May  2010, 27(2): 441-486. doi: 10.3934/dcds.2010.27.441

Mathematical strategies for filtering turbulent dynamical systems


Department of Mathematics and Center for Atmosphere and Ocean Science, Courant Institute for Mathematical Sciences, New York University, New York, NY 10012-1110, United States, United States


Depatment of Mathematics, North Carolina State University, Raleigh, NC 27695, United States

Received  October 2009 Revised  February 2010 Published  February 2010

The modus operandi of modern applied mathematics in developing very recent mathematical strategies for filtering turbulent dynamical systems is emphasized here. The approach involves the synergy of rigorous mathematical guidelines, exactly solvable nonlinear models with physical insight, and novel cheap algorithms with judicious model errors to filter turbulent signals with many degrees of freedom. A large number of new theoretical and computational phenomena such as "catastrophic filter divergence" in finite ensemble filters are reviewed here with the intention to introduce mathematicians, applied mathematicians, and scientists to this remarkable emerging scientific discipline with increasing practical importance.
Citation: Andrew J. Majda, John Harlim, Boris Gershgorin. Mathematical strategies for filtering turbulent dynamical systems. Discrete and Continuous Dynamical Systems, 2010, 27 (2) : 441-486. doi: 10.3934/dcds.2010.27.441

Jules Guillot, Emmanuel Frénod, Pierre Ailliot. Physics informed model error for data assimilation. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022059


Jiangqi Wu, Linjie Wen, Jinglai Li. Resampled ensemble Kalman inversion for Bayesian parameter estimation with sequential data. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 837-850. doi: 10.3934/dcdss.2021045


Dominique Chapelle, Philippe Moireau, Patrick Le Tallec. Robust filtering for joint state-parameter estimation in distributed mechanical systems. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 65-84. doi: 10.3934/dcds.2009.23.65


Z. G. Feng, Kok Lay Teo, N. U. Ahmed, Yulin Zhao, W. Y. Yan. Optimal fusion of sensor data for Kalman filtering. Discrete and Continuous Dynamical Systems, 2006, 14 (3) : 483-503. doi: 10.3934/dcds.2006.14.483


Laura Martín-Fernández, Gianni Gilioli, Ettore Lanzarone, Joaquín Míguez, Sara Pasquali, Fabrizio Ruggeri, Diego P. Ruiz. A Rao-Blackwellized particle filter for joint parameter estimation and biomass tracking in a stochastic predator-prey system. Mathematical Biosciences & Engineering, 2014, 11 (3) : 573-597. doi: 10.3934/mbe.2014.11.573


Gianni Gilioli, Sara Pasquali, Fabrizio Ruggeri. Nonlinear functional response parameter estimation in a stochastic predator-prey model. Mathematical Biosciences & Engineering, 2012, 9 (1) : 75-96. doi: 10.3934/mbe.2012.9.75


Andrea Arnold, Daniela Calvetti, Erkki Somersalo. Vectorized and parallel particle filter SMC parameter estimation for stiff ODEs. Conference Publications, 2015, 2015 (special) : 75-84. doi: 10.3934/proc.2015.0075


Sebastian Springer, Heikki Haario, Vladimir Shemyakin, Leonid Kalachev, Denis Shchepakin. Robust parameter estimation of chaotic systems. Inverse Problems and Imaging, 2019, 13 (6) : 1189-1212. doi: 10.3934/ipi.2019053


Azmy S. Ackleh, Jeremy J. Thibodeaux. Parameter estimation in a structured erythropoiesis model. Mathematical Biosciences & Engineering, 2008, 5 (4) : 601-616. doi: 10.3934/mbe.2008.5.601


Junyoung Jang, Kihoon Jang, Hee-Dae Kwon, Jeehyun Lee. Feedback control of an HBV model based on ensemble kalman filter and differential evolution. Mathematical Biosciences & Engineering, 2018, 15 (3) : 667-691. doi: 10.3934/mbe.2018030


Débora A. F. Albanez, Maicon J. Benvenutti. Continuous data assimilation algorithm for simplified Bardina model. Evolution Equations and Control Theory, 2018, 7 (1) : 33-52. doi: 10.3934/eect.2018002


Robert Azencott, Yutheeka Gadhyan. Accurate parameter estimation for coupled stochastic dynamics. Conference Publications, 2009, 2009 (Special) : 44-53. doi: 10.3934/proc.2009.2009.44


Alex Capaldi, Samuel Behrend, Benjamin Berman, Jason Smith, Justin Wright, Alun L. Lloyd. Parameter estimation and uncertainty quantification for an epidemic model. Mathematical Biosciences & Engineering, 2012, 9 (3) : 553-576. doi: 10.3934/mbe.2012.9.553


Alexander Bibov, Heikki Haario, Antti Solonen. Stabilized BFGS approximate Kalman filter. Inverse Problems and Imaging, 2015, 9 (4) : 1003-1024. doi: 10.3934/ipi.2015.9.1003


Russell Johnson, Carmen Núñez. The Kalman-Bucy filter revisited. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4139-4153. doi: 10.3934/dcds.2014.34.4139


John Maclean, Elaine T. Spiller. A surrogate-based approach to nonlinear, non-Gaussian joint state-parameter data assimilation. Foundations of Data Science, 2021, 3 (3) : 589-614. doi: 10.3934/fods.2021019


Gerasimos G. Rigatos, Efthymia G. Rigatou, Jean Daniel Djida. Change detection in the dynamics of an intracellular protein synthesis model using nonlinear Kalman filtering. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1017-1035. doi: 10.3934/mbe.2015.12.1017


Mojtaba F. Fathi, Ahmadreza Baghaie, Ali Bakhshinejad, Raphael H. Sacho, Roshan M. D'Souza. Time-resolved denoising using model order reduction, dynamic mode decomposition, and kalman filter and smoother. Journal of Computational Dynamics, 2020, 7 (2) : 469-487. doi: 10.3934/jcd.2020019


Wawan Hafid Syaifudin, Endah R. M. Putri. The application of model predictive control on stock portfolio optimization with prediction based on Geometric Brownian Motion-Kalman Filter. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021119


Chongyang Liu, Meijia Han, Zhaohua Gong, Kok Lay Teo. Robust parameter estimation for constrained time-delay systems with inexact measurements. Journal of Industrial and Management Optimization, 2021, 17 (1) : 317-337. doi: 10.3934/jimo.2019113

2020 Impact Factor: 1.392


  • PDF downloads (86)
  • HTML views (0)
  • Cited by (46)

Other articles
by authors

[Back to Top]