Ergodicity and long-time behavior of the Random Batch Method for interacting particle systems

TL;DR

This paper proves that the Random Batch Method for interacting particle systems converges exponentially to its invariant distribution independently of system size, and can accurately approximate the original system's distribution with less computation.

Contribution

It establishes geometric ergodicity and quantifies the approximation error of the Random Batch Method for large-scale interacting particle systems.

Findings

Exponential convergence to invariant distributions for both IPS and RB-IPS.

Convergence rate independent of particle number, batch size, and time step.

Bound on Wasserstein distance showing accurate sampling with reduced computational cost.

Abstract

We study the geometric ergodicity and the long time behavior of the Random Batch Method for interacting particle systems, which exhibits superior numerical performance in recent large-scale scientific computing experiments. We show that for both the interacting particle system (IPS) and the random batch interacting particle system (RB-IPS), the distribution laws converge to their respective invariant distributions exponentially, and the convergence rate does not depend on the number of particles $N$, the time step $\tau$ for batch divisions or the batch size $p$. Moreover, the Wasserstein distance between the invariant distributions of the IPS and the RB-IPS is bounded by $O(\sqrt{\tau})$, showing that the RB-IPS can be used to sample the invariant distribution of the IPS accurately with greatly reduced computational cost.