ...and 3 more sections

Key Result

Theorem 2.3

Let $f_0 \in \mathcal{P}_q( {\mathbb{R}^d})$, with $q>2$ for $d \leq 2$ and $q> d/(d-1)$ for $d>2$, and let $B$ be a kernel such that $B\in \mathcal{P}_{\infty}(\Theta)$. Consider $f_n$ to be the forward Euler discretization eq:euler for $\Delta t\in(0,1]$ to the Boltzmann-like equation eq:boltzmann with $M_q(f_0) = \int |v|^q f_0(v)\textup{d} v$, $\varepsilon(N)$ given by eq:eps1, and $C, \tilde{

Theorems & Definitions (21)

  • Remark 2.2
  • Theorem 2.3: Convergence of Nanbu method
  • Theorem 2.4: Convergence of forward Euler method
  • Corollary 2.5: Full error analysis
  • Theorem 2.6: Convergence of first order TRMC method
  • Remark 2.7
  • Remark 2.8
  • Theorem 3.1: fournier2015rate
  • Lemma 3.1: Moments estimate
  • proof
  • ...and 11 more