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

TL;DR

This work advances the numerical analysis of mean-field Langevin dynamics by proving that the Leimkuhler–Matthews non-Markovian Euler scheme achieves a weak convergence order of $3/2$ in the long-time limit when sampling the stationary distribution of a one-dimensional MV-SDE, with a rate that is uniform in the IPS dimension. The authors develop a Talay–Tubaro type expansion around the Kolmogorov backward equation and establish uniform-in-time decay for moments of the IPS and its variation processes, enabling a robust weak-error bound of $O(h^{3/2})$ as $T ofty$ while retaining the standard $O(h)$ rate for finite horizons. A detailed analysis of the variation processes up to the $6$-th order, together with decay estimates for the Kolmogorov equation derivatives, underpins the uniform-in-$N$ weak-error control. Numerical experiments on a linear MV-SDE confirm the theoretical gains, showing substantial improvements over Euler in both weak and entropy-based metrics and highlighting favorable uniform-in-time behavior for ergodic sampling in mean-field settings.

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.

Figures (2)

• Figure 1: Simulation of the linear MV-SDE \ref{['eq: example: linear 1']} with $\alpha =0.5, \sigma = 0.8$, $N=10^7, h = 0.16$, and $X_0 \sim \mathcal{N}(\pi,1)$ (a normal distribution with mean value $\pi$ and variance $1$). Both schemes run on the exact same samples of the initial condition and Brownian increments. (a) Entropy Error of the Euler method and non-Markovian method in log-scale over time. (b) $L_2$-Error of the Euler method and non-Markovian method in log-scale over time. (c) $L_2$-Error in particle size $N$ of the Euler method and non-Markovian method in log-scale with respect to different number of particles $N$ at $T=9$.
• Figure 2: Simulation of the linear MV-SDE \ref{['eq: example: linear 1']} with $\alpha =0.5, \sigma = 0.8$, $N=10^7$, and $X_0 \sim \mathcal{N}(1,1)$ (a normal distribution with mean value $1$ and variance $1$) with different choices of time step $h\in \{0.005,0.01,0.02,0.05,0.1\}$ at different time $T\in\{1,3,5\}$; the simulation at $T=7$ or $T=9$ is not different from that in (c) at $T=5$, we thus do not present them. Both schemes run on the exact same samples of the initial condition and Brownian increments. We show the slope of the regression line of the non-Markovian method.

Theorems & Definitions (57)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Definition 3.1
• Example 3.1
• Remark 3.1
• Remark 3.2: On point (2) of Assumption \ref{['assum:main_weak error']}
• Remark 3.3: On point (3) of Assumption \ref{['assum:main_weak error']}
• Lemma 3.2: Weak error expansion, Equation (3.17) in leimkuhler2014long