An optimal transport formulation of the Ensemble Kalman filter

Amirhossein Taghvaei, Prashant G. Mehta

Research output: Contribution to journalArticlepeer-review


Controlled interacting particle systems such as the ensemble Kalman filter (EnKF) and the feedback particle filter (FPF) are numerical algorithms to approximate the solution of the nonlinear filtering problem in continuous time. The distinguishing feature of these algorithms is that the Bayesian update step is implemented using a feedback control law. It has been noted in the literature that the control law is not unique. This is the main problem addressed in this article. To obtain a unique control law, the filtering problem is formulated here as an optimal transportation problem. An explicit formula for the (mean-field type) optimal control law is derived in the linear Gaussian setting. Comparisons are made with the control laws for different types of EnKF algorithms described in the literature. Via empirical approximation of the mean-field control law, a finite-N controlled interacting particle algorithm is obtained. For this algorithm, the equations for empirical mean and covariance are derived and shown to be identical to the Kalman filter. This allows strong conclusions on convergence and error properties based on the classical filter stability theory for the Kalman filter. It is shown that, under certain technical conditions, the mean squared error converges to zero even with a finite number of particles. A detailed propagation of chaos analysis is carried out for the finite-N algorithm. The analysis is used to prove weak convergence of the empirical distribution as N → ∞. For a certain simplified filtering problem, analytical comparison of the mse with the importance sampling-based algorithms is described. The analysis helps explain the favorable scaling properties of the control-based algorithms reported in several numerical studies in recent literature.

Original languageEnglish (US)
Article number9163273
Pages (from-to)3052-3067
Number of pages16
JournalIEEE Transactions on Automatic Control
Issue number7
StatePublished - Jul 2021


  • Approximation algorithms
  • Control systems
  • Convergence
  • Error analysis
  • Kalman filters
  • Monte Carlo methods
  • Symmetric matrices
  • Kalman filter
  • Stochastic processes
  • Filtering algorithms

ASJC Scopus subject areas

  • Electrical and Electronic Engineering
  • Control and Systems Engineering
  • Computer Science Applications


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