Speaker
Description
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. The number of available samples is often far less than the dimension of the underlying state space because of computational constraints. Under these circumstances, extracting meaningful covariance information requires that the noisy statistics of the sample are regularized with a structural assumption. This talk will describe a regularization technique that exploits the rank structure of submatrices representing cross-covariances between well-separated domains of space. We will establish that low-rank truncations of these submatrices can be estimated from fewer samples than the submatrices themselves. We will then show how this fact can be used to encode physics-informed regularizing assumptions onto the sample statistics, resulting in a hierarchically rank-structured covariance estimator. Through numerical experiments with a variety of dynamical systems, we will demonstrate that these techniques are effective at reducing sampling errors in the covariance.