CMX Lunch Seminar

This talk aims to demonstrate how existing and new results in high and infinite-dimensional statistical theory can bring novel understanding to ensemble Kalman methods.

The first part of the talk develops a non-asymptotic analysis of ensemble Kalman updates in high dimension that rigorously explains why a small ensemble size suffices if the prior covariance has moderate effective dimension due to fast spectrum decay or approximate sparsity. We present our theory in a unified framework, comparing several implementations of ensemble Kalman updates that use perturbed observations, square root filtering, and localization. As part of our analysis, we develop new dimension-free covariance matrix estimation bounds for approximately sparse matrices that may be of independent interest.

The second part of the talk studies covariance operator estimation in infinite dimension. We introduce a notion of sparsity for infinite-dimensional covariance operators and a family of thresholded estimators which exploits it. In a small lengthscale regime, we show that thresholded estimators achieve an exponential improvement in sample complexity over the standard sample covariance estimator. Our analysis explains the importance of using covariance localization techniques in ensemble Kalman methods for global data assimilation.