On Random Batch Methods (RBM) for interacting particle systems driven by Lévy processes

TL;DR

This work introduces RBM-Lévy, a random batch strategy for simulating interacting particle systems driven by Lévy noise. By randomly partitioning particles into batches of size $p$ and updating only intra-batch interactions, the method reduces per-step complexity from $O(N^2)$ to $O(pN)$ while provably converging to the original Lévy-driven system as the time step $\kappa \to 0$. The convergence is established in Wasserstein distances for Lévy measures with both finite and infinite second moments, employing synchronous coupling and regularization to handle jump discontinuities; corollaries extend to rotationally invariant $\alpha$-stable noise. Numerical experiments on simple 1D tests and stochastic Cucker-Smale models confirm the predicted $O(\sqrt{\kappa})$ rate and demonstrate substantial computational savings, highlighting RBM-Lévy's practicality for heavy-tailed, jump-driven systems.

Abstract

In many real-world scenarios, the underlying random fluctuations are non-Gaussian, particularly in contexts where heavy-tailed data distributions arise. A typical example of such non-Gaussian phenomena calls for Lévy noise, which accommodates jumps and extreme variations. We propose the Random Batch Method for interacting particle systems driven by Lévy noises (RBM-Lévy), which can be viewed as an extension of the original RBM algorithm in [Jin et al. J Compt Phys, 2020]. In our RBM-Lévy algorithm, $N$ particles are randomly grouped into small batches of size $p$, and interactions occur only within each batch for a short time. Then one reshuffles the particles and continues to repeat this shuffle-and-interact process. In other words, by replacing the strong interacting force by the weak interacting force, RBM-Lévy dramatically reduces the computational cost from $O(N^2)$ to $O(pN)$ per time step. Meanwhile, the resulting dynamics converges to the original interacting particle system, even at the appearance of the Lévy jump. We rigorously prove this convergence in Wasserstein distance, assuming either a finite or infinite second moment of the Lévy measure. Some numerical examples are given to verify our convergence rate.

Figures (6)

• Figure 1: Error of RBM-Lévy versus time step at terminal time $T=1$ (log-log scale): (a):$a = 1$ (with confining condition), (b):$a=0$ (without confining condition). The particle number $N$ is $50, 100, 1000$ in each figure.
• Figure 2: (a): Error of RBM-Lévy versus time $T$ with configuration $T \in \{1,2,4,8,16 \}$. $\kappa = 2^{-7}$, $N=100$. (b): CPU running time versus particle number $N$. $a=1$, $\kappa = 2^{-7}$, $T=1$.
• Figure 3: $\sigma = 1$, $\lambda = 0.1$ (white noise + jump). flocking.(a): velocity evolution of each particle without randam batch. (b): velocity evolution of each particle with random batch. (c): time evolution of $D_v(t)$(d): time evolution of $D_x(t)$.
• Figure 4: $\sigma = 1$, $\lambda = 0$ (white noise only). flocking.(a): velocity evolution of each particle without randam batch. (b): velocity evolution of each particle with random batch. (c): time evolution of $D_v(t)$(d): time evolution of $D_x(t)$.
• Figure 5: $\sigma = 0$, $\lambda = 0.1$ (jump only). unflocking.(a): velocity evolution of each particle without randam batch. (b): velocity evolution of each particle with random batch. (c): time evolution of $D_v(t)$(d): time evolution of $D_x(t)$.

Theorems & Definitions (12)

• Theorem 1
• proof
• Theorem 2
• proof
• Corollary 1: rotational invariant $\alpha$-stable case
• Corollary 2
• Lemma 1: uniform-in-time moment estimate
• proof
• Lemma 2: Hölder's continuity for the process $Z^1$
• proof