2024/05/02 by Xingyuan Chen, Gonçalo dos Reis, Chen, Xingyuan +5
Computer Science · Mathematics · #Applied mathematics #Convergence (economics) #Economics #FOS: Mathematics #Field (mathematics) #Langevin dynamics #Langevin equation #Mathematics #Mean field theory #Numerical Analysis (math.NA) #Physics #Probability (math.PR) #Pure mathematics #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum mechanics #Statistical physics #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.2405.01346
openalex publication_date 2024/05/02 · openalex created_date 2024/05/05 · openalex updated_date 2026/08/01
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.