Improved weak convergence for the long time simulation of Mean-field Langevin equations

TL;DR

This paper demonstrates that the Leimkuhler–Matthews method achieves higher-order weak convergence for long-time simulation of mean-field Langevin equations, outperforming the standard Euler method, with convergence rates independent of system dimension.

Contribution

The paper introduces a higher-order weak convergence analysis for the Leimkuhler–Matthews method applied to mean-field Langevin equations, showing improved long-time accuracy under strong convexity.

Findings

Achieves weak order 3/2 in long-time limit

Convergence rate independent of system dimension

Supported by numerical validation

Abstract

We study the weak convergence behaviour of the Leimkuhler--Matthews method, a non-Markovian Euler-type scheme with the same computational cost as the Euler scheme, for the approximation of the stationary distribution of a one-dimensional McKean--Vlasov Stochastic Differential Equation (MV-SDE). The particular class under study is known as mean-field (overdamped) Langevin equations (MFL). We provide weak and strong error results for the scheme in both finite and infinite time. We work under a strong convexity assumption. Based on a careful analysis of the variation processes and the Kolmogorov backward equation for the particle system associated with the MV-SDE, we show that the method attains a higher-order approximation accuracy in the long-time limit (of weak order convergence rate $3/2$) than the standard Euler method (of weak order $1$). While we use an interacting particle system (IPS) to approximate the MV-SDE, we show the convergence rate is independent of the dimension of the IPS and this includes establishing uniform-in-time decay estimates for moments of the IPS, the Kolmogorov backward equation and their derivatives. The theoretical findings are supported by numerical tests.