Improved weak convergence for the long time simulation of Mean-field Langevin equations
<|reference_start|>Improved weak convergence for the long time simulation of Mean-field Langevin equations: 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.<|reference_end|>
@article{chen2024improved,
    title={Improved weak convergence for the long time simulation of Mean-field
  Langevin equations},
    author={Xingyuan Chen and Goncalo dos Reis and Wolfgang Stockinger and Zac
  Wilde},
    journal={arXiv preprint arXiv:2405.01346},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.01346},
    primaryClass={math.NA cs.NA math.PR}
}