Yousef Marzouk, MIT
TITLE: "Nonlinear filtering and smoothing with transport maps"
Bayesian inference for non-Gaussian state-space models is a ubiquitous problem, arising in applications from geophysical data assimilation to mathematical finance. We will present a broad introduction to these problems and then focus on high dimensional models with nonlinear (potentially chaotic) dynamics and sparse observations in space and time. While the ensemble Kalman filter (EnKF) yields robust ensemble approximations of the filtering distribution in this setting, 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 local nonlinear couplings. These couplings are based on transport maps that can be computed quickly using convex optimization, and that can be enriched in complexity to reduce the intrinsic bias of the EnKF. We discuss the low-dimensional structure inherited by the transport maps from the filtering problem, including decay of correlations, conditional independence, and local likelihoods. We then exploit this structure to regularize the estimation of the maps in high dimensions and with a limited ensemble size.
We also present variational methods---again based on transport maps---for smoothing and sequential parameter estimation in non-Gaussian state-space models. These methods rely on results linking the Markov properties of a target measure to the existence of low-dimensional couplings, induced by transport maps that are decomposable. The resulting algorithms can be understood as a generalization, to the non-Gaussian case, of the square-root Rauch--Tung--Striebel Gaussian smoother.
This is joint work with Ricardo Baptista, Daniele Bigoni, and Alessio Spantini.