Ensemble Kalman Sampler: Mean-field Limit and Convergence Analysis
Zhiyan Ding, Qin Li
Abstract
The ensemble Kalman sampler (EKS) is a method introduced in [Garbuno-Inigo et al., SIAM J. Appl. Dyn. Syst., 19 (2020), pp. 412--441] to find approximately independent and identically distributed samples from a target distribution. As of today, why the algorithm works and how it converges is mostly unknown. The continuous version of the algorithm is a set of coupled stochastic differential equations (SDEs). In this paper, we prove the well posedness of the SDE system and justify its mean-field limit is a Fokker--Planck equation, whose long time equilibrium is the target distribution. We further demonstrate that the convergence rate is near optimal ($J^{-1/2}$ with $J$ being the number of particles). These results, combined with the in-time convergence of the Fokker--Planck equation to its equilibrium [J. A. Carrillo and U. Vaes, preprint, arXiv:1910.07555, 2019] justify the validity of the EKS, and provide the convergence rate as a sampling method.