The paper studies the weak convergence behavior of the Leimkuhler-Matthews non-Markovian Euler-type scheme for approximating the stationary distribution of a one-dimensional mean-field (overdamped) Langevin equation (MFL).
The key highlights and insights are:
The authors provide weak and strong error results for the non-Markovian Euler scheme in both finite and infinite time horizons, under a strong convexity assumption on the potentials.
Through a careful analysis of the variation processes and the Kolmogorov backward equation for the particle system associated with the mean-field SDE, the authors show that the non-Markovian Euler scheme attains a higher-order weak approximation accuracy in the long-time limit (weak order 3/2) compared to the standard Euler method (weak order 1).
The convergence rate is shown to be independent of the dimension of the interacting particle system (IPS) used to approximate the mean-field SDE. This is achieved by 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.
The analysis involves several technical challenges, including deriving suitable Lp-norm estimates for the solution to the Kolmogorov backward equation that decay exponentially in time in a non-explosive way in the number of particles, as well as establishing Lp-norm estimates for the variation processes of the IPS flow that are uniform in the number of particles and time.
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arxiv.org
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