High-Dimensional Ensemble Filtering with Nonlinear Couplings

We consider the Bayesian filtering problem for high dimensional non-Gaussian state-space models with challenging nonlinear dynamics, and sparse measurements in space and time. While the Ensemble Kalman filter (EnKF) yields robust approximations of the filtering distribution, it is limited by linear forecast-to-analysis transformations. To generalize the EnKF, we propose a methodology that transforms the non-Gaussian forecast ensemble at each assimilation step into samples from the current filtering distribution via a sequence of nonlinear local couplings. These couplings are based on transport maps that can be computed quickly using convex optimization and inherit low-dimensional structure from the filtering problem (e.g., decay of correlation, conditional independence, and local likelihoods). In this presentation, we exploit this structure to regularize the estimation of these maps in high dimensions and reduce their ensemble size requirements. The numerical performance of our algorithms will be presented in the context of chaotic dynamical systems (e.g., Lorenz 96).