Non-Asymptotic Analysis of Ensemble Kalman Updates: Effective Dimension and Localization

Abstract

Many modern algorithms for inverse problems and data assimilation rely on ensemble Kalman updates to blend prior predictions with observed data. Ensemble Kalman methods often perform well with a small ensemble size, which is essential in applications where generating each particle is costly. This paper develops a non-asymptotic analysis of ensemble Kalman updates 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 estimation bounds for approximately sparse matrices that may be of independent interest.

References in corpus (12)

• Regularized estimation of large covariance matrices
• Covariance regularization by thresholding
• The Ensemble Kalman Filter for Inverse Problems
• Curse-of-dimensionality revisited: Collapse of the particle filter in very large scale systems
• Operator norm consistent estimation of large-dimensional sparse covariance matrices
• Optimal rates of convergence for sparse covariance matrix estimation
• Adaptive covariance matrix estimation through block thresholding
• Convergence of the Square Root Ensemble Kalman Filter in the Large Ensemble Limit
• Localization in Ensemble Kalman inversion
• Ensemble Kalman Methods: A Mean Field Perspective
• Mean field limit of Ensemble Square Root Filters -- discrete and continuous time
• Ensemble Kalman Filters with Resampling

Cited by in corpus (3)

• Localization in Ensemble Kalman inversion
• Uniform error bounds of the ensemble transform Kalman filter for infinite-dimensional dynamics with multiplicative covariance inflation
• Covariance Operator Estimation via Adaptive Thresholding