On the continuous time limit of the ensemble Kalman filter

Abstract

We present recent results on the existence of a continuous time limit for Ensemble Kalman Filter algorithms. In the setting of continuous signal and observation processes, we apply the original Ensemble Kalman Filter algorithm proposed by Burgers, van Leeuwen, and Evensen [Monthly Weather Review 126 (1998), pp. 1719–1724] as well as a recent variant of de Wiljes, Reich, and Stannat [SIAM J. Appl. Dyn. Syst. 17 (2018), no. 2, pp. 1152–1181] to the respective discretizations and show that in the limit of decreasing stepsize the filter equations converge to an ensemble of interacting (stochastic) differential equations in the ensemble-mean-square sense. Our analysis also allows for the derivation of convergence rates with respect to the stepsize. An application of our analysis is the rigorous derivation of continuous-time ensemble filtering algorithms consistent with discrete-time approximation schemes. Conversely, the continuous time limit allows for a better qualitative and quantitative analysis of the discrete-time counterparts using the rich theory of dynamical systems in continuous time.