On the continuous time limit of Ensemble Square Root Filters

TL;DR

This paper analyzes the continuous time limit of Ensemble Square Root Filters, showing convergence to the Kalman-Bucy Filter and establishing a universal limiting behavior for ensemble members, thus providing a rigorous theoretical foundation.

Contribution

It offers a rigorous continuous time limit analysis for Ensemble Square Root Filters, including convergence conditions and a universal limiting equation for ensemble members.

Findings

Empirical mean and covariance converge to Kalman-Bucy Filter in linear case.

Ensemble members' dynamics converge to Ensemble Kalman-Bucy equations.

Results apply to nonlinear Lipschitz-continuous models.

Abstract

We provide a continuous time limit analysis for the class of Ensemble Square Root Filter algorithms with deterministic model perturbations. In the particular linear case, we specify general conditions on the model perturbations implying convergence of the empirical mean and covariance matrix towards their respective counterparts of the Kalman-Bucy Filter. As a second main result we identify additional assumptions for the convergence of the whole ensemble towards solutions of the Ensemble Kalman-Bucy filtering equations introduced in [1]. The latter result can be generalized to nonlinear Lipschitz-continuous model operators. A striking implication of our results is the fact that the limiting equations for the ensemble members are universal for a large class of Ensemble Square Root Filters. This yields a mathematically rigorous justification for the analysis of these algorithms with the help of the Ensemble Kalman-Bucy Filter. [1] de Wiljes, Jana, Reich, Sebastian, Stannat, Wilhelm, Long-Time Stability and Accuracy of the Ensemble Kalman-Bucy Filter for Fully Observed Processes and Small Measurement Noise. SIAM Journal on Applied Dynamical Systems, Vol. 17, No. 2, 1152-1181, 2018