Mean field limit of Ensemble Square Root Filters -- discrete and continuous time

Abstract

Consider the class of Ensemble Square Root filtering algorithms for the numerical approximation of the posterior distribution of nonlinear Markovian signals partially observed with linear observations corrupted with independent measurement noise. We analyze the asymptotic behavior of these algorithms in the large ensemble limit both in discrete and continuous time. We identify limiting mean-field processes on the level of the ensemble members, prove corresponding propagation of chaos results and derive associated convergence rates in terms of the ensemble size. In continuous time we also identify the stochastic partial differential equation driving the distribution of the mean-field process and perform a comparison with the Kushner-Stratonovich equation.

arXiv commentAccepted for publication in Foundations of Data Science. Revised version: corrected typos, added more explanations, updated Theorem 3.1 (holds for all p >=2)

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• Analysis of the Ensemble Kalman--Bucy Filter for correlated observation noise
• Analysis of the feedback particle filter with diffusion map based approximation of the gain