Multilevel Ensemble Kalman–Bucy Filters
Neil K. Chada, Ajay Jasra, Fangyuan Yu · SIAM/ASA Journal on Uncertainty Quantification · 2022
Abstract. In this article we consider the linear filtering problem in continuous time. We develop and apply multilevel Monte Carlo (MLMC) strategies for ensemble Kalman–Bucy filters (EnKBFs). These filters can be viewed as approximations of conditional McKean–Vlasov-type diffusion processes. They are also interpreted as the continuous-time analogue of the ensemble Kalman filter, which has proven to be successful due to its applicability and computational cost. We prove that an ideal version of our multilevel EnKBF can achieve a mean square error (MSE) of [Formula: see text], [Formula: see text], with a cost of order [Formula: see text]. In order to prove this result we provide a Monte Carlo convergence and approximation bounds associated to time-discretized EnKBFs. This implies a reduction in cost compared to the (single level) EnKBF which requires a cost of [Formula: see text] to achieve an MSE of [Formula: see text]. We test our theory on a linear Ornstein–Uhlenbeck process, which we motivate through high-dimensional examples of order [Formula: see text] and [Formula: see text], where we also numerically test an alternative deterministic counterpart of the EnKBF.