We consider filtering in high-dimensional non-Gaussian state-space models with intractable transition kernels, nonlinear and possibly chaotic dynamics, and sparse observations in space and time. We propose a novel filtering methodology that harnesses transportation of measures, convex optimization, and ideas from probabilistic graphical models to yield robust ensemble approximations of the filtering distribution in high dimensions. Our approach can be understood as the natural generalization of the ensemble Kalman filter (EnKF) to nonlinear updates, using stochastic or deterministic couplings. The use of nonlinear updates can reduce the intrinsic bias of the EnKF at a marginal increase in computational cost. We avoid any form of importance sampling and introduce non-Gaussian localization approaches for dimension scalability. Our framework achieves state-of-the-art tracking performance on challenging configurations of the Lorenz-96 model in the chaotic regime.


  1. nonlinear filtering
  2. state-space models
  3. couplings
  4. transport maps
  5. ensemble Kalman filter
  6. graphical models
  7. localization
  8. approximate Bayesian computation

MSC codes

  1. 93E11
  2. 62M20
  3. 60G35
  4. 62L12

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Supplementary Material

PLEASE NOTE: These supplementary files have not been peer-reviewed.

Index of Supplementary Materials

Title of paper: Coupling Techniques for Nonlinear Ensemble Filtering

Authors: Alessio Spantini, Ricardo Baptista, and Youssef Marzouk

File: M131220_supplementary.pdf

Type: PDF

Contents: In the supplementary text, we provide more details on the numerical computation of triangular maps in Section SM1. We also describe a variant of the deterministic map filtering algorithm from Section 10 for the setting of conditionally independent and local observations in Section SM2. Lastly, we provide additional numerical results using the stochastic map algorithm in Sections SM3-SM6.


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Information & Authors


Published In

cover image SIAM Review
SIAM Review
Pages: 921 - 953
ISSN (online): 1095-7200


Submitted: 10 January 2020
Accepted: 16 November 2021
Published online: 3 November 2022


  1. nonlinear filtering
  2. state-space models
  3. couplings
  4. transport maps
  5. ensemble Kalman filter
  6. graphical models
  7. localization
  8. approximate Bayesian computation

MSC codes

  1. 93E11
  2. 62M20
  3. 60G35
  4. 62L12



Funding Information

Natural Sciences and Engineering Research Council of Canada : PGSD-D Fellowship
Air Force Office of Scientific Research https://doi.org/10.13039/100000181 : FA9550-15-1-0038
Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : CRC 1294
U.S. Department of Energy https://doi.org/10.13039/100000015 : AEOLUS

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