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The purpose of this review is to present a comprehensive overview of the theory of ensemble Kalman-Bucy filtering for continuous-time, linear-Gaussian signal and observation models.
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M-A. Poubelle, I.R. Petersen, M.R. Gevers, and R.R. Bitmead. A Miscellany of Results on an Equation of Count J. F. Riccati. IEEE Transactions on Automatic Control. vol. 31, no. 7. pp. 651–654 (1986)
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J.S. Baras, A. Bensoussan, and M.R. James. Dynamic observers as asymptotic limits of recursive filters: Special cases. SIAM Journal on Applied Mathematics. vol. 48, no. 5. pp. 1147–1158 (1988)
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P. Bougerol. Kalman filtering with random coefficients and contractions. SIAM Journal on Control and Optimization. vol. 31, no. 4. pp. 942–959 (1993)
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N. Gordon, J. Salmond and A. Smith. A novel approach to non-linear/non-Gaussian Bayesian state estimation. IEE Proceedings on Radar and Signal Processing. vol. 140, no. 2. pp. 107–113 (1993)
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G. Kitagawa. A Monte Carlo Filtering and Smoothing Method for Non-Gaussian Nonlinear State Space Models. Proceedings of the 2nd U.S.-Japan Joint Seminar on Statistical Time Series Analysis: pp. 110–131 (1993)
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G. Evensen. Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. Journal of Geophysical Research: Oceans. vol. 99, no. C5. pp. 10143–10162 (1994)
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G.M. Krause. Bounds for the Variation of Matrix Eigenvalues and Polynomial Roots. Linear Algebra and its Applications. vol. 208-209. pp. 73–82 (1994)
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F.M. Callier and J. Winkin. Convergence of the Time-Invariant Riccati Differential Equation towards Its Strong Solution for Stabilizable Systems. Journal of Mathematical Analysis and Applications. vol. 192, no. 1. pp. 230–257 (1995)
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P. Lancaster and L. Rodman. Algebraic Riccati Equations. Oxford University Press (1995)
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B.F. La Scala, R.R. Bitmead, and M.R. James. Conditions for stability of the extended Kalman filter and their application to the frequency tracking problem. Mathematics of Control, Signals and Systems. vol. 8, no. 1. (1995)
1995
Earlier work this paper cites.
P. Del Moral. Non Linear Filtering: Interacting Particle Solution. Markov Processes and Related Fields. vol. 2, no. 4, pp. 555–580 (1996)
1996
Earlier work this paper cites.
I. Karatzas and S.E. Shreve. Brownian Motion and Stochastic Calculus. Springer (1996)
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Earlier work this paper cites.
G. Kitagawa. Monte Carlo filter and smoother for non-Gaussian nonlinear state space models. Journal of Computational and Graphical Statistics. vol. 5, no.1, pp 1–25 (1996)
1996
Earlier work this paper cites.
D. Ocone and E. Pardoux. Asymptotic stability of the optimal filter with respect to its initial condition. SIAM Journal on Control and Optimization. vol. 34, no. 1. pp. 226-243 (1996)
1996
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P. Park and T. Kailath. Convergence of the DRE solution to the ARE strong solution. IEEE Transactions on Automatic Control. vol. 42, no. 4. 573–578 (1997)
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R. Atar. Exponential stability for nonlinear filtering of diffusion processes in a noncompact domain. Annals of Probability. pp. 1552–1574 (1998)
1998
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G. Burgers, P.J. van Leeuwen, and G. Evensen. Analysis Scheme in the Ensemble Kalman Filter. Monthly Weather Review, vol. 126, no. 6. pp. 1719–1724 (1998)
1998
Earlier work this paper cites.
P. Del Moral. Measure-valued processes and interacting particle systems. Application to nonlinear filtering problems. The Annals of Applied Probability. vol. 8, no. 2, pp. 438–495 (1998)
1998
Earlier work this paper cites.
P.L. Houtekamer and H.L. Mitchell. Data assimilation using an ensemble Kalman filter technique. Monthly Weather Review. vol. 126, no. 3. pp. 796–811 (1998)
1998
Earlier work this paper cites.
J.L. Anderson and S.L. Anderson. A Monte Carlo Implementation of the Nonlinear Filtering Problem to Produce Ensemble Assimilations and Forecasts. Monthly Weather Review. vol. 127, no. 12. pp. 2741–2758 (1999)
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Earlier work this paper cites.
A. Budhiraja and H.J. Kushner. Approximation and limit results for nonlinear filters over an infinite time interval. SIAM Journal on Control and Optimization. vol. 37, no. 6. pp. 1946–1979 (1999)
1999
Earlier work this paper cites.
P. Del Moral, A. Guionnet. On the stability of measure valued processes with applications to filtering. Comptes Rendus de l’Académie des Sciences-Series I-Mathematics. vol. 329, no. 5. pp. 429–434 (1999)
1999
Earlier work this paper cites.
A.G. Bhatt, A. Budhiraja, and R.L. Karandikar. Markov property and ergodicity of the nonlinear filter. SIAM Journal on Control and Optimization. vol. 39, no. 3. pp. 928–949 (2000)
2000
Earlier work this paper cites.
P. Del Moral, L. Miclo. Branching and interacting particle systems approximations of Feynman-Kac formulae with applications to non-linear filtering. Séminaire de Probabilités XXXIV, pp. 1–145 (2000)
2000
Earlier work this paper cites.
A. Doucet, S. Godsill and C. Andrieu. On sequential Monte Carlo sampling methods for Bayesian filtering. Statistics and Computing. vol. 10, no. 3. pp. 197–208 (2000)
2000
Earlier work this paper cites.
K. Reif, S. Gunther, E. Yaz, and R. Unbehauen. Stochastic stability of the continuous-time extended Kalman filter. IEE Proceedings – Control Theory and Applications. vol. 147, no. 1. pp. 45–52 (2000)
2000
Earlier work this paper cites.
J.L. Anderson. An ensemble adjustment Kalman filter for data assimilation. Monthly Weather Review. vol. 129, no. 12. pp. 2884–2903 (2001)
2001
Earlier work this paper cites.
C.H. Bishop, B.J. Etherton, and S.J. Majumdar. Adaptive sampling with the ensemble transform Kalman filter. Part I: Theoretical aspects. Monthly Weather Review vol. 129, no. 3. pp. 420–436 (2001)
2001
Earlier work this paper cites.
A. Doucet, N. de Freitas and N.J. Gordon (editors). Sequential Monte Carlo Methods in Practice. Springer (2001)
2001
Earlier work this paper cites.
T.M. Hamill, J.S. Whitaker, and C. Snyder. Distance-Dependent Filtering of Background Error Covariance Estimates in an Ensemble Kalman Filter. Monthly Weather Review. vol. 129, no. 11. pp. 2776–2790 (2001)
2001
Earlier work this paper cites.
P.L. Houtekamer and H.L. Mitchell. A Sequential Ensemble Kalman Filter for Atmospheric Data Assimilation. Monthly Weather Review. vol. 129, no. 1. pp. 123–137 (2001)
2001
Earlier work this paper cites.
H.L. Mitchell, P.L. Houtekamer and G. Pellerin. Ensemble size, balance, and model-error representation in an ensemble Kalman filter. Monthly Weather Review. vol. 130, no. 11. pp. 2791–2808 (2002)
2002
Earlier work this paper cites.
J.S. Whitaker and T.M. Hamill. Ensemble data assimilation without perturbed observations. Monthly Weather Review. vol. 130, no. 7. pp. 1913–1924 (2002)
2002
Earlier work this paper cites.
J.I. Allen, M. Eknes and G. Evensen. An ensemble Kalman filter with a complex marine ecosystem model: Hindcasting phytoplankton in the Cretan Sea. Annales Geophysicae. vol. 21. pp. 399–411 (2003)
2003
Earlier work this paper cites.
J.L. Anderson. A local least squares framework for ensemble filtering. Monthly Weather Review. vol. 131, no. 4. pp 634–642 (2003)
2003
Cited alongside, same era.
A. Budhiraja. Asymptotic stability, ergodicity and other asymptotic properties of the nonlinear filter. Annales de l’IHP Probabilites et Statistiques. vol. 39, no. 6. pp. 919–941 (2003)
2003
Cited alongside, same era.
G. Evensen. The Ensemble Kalman Filter: Theoretical Formulation and Practical Implementation. Ocean Dynamics. vol. 53, no. 4. pp. 343–367 (2003)
2003
Cited alongside, same era.
Y. Hu and X.Y. Zhou. Indefinite stochastic Riccati equations. SIAM Journal on Control Optimization. vol. 42, no. 1. pp. 123–137 (2003)
2003
Cited alongside, same era.
E. Kalnay. Atmospheric Modelling, Data Assimilation and Predictability. Cambridge University Press (2003)
2003
D. Kelly, A.J. Majda, and X.T. Tong. Concrete ensemble Kalman filters with rigorous catastrophic filter divergence. Proceedings of the National Academy of Sciences of the United States of America. vol 112, no. 34. pp. 10589–10594 (2015)
2015
Later among the works it cites.
E. Kwiatkowski and J. Mandel. Convergence of the Square Root Ensemble Kalman Filter in the Large Ensemble Limit. SIAM/ASA Journal on Uncertainty Quantification. vol. 3, no. 1. pp. 1–17 (2015)
2015
Later among the works it cites.
K.J.H. Law, A.M. Stuart and K. Zygalakis. Data Assimilation: A Mathematical Introduction. Springer (2015)
2015
Later among the works it cites.
P. Rebeschini and R. Van Handel. Can local particle filters beat the curse of dimensionality? The Annals of Applied Probability. vol. 25, no. 5. pp. 2809–2866 (2015)
2015
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Cited alongside, same era.
M. Kohlmann and S. Tang. Multidimensional backward stochastic Riccati equations and applications. SIAM Journal on Control Optimization. vol. 41, no. 6. pp. 1696–1721 (2003)
2003
Cited alongside, same era.
K.A. Lisaeter, J. Rosanova and G. Evensen. Assimilation of ice concentration in a coupled ice-ocean model using the Ensemble Kalman Filter. Ocean Dynamics. vol. 53, no. 4. pp. 368–388 (2003)
2003
Cited alongside, same era.
G. Naevdal, L.M. Johnsen, S.I. Aanonsen and E.H. Vefring. Reservoir monitoring and continuous model updating using ensemble Kalman filter. In Proceedings of the 2003 SPE Annual Technical Conference and Exhibition, Denver, Colorado (October, 2003)
2003
Cited alongside, same era.
M.K. Tippett, J.L. Anderson, C.H. Bishop, T.M. Hamill, and J.S. Whitaker. Ensemble square root filters. Monthly Weather Review. vol. 131, no. 7. pp. 1485–1490 (2003)
2003
Cited alongside, same era.
P. Baxendale, P. Chigansky, and R. Liptser. Asymptotic stability of the Wonham filter: ergodic and nonergodic signals. SIAM Journal on Control and Optimization. vol. 43, no. 2. pp. 643–669 (2004)
2004
Cited alongside, same era.
N. Chopin. Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference. The Annals of Statistics. vol. 32, no. 6. pp. 2385–2411 (2004)
2004
Cited alongside, same era.
P. Del Moral. Feynman-Kac Formulae. Springer (2004)
2004
Cited alongside, same era.
2015
Later among the works it cites.
P. Bunch and S. Godsill. Approximations of the optimal importance density using Gaussian particle flow importance sampling. Journal of the American Statistical Association. vol. 111, no. 514. pp. 748–762 (2016)
2016
Later among the works it cites.
H. Hoel, K.J.H. Law, and R. Tempone. Multilevel ensemble Kalman filtering. SIAM Journal on Numerical Analysis. vol. 54, no. 3. pp. 1813–1839 (2016)
2016
Later among the works it cites.
K.J.H. Law, H. Tembine and R. Tempone. Deterministic Mean-Field Ensemble Kalman Filtering. SIAM Journal on Scientific Computing. vol. 38, no. 3. pp. A1251-A1279 (2016)
2016
Later among the works it cites.
B.C. Levy, and M. Zorzi. A contraction analysis of the convergence of risk-sensitive filters. SIAM Journal on Control and Optimization. vol. 54, no. 4. pp. 2154–2173 (2016)
2016
Later among the works it cites.
A. Taghvaei and P.G. Mehta. An optimal transport formulation of the linear feedback particle filter. In Proc. of the 2016 American Control Conference (ACC), Boston, USA (July, 2016)
2016
Later among the works it cites.
X.T. Tong, A.J. Majda, and D. Kelly. Nonlinear stability and ergodicity of ensemble based Kalman filters. Nonlinearity. vol. 29, no. 2. pp 657–691 (2016)
2016
Later among the works it cites.
X.T. Tong, A.J. Majda, and D. Kelly. Nonlinear stability of the ensemble Kalman filter with adaptive covariance inflation. Communications in Mathematical Sciences. vol. 14, no. 5. pp. 1283–1313 (2016)
2016
Later among the works it cites.
T. Yang, R.S. Laugesen, P.G. Mehta, and S.P. Meyn. Multivariable feedback particle filter. Automatica. vol. 71. pp. 10–23 (2016)
2016
Later among the works it cites.
2017
Later among the works it cites.
P. Del Moral, A. Kurtzmann, and J. Tugaut. On the Stability and the Uniform Propagation of Chaos of a Class of Extended Ensemble Kalman–Bucy Filters. SIAM Journal on Control and Optimization. vol. 55, no.1. pp. 119–155 (2017)
2017
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C. Schillings and A.M. Stuart. Convergence analysis of ensemble Kalman inversion: the linear, noisy case. Applicable Analysis. vol. 97, no. 1. pp. 107–123 (2017)
2017
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C. Schillings and A.M. Stuart. Analysis of the ensemble Kalman filter for inverse problems. SIAM Journal on Numerical Analysis. vol. 55, no. 3. pp. 1264–1290 (2017)
2017
Later among the works it cites.
2018
Later among the works it cites.
A.N. Bishop and P. Del Moral. On the robustness of Riccati flows to complete model misspecification. Journal of the Franklin Institute. vol. 355, no. 15. pp 7178–7200 (2018)
2018
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2018
Later among the works it cites.
A.N. Bishop, P. Del Moral and S. Pathiraja. Perturbations and Projections of Kalman-Bucy Semigroups. Stochastic Processes and their Applications. vol. 128, no. 9. pp. 2857–2904. (2018)
2018
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N.K. Chada, M.A. Iglesias, L. Roininen, and A.M. Stuart. Parameterizations for ensemble Kalman inversion. Inverse Problems. vol. 34, no. 5. (2018)
2018
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J. de Wiljes, S. Reich, and W. Stannat. Long-Time Stability and Accuracy of the Ensemble Kalman-Bucy Filter for Fully Observed Processes and Small Measurement Noise. SIAM Journal on Applied Dynamical Systems. vol. 17, no. 2. pp. 1152–1181 (2018)
2018
Later among the works it cites.
P. Del Moral and J. Tugaut. On the stability and the uniform propagation of chaos properties of ensemble Kalman-Bucy filters. Annals of Applied Probability. vol. 28, no. 2. pp 790–850 (2018)
2018
Later among the works it cites.
2018
Later among the works it cites.
A.J. Majda and X.T. Tong. Performance of Ensemble Kalman Filters in Large Dimensions. Communications in Mathematical Sciences. vol. 71, no. 5. pp. 892–937 (2018)
2018
Later among the works it cites.
C. Zhang, A. Taghvaei, and P.G. Mehta. A mean-field optimal control formulation for global optimization. IEEE Transactions on Automatic Control. vol. 64, no. 1. pp. 282–289 (2018)
2018
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2019
Later among the works it cites.
2019
Later among the works it cites.
2019
Later among the works it cites.
2019
Later among the works it cites.
2020
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A. Taghvaei, P.G. Mehta, and S.P. Meyn. Diffusion map-based algorithm for gain function approximation in the feedback particle filter. SIAM/ASA Journal on Uncertainty Quantification. vol. 8, no. 3. pp. 1090–1117 (2020)
2020
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A. Tanwani and O. Yufereva. Error covariance bounds for suboptimal filters with Lipschitzian drift and Poisson-sampled measurements. Automatica. vol. 122. (2020)
2020
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2021
Closest in time.
T. Lange. Derivation of ensemble Kalman-Bucy filters with unbounded nonlinear coefficients. Nonlinearity. vol. 35, no. 2. (2021)
2021
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T. Lange and W. Stannat. Mean field limit of Ensemble Square Root Filters–discrete and continuous time. Foundations of Data Science. vol. 3, no. 3. pp. 563–588 (2021)
2021
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T. Lange and W. Stannat. On the continuous time limit of the ensemble Kalman filter. Mathematics of Computation. vol. 90, no. 327. pp. 233–265 (2021)
2021
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T. Lange and W. Stannat. On the continuous time limit of ensemble square root filters. Communications in Mathematical Sciences. vol. 19, no. 7. (2021)
2021
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S. Pathiraja, S. Reich, and W. Stannat. McKean-Vlasov SDEs in nonlinear filtering. SIAM Journal on Control and Optimization. vol. 59, no. 6. pp. 4188–4215 (2021)
2021
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2022
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N.K. Chada, A. Jasra, and F. Yu. Multilevel ensemble Kalman-Bucy filters. SIAM/ASA Journal on Uncertainty Quantification. vol. 10, no. 2. pp. 584–618 (2022)
2022
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D. Crisan, P. Del Moral, A. Jasra, and H. Ruzayqat. Log-normalization constant estimation using the ensemble Kalman-Bucy filter with application to high-dimensional models. Advances in Applied Probability. vol. 54, no. 4. pp. 1139–1163 (2022)
2022
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H. Hoel, G. Shaimerdenova, and R. Tempone. Multi-index ensemble Kalman filtering. Journal of Computational Physics. vol. 470. (2022)
2022
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2022
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H. Ruzayqat, N.K. Chada, and A. Jasra. Multilevel estimation of normalization constants using ensemble Kalman-Bucy filters. Statistics and Computing. vol. 32, no. 3. pp. 1–25 (2022)
2022
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2023
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