High-dimensional estimation of nonlinear transformations for Bayesian filtering
Bayesian filtering algorithms for high-dimensional systems with challenging nonlinear dynamics include ensemble Kalman filters and their recent generalization to parametric nonlinear updates [Spantini et. al.]. These algorithms require estimating a high dimensional transformation that pushes forward a non-Gaussian forecast ensemble into samples from the current filtering distribution. In this presentation, we will analyze the consistency of a constrained maximum likelihood estimator for these nonlinear transformations, leveraging the framework of transport maps. We will study the theoretical performance of this estimator and thus its ability to capture the filtering distribution as a function of the map complexity and the number of ensembles. In addition, we will explore several regularization strategies to reduce the sample size requirements for estimating the map under marginal and conditional independence assumptions on the filtering distribution, as well as sparsity and low-rank structural properties for the map. The numerical performance of these nonlinear filtering algorithms will be presented in the context of chaotic (e.g., Lorenz 96) dynamical systems.
Mr. Ricardo Baptista Massachusetts Institute of Technology