An example of Ensemble Kalman Filter with resampling
This paper introduces the Exact Ensemble Kalman Filter (ExEnKF), a novel algorithm for state estimation in discrete-time nonlinear filtering problems with linear observations. Unlike traditional Ensemble Kalman Filters (EnKFs), which approximate the filtering distribution using ensembles of Dirac measures, the ExEnKF employs Gaussian measures, enabling more efficient exploration of the state space and potentially alleviating the curse of dimensionality. We prove the algorithm's asymptotic consistency with the optimal filter (Theorem 3.1), establishing a convergence rate of order 1/ $\sqrt$ N for N particles. Numerical experiments on the Lorenz-96 multiscale model demonstrate that the ExEnKF outperforms the standard EnKF under model misspecification and poor initialization, particularly in highly stochastic regimes. The algorithm's robustness is further highlighted by its ability to track hidden components of the true signal, even when observations are generated from a different model (e.g., multiscale vs. single-scale). This work advances the theoretical understanding of ensemble methods in nonlinear filtering and provides a practical alternative to sequential Monte Carlo methods for high-dimensional systems