Performance of Ensemble Kalman Filters in Large Dimensions

Andrew J. Majda, Xin T. Tong

Research output: Contribution to journalArticlepeer-review


Contemporary data assimilation often involves more than a million prediction variables. Ensemble Kalman filters (EnKF) have been developed by geoscientists. They are successful indispensable tools in science and engineering, because they allow for computationally cheap low-ensemble-state approximation for extremely large-dimensional turbulent dynamical systems. The practical finite ensemble filters like EnKF necessarily involve modifications such as covariance inflation and localization, and it is a genuine mystery why they perform so well with small ensemble sizes in large dimensions. This paper provides the first rigorous stochastic analysis of the accuracy and covariance fidelity of EnKF in the practical regime where the ensemble size is much smaller than the large ambient dimension for EnKFs with random coefficients. A challenging issue overcome here is that EnKF in huge dimensions introduces unavoidable bias and model errors that need to be controlled and estimated.

Original languageEnglish (US)
Pages (from-to)892-937
Number of pages46
JournalCommunications on Pure and Applied Mathematics
Issue number5
StatePublished - May 2018

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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