Robin Armstrong (Cornell University)
Estimating and Manipulating High-Dimensional Covariance Operators for Ensemble Data Assimilation.
Abstract: Many algorithms in data assimilation and model order reduction rely on sample-based estimates for a covariance matrix associated with the trajectory of a high-dimensional dynamical system. Due to computational constraints, the number of available samples is often far less than the dimension of the underlying state space, meaning that the noisy statistics of the sample must be regularized using a priori knowledge of the covariance structure. This talk will first describe a new technique for estimating a linear operator representation of a high-dimensional covariance matrix using the rank structure of cross-covariance submatrices associated with well-separated subdomains of space. We will establish that low-rank truncations of these submatrices can be estimated with low sample complexity, and we will then show how this can be used to estimate the linear operator action of the covariance as a hierarchically rank-structured matrix. We will then introduce a novel formulation of the ensemble square-root filter (ESRF) that can efficiently assimilate data given only linear operator access to the prior covariance matrix. This filter combines techniques from Krylov subspace iteration and numerical quadrature to avoid evaluating the square-root of the prior covariance, a burdensome calculation that plagues high-dimensional implementations of the ESRF. We will conclude by discussing potential future research directions for these projects.
