| 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, necessitating the use of regularization techniques such as spatial localization. This talk will examine systems whose behavior is governed by a range of widely separated correlation lengthscales, giving rise to a covariance structure that is not spatially localized. We will show how to estimate these covariance matrices from a small number of samples by exploiting the rank structure of submatrices representing cross-covariances between well-separated domains of space. Our techniques produce covariance estimators that are hierarchically rank-structured; as a result, they have a small memory footprint and support highly efficient matrix computations for downstream tasks in data assimilation and model reduction. Using both theoretical error bounds and numerical experiments with a variety of dynamical systems, we will demonstrate that our estimators can accurately recover the underlying covariance matrix from a very small number of samples. |
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