Deterministic Mean-Field Ensemble Kalman Filtering
Abstract
The proof of convergence of the standard ensemble Kalman filter (EnKF) from Le Gland, Monbet, and 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] is extended to non-Gaussian state-space models. In this paper, a density-based deterministic approximation of the mean-field limit EnKF (DMFEnKF) is proposed, consisting of a PDE solver and a quadrature rule. Given a certain minimal order of convergence κ between the two, this extends to the deterministic filter approximation, which is therefore asymptotically superior to standard EnKF for dimension d < 2κ. The fidelity of approximation of the true distribution is also established using an extension of the total variation metric to random measures. Lastly, this is limited by a Gaussian bias term arising from nonlinearity/non-Gaussianity of the model, which arises in both deterministic and standard EnKF. Numerical results support and extend the theory.
- Authors:
-
- King Abdullah Univeristy of Science and Technology (KAUST) SRI-UQ Center, Thuwal (Saudi Arabia); Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
- King Abdullah Univeristy of Science and Technology (KAUST) SRI-UQ Center, Thuwal (Saudi Arabia)
- Publication Date:
- Research Org.:
- Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)
- Sponsoring Org.:
- USDOE Laboratory Directed Research and Development (LDRD) Program
- OSTI Identifier:
- 1311261
- Grant/Contract Number:
- AC05-00OR22725; 32112580
- Resource Type:
- Accepted Manuscript
- Journal Name:
- SIAM Journal on Scientific Computing
- Additional Journal Information:
- Journal Volume: 38; Journal Issue: 3; Journal ID: ISSN 1064-8275
- Publisher:
- SIAM
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; filtering; Fokker-Planck; EnKF
Citation Formats
Law, Kody J. H., Tembine, Hamidou, and Tempone, Raul. Deterministic Mean-Field Ensemble Kalman Filtering. United States: N. p., 2016.
Web. doi:10.1137/140984415.
Law, Kody J. H., Tembine, Hamidou, & Tempone, Raul. Deterministic Mean-Field Ensemble Kalman Filtering. United States. https://doi.org/10.1137/140984415
Law, Kody J. H., Tembine, Hamidou, and Tempone, Raul. Tue .
"Deterministic Mean-Field Ensemble Kalman Filtering". United States. https://doi.org/10.1137/140984415. https://www.osti.gov/servlets/purl/1311261.
@article{osti_1311261,
title = {Deterministic Mean-Field Ensemble Kalman Filtering},
author = {Law, Kody J. H. and Tembine, Hamidou and Tempone, Raul},
abstractNote = {The proof of convergence of the standard ensemble Kalman filter (EnKF) from Le Gland, Monbet, and 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] is extended to non-Gaussian state-space models. In this paper, a density-based deterministic approximation of the mean-field limit EnKF (DMFEnKF) is proposed, consisting of a PDE solver and a quadrature rule. Given a certain minimal order of convergence κ between the two, this extends to the deterministic filter approximation, which is therefore asymptotically superior to standard EnKF for dimension d < 2κ. The fidelity of approximation of the true distribution is also established using an extension of the total variation metric to random measures. Lastly, this is limited by a Gaussian bias term arising from nonlinearity/non-Gaussianity of the model, which arises in both deterministic and standard EnKF. Numerical results support and extend the theory.},
doi = {10.1137/140984415},
journal = {SIAM Journal on Scientific Computing},
number = 3,
volume = 38,
place = {United States},
year = {Tue May 03 00:00:00 EDT 2016},
month = {Tue May 03 00:00:00 EDT 2016}
}
Web of Science
Works referenced in this record:
An exploration of the equivalent weights particle filter
journal, August 2012
- Ades, M.; van Leeuwen, P. J.
- Quarterly Journal of the Royal Meteorological Society, Vol. 139, Issue 672
Data assimilation: Mathematical and statistical perspectives
journal, January 2008
- Apte, A.; Jones, C. K. R. T.; Stuart, A. M.
- International Journal for Numerical Methods in Fluids, Vol. 56, Issue 8
Polynomial chaos for the approximation of uncertainties: Chances and limits
journal, March 2008
- Augustin, F.; Gilg, A.; Paffrath, M.
- European Journal of Applied Mathematics, Vol. 19, Issue 02
A Hybrid Sparse-Grid Approach for Nonlinear Filtering Problems Based on Adaptive-Domain of the Zakai Equation Approximations
journal, January 2014
- Bao, Feng; Cao, Yanzhao; Webster, Clayton
- SIAM/ASA Journal on Uncertainty Quantification, Vol. 2, Issue 1
Accuracy and stability of the continuous-time 3DVAR filter for the Navier–Stokes equation
journal, July 2013
- Blömker, D.; Law, K.; Stuart, A. M.
- Nonlinearity, Vol. 26, Issue 8
Fundamental limitations of polynomial chaos for uncertainty quantification in systems with intermittent instabilities
journal, January 2013
- Branicki, Michal; Majda, Andrew J.
- Communications in Mathematical Sciences, Vol. 11, Issue 1
Accuracy and stability of filters for dissipative PDEs
journal, February 2013
- Brett, C. E. A.; Lam, K. F.; Law, K. J. H.
- Physica D: Nonlinear Phenomena, Vol. 245, Issue 1
Analysis Scheme in the Ensemble Kalman Filter
journal, June 1998
- Burgers, Gerrit; Jan van Leeuwen, Peter; Evensen, Geir
- Monthly Weather Review, Vol. 126, Issue 6
Implicit particle filters for data assimilation
journal, January 2010
- Chorin, Alexandre; Morzfeld, Matthias; Tu, Xuemin
- Communications in Applied Mathematics and Computational Science, Vol. 5, Issue 2
Dimensional reduction for a Bayesian filter
journal, October 2004
- Chorin, A. J.; Krause, P.
- Proceedings of the National Academy of Sciences, Vol. 101, Issue 42
On the stability of interacting processes with applications to filtering and genetic algorithms
journal, March 2001
- Del Moral, P.
- Annales de l'Institut Henri Poincare (B) Probability and Statistics, Vol. 37, Issue 2
Central Limit Theorem for Nonstationary Markov Chains. I
journal, January 1956
- Dobrushin, R. L.
- Theory of Probability & Its Applications, Vol. 1, Issue 1
Central Limit Theorem for Nonstationary Markov Chains. II
journal, January 1956
- Dobrushin, R. L.
- Theory of Probability & Its Applications, Vol. 1, Issue 4
Bayesian inference with optimal maps
journal, October 2012
- El Moselhy, Tarek A.; Marzouk, Youssef M.
- Journal of Computational Physics, Vol. 231, Issue 23
Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics
journal, January 1994
- Evensen, Geir
- Journal of Geophysical Research, Vol. 99, Issue C5
A mean field approximation in data assimilation for nonlinear dynamics
journal, August 2004
- Eyink, Gregory L.; Restrepo, Juan M.; Alexander, Francis J.
- Physica D: Nonlinear Phenomena, Vol. 195, Issue 3-4
Discrete data assimilation in the Lorenz and 2D Navier–Stokes equations
journal, September 2011
- Hayden, Kevin; Olson, Eric; Titi, Edriss S.
- Physica D: Nonlinear Phenomena, Vol. 240, Issue 18
Particle Kalman Filtering: A Nonlinear Bayesian Framework for Ensemble Kalman Filters
journal, February 2012
- Hoteit, Ibrahim; Luo, Xiaodong; Pham, Dinh-Tuan
- Monthly Weather Review, Vol. 140, Issue 2
A New Approach to Linear Filtering and Prediction Problems
journal, March 1960
- Kalman, R. E.
- Journal of Basic Engineering, Vol. 82, Issue 1
Well-posedness and accuracy of the ensemble Kalman filter in discrete and continuous time
journal, September 2014
- Kelly, D. T. B.; Law, K. J. H.; Stuart, A. M.
- Nonlinearity, Vol. 27, Issue 10
Convergence of the Square Root Ensemble Kalman Filter in the Large Ensemble Limit
journal, January 2015
- Kwiatkowski, Evan; Mandel, Jan
- SIAM/ASA Journal on Uncertainty Quantification, Vol. 3, Issue 1
Memoir on the Probability of the Causes of Events
journal, August 1986
- Laplace, Pierre Simon
- Statistical Science, Vol. 1, Issue 3
A generalized polynomial chaos based ensemble Kalman filter with high accuracy
journal, August 2009
- Li, Jia; Xiu, Dongbin
- Journal of Computational Physics, Vol. 228, Issue 15
On the convergence of the ensemble Kalman filter
journal, December 2011
- Mandel, Jan; Cobb, Loren; Beezley, Jonathan D.
- Applications of Mathematics, Vol. 56, Issue 6
Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise
journal, October 2002
- Mattingly, J. C.; Stuart, A. M.; Higham, D. J.
- Stochastic Processes and their Applications, Vol. 101, Issue 2
A deterministic filter for non-Gaussian Bayesian estimation— Applications to dynamical system estimation with noisy measurements
journal, April 2012
- Pajonk, Oliver; Rosić, Bojana V.; Litvinenko, Alexander
- Physica D: Nonlinear Phenomena, Vol. 241, Issue 7
Sampling-free linear Bayesian updating of model state and parameters using a square root approach
journal, June 2013
- Pajonk, Oliver; Rosić, Bojana V.; Matthies, Hermann G.
- Computers & Geosciences, Vol. 55
A Gaussian-mixture ensemble transform filter
journal, August 2011
- Reich, Sebastian
- Quarterly Journal of the Royal Meteorological Society, Vol. 138, Issue 662
A hybrid grid/particle filter for Lagrangian data assimilation. I: Formulating the passive scalar approximation
journal, July 2008
- Salman, H.
- Quarterly Journal of the Royal Meteorological Society, Vol. 134, Issue 635
Two-Stage Filtering for Joint State-Parameter Estimation
journal, May 2015
- Santitissadeekorn, Naratip; Jones, Christopher
- Monthly Weather Review, Vol. 143, Issue 6
Blended reduced subspace algorithms for uncertainty quantification of quadratic systems with a stable mean state
journal, September 2013
- Sapsis, Themistoklis P.; Majda, Andrew J.
- Physica D: Nonlinear Phenomena, Vol. 258
Developing a dynamically based assimilation method for targeted and standard observations
journal, January 2005
- Uboldi, F.; Trevisan, A.; Carrassi, A.
- Nonlinear Processes in Geophysics, Vol. 12, Issue 1
Data Assimilation in the Low Noise Regime with Application to the Kuroshio
journal, June 2013
- Vanden-Eijnden, Eric; Weare, Jonathan
- Monthly Weather Review, Vol. 141, Issue 6
Statistical and Computational Inverse Problems
book, January 2005
- Kaipio, Jari P.; Somersalo, Erkki
- Applied Mathematical Sciences
Bayesian inference with optimal maps
journal, October 2012
- El Moselhy, Tarek A.; Marzouk, Youssef M.
- Journal of Computational Physics, Vol. 231, Issue 23
Can local particle filters beat the curse of dimensionality?
text, January 2013
- Rebeschini, Patrick; van Handel, Ramon
- arXiv
Two-state filtering for joint state-parameter estimation
preprint, January 2014
- Santitissadeekorn, Naratip; Jones, Chris
- arXiv
Works referencing / citing this record:
Performance Analysis of Local Ensemble Kalman Filter
journal, March 2018
- Tong, Xin T.
- Journal of Nonlinear Science, Vol. 28, Issue 4
Ensemble Kalman Methods for High-Dimensional Hierarchical Dynamic Space-Time Models
journal, March 2019
- Katzfuss, Matthias; Stroud, Jonathan R.; Wikle, Christopher K.
- Journal of the American Statistical Association
Well posedness and convergence analysis of the ensemble Kalman inversion
journal, July 2019
- Blömker, Dirk; Schillings, Claudia; Wacker, Philipp
- Inverse Problems, Vol. 35, Issue 8
Ergodicity and Accuracy of Optimal Particle Filters for Bayesian Data Assimilation
journal, September 2019
- Kelly, David; Stuart, Andrew M.
- Chinese Annals of Mathematics, Series B, Vol. 40, Issue 5
Gaussian approximations of small noise diffusions in Kullback–Leibler divergence
journal, January 2017
- Sanz-Alonso, Daniel; Stuart, Andrew M.
- Communications in Mathematical Sciences, Vol. 15, Issue 7