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

TL;DR

This paper analyzes the large ensemble limit of Ensemble Square Root Filters for nonlinear signal estimation, establishing mean-field limits, propagation of chaos, and convergence rates in both discrete and continuous time.

Contribution

It provides a rigorous mean-field analysis of Ensemble Square Root Filters, including convergence rates and the derivation of a stochastic PDE in continuous time.

Findings

Identification of limiting mean-field processes

Propagation of chaos results with convergence rates

Derivation of a stochastic PDE in 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.