Convergence of Random Batch Method for interacting particles with disparate species and weights

TL;DR

This paper proves the convergence properties of the Random Batch Method for interacting particles with different species and weights, showing strong and weak error bounds that are uniform in the number of particles.

Contribution

It extends previous convergence analysis of the Random Batch Method to cases with disparate species and weights, providing new error bounds and refined proofs.

Findings

Strong error is of order $O(\sqrt{ au})$

Weak error is of order $O( au)$

Convergence is uniform in the number of particles $N$

Abstract

We consider in this work the convergence of Random Batch Method proposed in our previous work [Jin et al., J. Comput. Phys., 400(1), 2020] for interacting particles to the case of disparate species and weights. We show that the strong error is of $O(\sqrt{\tau})$ while the weak error is of $O(\tau)$ where $\tau$ is the time step between two random divisions of batches. Both types of convergence are uniform in $N$, the number of particles. The proof of strong convergence follows closely the proof in [Jin et al., J. Comput. Phys., 400(1), 2020] for indistinguishable particles, but there are still some differences: since there is no exchangeability now, we have to use a certain weighted average of the errors; some refined auxiliary lemmas have to be proved compared with our previous work. To show that the weak convergence of empirical measure is uniform in $N$, certain sharp estimates for the derivatives of the backward equations have been used. The weak convergence analysis is also illustrating for the convergence of Random Batch Method for $N$-body Liouville equations.