Nonlinear filtering with local couplings
We consider the problem of filtering a high dimensional non-Gaussian state-space model, with intractable transition kernels, challenging nonlinear dynamics (e.g., chaotic), and sparse measurements in space and time. We propose a novel filtering methodology, which leverages elements of measure transport, convex optimization, and non-Gaussian graphical models, to yield robust ensemble approximations of the filtering distribution in high dimensions. We provide a generalization of the Ensemble Kalman filter (EnKF) to nonlinear updates. The idea is to transform the forecast ensemble into samples from the current filtering distribution via a sequence of local (in state-space) nonlinear couplings, i.e., low-dimensional transport maps that can be computed via fast convex optimization. The computation of the maps is regularized by leveraging potential structure in the filtering problem e.g., decay of correlations, approximate conditional independence, and local likelihoods thus extending notions of localization to nonlinear updates. These transformations implicitly approximate the projection of the filtering distribution onto a manifold of sparse and non-Gaussian Markov random fields. Many popular EnKF algorithms can be derived as special instances of the proposed framework, when we restrict our attention to linear transformations. We consider applications to chaotic dynamical systems.
Dr. Alessio Spantini Massachusetts Institute of Technology