Communications in Mathematical Sciences

Volume 16 (2018)

Number 4

Rigorous accuracy and robustness analysis for two-scale reduced random Kalman filters in high dimensions

Pages: 1095 – 1132



Andrew J. Majda (Courant Institute of Mathematical Sciences, New York University, New York, N.Y, U.S.A.)

Xin T. Tong (Department of Mathematics, National University of Singapore)


Contemporary data assimilation often involves millions of prediction variables. The classical Kalman filter is no longer computationally feasible in such a high dimensional context. This problem can often be resolved by exploiting the underlying multiscale structure, applying the full Kalman filtering procedures only to the large scale variables, and estimating the small scale variables with proper statistical strategies, including multiplicative inflation, representation model error in the observations, and crude localization. The resulting two-scale reduced filters can have close to optimal numerical filtering skill based on previous numerical evidence. Yet, no rigorous explanation exists for this success, because these modifications create unavoidable bias and model error. This paper contributes to this issue by establishing a new error analysis framework for two different reduced random Kalman filters, valid independent of the large dimension. The first part of our results examines the fidelity of the covariance estimators, which is essential for accurate uncertainty quantification. In a simplified setting, this is demonstrated by showing the true error covariance is dominated by its estimators. In general settings, the Mahalanobis error and its intrinsic dissipation can be used as simplified quantification of the same property. The second part develops upper bounds for the covariance estimators by comparing with proper Kalman filters. Combining both results, the classical tools for Kalman filters can be used as a-priori performance criteria for the reduced filters. In applications, these criteria guarantee the reduced filters are robust, and accurate for small noise systems. They also shed light on how to tune the reduced filters for stochastic turbulence.


model reduction, reduced Kalman filters, filter robustness, filter accuracy

2010 Mathematics Subject Classification

60G35, 76F65, 93E10, 94B99

Received 3 April 2017

Accepted 30 March 2018

Published 31 October 2018