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

Understand

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.