Open access
Methods and Algorithms for Scientific Computing

Multilevel Ensemble Transform Particle Filtering


This paper extends the multilevel Monte Carlo variance reduction technique to nonlinear filtering. In particular, multilevel Monte Carlo is applied to a certain variant of the particle filter, the ensemble transform particle filter (EPTF). A key aspect is the use of optimal transport methods to re-establish correlation between coarse and fine ensembles after resampling; this controls the variance of the estimator. Numerical examples present a proof of concept of the effectiveness of the proposed method, demonstrating significant computational cost reductions (relative to the single-level ETPF counterpart) in the propagation of ensembles.


  1. multilevel Monte Carlo
  2. sequential data assimilation
  3. optimal transport

MSC codes

  1. 65C05
  2. 62M20
  3. 93E11
  4. 93B40
  5. 90C05

Formats available

You can view the full content in the following formats:


J. L. Anderson, Localization and sampling error correction in ensemble Kalman filter data assimilation, Mon. Weather Rev., 140 (2012), pp. 2359--2371,
A. Barth and A. Lang, Multilevel Monte Carlo method with applications to stochastic partial differential equations, Int. J. Comput. Math., 89 (2012), pp. 2479--2498,
A. Beskos, A. Jasra, K. Law, R. Tempone, and Y. Zhou, Multilevel Sequential Monte Carlo Samplers, preprint,, 2015.
O. Cappé, S. J. Godsill, and E. Moulines, An overview of existing methods and recent advances in sequential Monte Carlo, Proc. IEEE, 95 (2007), pp. 899--924,
Y. Cheng and S. Reich, A McKean Optimal Transportation Perspective on Feynman-Kac Formulae with Application to Data Assimilation, preprint,, 2013.
K. A. Cliffe, M. B. Giles, R. Scheichl, and A. L. Teckentrup, Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients, Comput. Vis. Sci., 14 (2011), pp. 3--15,
A. Doucet and A. M. Johansen, A tutorial on particle filtering and smoothing: Fifteen years later, in The Oxford Handbook of Nonlinear Filtering, Oxford University Press, Oxford, UK, 2011, pp. 656--704,
M. Giles, Multilevel Monte Carlo methods, in Monte Carlo and Quasi-Monte Carlo Methods 2012, Springer Proc. Math. Stat. 65, Springer, Heidelberg, 2013, pp. 83--103,
M. B. Giles, Multilevel Monte Carlo path simulation, Oper. Res., 56 (2008), pp. 607--617,
M. B. Giles and B. J. Waterhouse, Multilevel quasi-Monte Carlo path simulation, in Advanced Financial Modelling, Radon Ser. Comput. Appl. Math. 8, Walter de Gruyter, Berlin, 2009, pp. 165--181.
H. Hoel, K. J. H. Law, and R. Tempone, Multilevel Ensemble Kalman Filtering, preprint,, 2015.
A. Jasra, K. Kamatani, K. J. H. Law, and Y. Zhou, Multilevel Particle Filter, preprint,, 2015.
C. Ketelsen, R. Scheichl, and A. L. Teckentrup, A Hierarchical Multilevel Markov Chain Monte Carlo Algorithm with Applications to Uncertainty Quantification in Subsurface Flow, preprint, Center for Applied Scientific Computing, Livermore, CA, 2013.
F. Le Gland, V. Monbet, and V. Tran, Large sample asymptotics for the ensemble Kalman filter, in The Oxford Handbook of Nonlinear Filtering, Oxford University Press, Oxford, UK, 2011, pp. 598--631.
J. Munkres, Algorithms for the assignment and transportation problems, J. Soc. Indust. Appl. Math., 5 (1957), pp. 32--38,
O. Pele and M. Werman, Fast and robust earth mover's distances, in Proceedings of the 12th IEEE International Conference on Computer Vision, 2009, pp. 460--467,
R. Ravi, Iterative methods in combinatorial optimization, in Proceedings of the 29th Symposium on Theoretical Aspects of Computer Science, Vol. 14, 2012, p. 24,
P. Rebeschini and R. Van Handel, Can local particle filters beat the curse of dimensionality?, Ann. Appl. Probab., 25 (2015), pp. 2809--2866,
S. Reich, A nonparametric ensemble transform method for Bayesian inference, SIAM J. Sci. Comput., 35 (2013), pp. A2013--A2024,
S. Reich and C. J. Cotter, Probabilistic Forecasting and Bayesian Data Assimilation, Cambridge University Press, New York, 2015.
R. Schefzik, T. L. Thorarinsdottir, and T. Gneiting, Uncertainty quantification in complex simulation models using ensemble copula coupling, Statist. Sci., 28 (2013), pp. 616--640,
C. Villani, Optimal Transport: Old and New, Grundlehren Math. Wiss. 338, Springer, Berlin, 2009.

Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: A1317 - A1338
ISSN (online): 1095-7197


Submitted: 3 September 2015
Accepted: 22 February 2016
Published online: 3 May 2016


  1. multilevel Monte Carlo
  2. sequential data assimilation
  3. optimal transport

MSC codes

  1. 65C05
  2. 62M20
  3. 93E11
  4. 93B40
  5. 90C05



Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By







Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account