
-
Previous Article
Analysis of the feedback particle filter with diffusion map based approximation of the gain
- FoDS Home
- This Issue
-
Next Article
Mean field limit of Ensemble Square Root filters - discrete and continuous time
A surrogate-based approach to nonlinear, non-Gaussian joint state-parameter data assimilation
1. | School of Mathematical Sciences, University of Adelaide, SA 5005, Australia |
2. | Department of Mathematical and Statistical Sciences, Marquette University, Milwaukee, WI 53201, USA |
Many recent advances in sequential assimilation of data into nonlinear high-dimensional models are modifications to particle filters which employ efficient searches of a high-dimensional state space. In this work, we present a complementary strategy that combines statistical emulators and particle filters. The emulators are used to learn and offer a computationally cheap approximation to the forward dynamic mapping. This emulator-particle filter (Emu-PF) approach requires a modest number of forward-model runs, but yields well-resolved posterior distributions even in non-Gaussian cases. We explore several modifications to the Emu-PF that utilize mechanisms for dimension reduction to efficiently fit the statistical emulator, and present a series of simulation experiments on an atypical Lorenz-96 system to demonstrate their performance. We conclude with a discussion on how the Emu-PF can be paired with modern particle filtering algorithms.
References:
[1] |
M. J. Bayarri, J. O. Berger, J. Cafeo, G. Garcia-Donato and F. Liu,
Computer model validation with functional output, Ann. Statist., 35 (2007), 1874-1906.
doi: 10.1214/009053607000000163. |
[2] |
J. Betancourt, F. Bachoc, T. Klein, D. Idier, R. Pedreros and J. Rohmer, Gaussian process metamodeling of functional-input code for coastal flood hazard assessment, Reliability Engineering & System Safety, 198 (2020).
doi: 10.1016/j.ress.2020.106870. |
[3] |
M. Bocquet, J. Brajard, A. Carrassi and L. Bertino,
Bayesian inference of chaotic dynamics by merging data assimilation, machine learning and expectation-maximization, Foundations of Data Science, 2 (2020), 55-80.
doi: 10.3934/fods.2020004. |
[4] |
J. Brajard, A. Carassi, M. Bocquet and L. Bertino, Combining data assimilation and machine learning to emulate a dynamical model from sparse and noisy observations: A case study with the Lorenz 96 model, J. Comput. Sci., 44 (2020), 11pp.
doi: 10.1016/j.jocs.2020.101171. |
[5] |
A. Carrassi, M. Bocquet, L. Bertino and G. Evensen, Data assimilation in the geosciences: An overview of methods, issues, and perspectives, Wiley Interdisciplinary Reviews: Climate Change, 9 (2018).
doi: 10.1002/wcc.535. |
[6] |
E. Cleary, A. Garbuno-Inigo, S. Lan, T. Schneider and A. M. Stuart, Calibrate, emulate, sample, J. Comput. Phys., 424 (2021), 20pp.
doi: 10.1016/j.jcp.2020.109716. |
[7] |
D. Crisan and K. Li,
Generalised particle filters with Gaussian mixtures, Stochastic Process. Appl., 125 (2015), 2643-2673.
doi: 10.1016/j.spa.2015.01.008. |
[8] |
A. Doucet, N. de Freitas and N. Gordon, Sequential Monte Carlo Methods in Practice, Statistics for Engineering and Information Science, Springer-Verlag, New York, 2001.
doi: 10.1007/978-1-4757-3437-9. |
[9] |
G. Evensen, Data Assimilation. The Ensemble Kalman Filter, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-642-03711-5. |
[10] |
G. Evensen,
The ensemble Kalman filter: Theoretical formulation and practical implementation, Ocean Dynamics, 53 (2003), 343-367.
doi: 10.1007/s10236-003-0036-9. |
[11] |
G. A. Gottwald and S. Reich, Supervised learning from noisy observations: Combining machine-learning techniques with data assimilation, Phys. D, 423 (2021), 15pp.
doi: 10.1016/j.physd.2021.132911. |
[12] |
M. Gu and J. O. Berger,
Parallel partial Gaussian process emulation for computer models with massive output, Ann. Appl. Stat., 10 (2016), 1317-1347.
doi: 10.1214/16-AOAS934. |
[13] |
M. Gu, J. Palomo and J. O. Berger,
RobustGaSP: Robust Gaussian Stochastic Process Emulation in R, The R Journal, 11 (2019), 112-136.
doi: 10.32614/RJ-2019-011. |
[14] |
M. E. Johnson, L. M. Moore and D. Ylvisaker,
Minimax and maximin distance designs, J. Statist. Plann. Inference, 26 (1990), 131-148.
doi: 10.1016/0378-3758(90)90122-B. |
[15] |
K. Law, A. Stuart and K. Zygalakis, Data Assimilation. A Mathematical Introduction, Texts in Applied Mathematics, 62, Springer, Cham, 2015.
doi: 10.1007/978-3-319-20325-6. |
[16] |
J. Liu and M. West, Combined parameter and state estimation in simulation-based filtering, in Sequential Monte Carlo Methods in Practice, Stat. Eng. Inf. Sci., Springer, New York, 2001,197–223.
doi: 10.1007/978-1-4757-3437-9_10. |
[17] |
J. S. Liu and R. Chen,
Sequential Monte Carlo methods for dynamic systems, J. Amer. Statist. Assoc., 93 (1998), 1032-1044.
doi: 10.1080/01621459.1998.10473765. |
[18] |
X. Liu and S. Guillas,
Dimension reduction for Gaussian process emulation: An application to the influence of bathymetry on tsunami heights, SIAM/ASA J. Uncertain. Quantif., 5 (2017), 787-812.
doi: 10.1137/16M1090648. |
[19] |
E. N. Lorenz, Predictability - A problem partly solved, in Proceedings of Seminar on Predictability, Cambridge University Press, Reading, UK, 1996.
doi: 10.1017/CBO9780511617652.004.![]() ![]() |
[20] |
J. Maclean and E. S. V. Vleck,
Particle filters for data assimilation based on reduced-order data models, Q. J. Roy. Meteor. Soc., 147 (2021), 1892-1907.
doi: 10.1002/qj.4001. |
[21] |
M. Morzfeld and D. Hodyss, Gaussian approximations in filters and smoothers for data assimilation, Tellus A, 71 (2019).
doi: 10.1080/16000870.2019.1600344. |
[22] |
S. Nakano, G. Ueno and T. Higuchi,
Merging particle filter for sequential data assimilation, Nonlin. Processes Geophys., 14 (2007), 395-408.
doi: 10.5194/npg-14-395-2007. |
[23] |
D. Orrell and L. A. Smith,
Visualizing bifurcations in high dimensional systems: The spectral bifurcation diagram, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 3015-3027.
doi: 10.1142/S0218127403008387. |
[24] |
J. Poterjoy,
A localized particle filter for high-dimensional nonlinear systems, Monthly Weather Review, 144 (2016), 59-76.
doi: 10.1175/MWR-D-15-0163.1. |
[25] |
R. Potthast, A. Walter and A. Rhodin,
A localized adaptive particle filter within an operational NWP framework, Monthly Weather Review, 147 (2019), 345-362.
doi: 10.1175/MWR-D-18-0028.1. |
[26] |
C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning, Adaptative Computation and Machine Learning, MIT Press, Cambridge, MA, 2006. Available from: http://www.gaussianprocess.org/gpml/chapters. |
[27] |
S. Reich and C. Cotter, Probabilistic Forecasting and Bayesian Data Assimilation, Cambridge University Press, New York, 2015.
doi: 10.1017/CBO9781107706804.![]() ![]() ![]() |
[28] |
J. Sacks, W. J. Welch, T. J. Mitchell and H. P. Wynn,
Design and analysis of computer experiments, Statist. Sci., 4 (1989), 409-423.
doi: 10.1214/ss/1177012413. |
[29] |
N. Santitissadeekorn and C. Jones,
Two-stage filtering for joint state-parameter estimation, Monthly Weather Review, 143 (2015), 2028-2042.
doi: 10.1175/MWR-D-14-00176.1. |
[30] |
T. J. Santner, B. J. Williams and W. I. Notz, The Design and Analysis of Computer Experiments, Springer Series in Statistics, Springer, New York, 2018.
doi: 10.1007/978-1-4939-8847-1. |
[31] |
C. Snyder, Particle filters, the "optimal" proposal and high-dimensional systems, in Proceedings of the ECMWF Seminar on Data Assimilation for Atmosphere and Ocean, 2011, 1–10. Available from: https://www.ecmwf.int/sites/default/files/elibrary/2012/12354-particle-filters-optimal-proposal-and-high-dimensional-systems.pdf. |
[32] |
C. Snyder, T. Bengtsson, P. Bickel and J. Anderson,
Obstacles to high-dimensional particle filtering, Monthly Weather Review, 136 (2008), 4629-4640.
doi: 10.1175/2008MWR2529.1. |
[33] |
P. J. van Leeuwen,
Nonlinear data assimilation in geosciences: An extremely efficient particle filter, Q. J. Roy. Meteor. Soc., 136 (2010), 1991-1999.
doi: 10.1002/qj.699. |
[34] |
P. J. van Leeuwen, H. R. Künsch, L. Nerger, R. Potthast and S. Reich,
Particle filters for high-dimensional geoscience applications: A review, Q. J. Roy. Meteor. Soc., 145 (2019), 2335-2365.
doi: 10.1002/qj.3551. |
[35] |
W. J. Welch, R. J. Buck, J. Sacks, H. P. Wynn, T. J. Mitchell and M. D. Morris,
Screening, predicting, and computer experiments, Technometrics, 34 (1992), 15-25.
doi: 10.2307/1269548. |
show all references
References:
[1] |
M. J. Bayarri, J. O. Berger, J. Cafeo, G. Garcia-Donato and F. Liu,
Computer model validation with functional output, Ann. Statist., 35 (2007), 1874-1906.
doi: 10.1214/009053607000000163. |
[2] |
J. Betancourt, F. Bachoc, T. Klein, D. Idier, R. Pedreros and J. Rohmer, Gaussian process metamodeling of functional-input code for coastal flood hazard assessment, Reliability Engineering & System Safety, 198 (2020).
doi: 10.1016/j.ress.2020.106870. |
[3] |
M. Bocquet, J. Brajard, A. Carrassi and L. Bertino,
Bayesian inference of chaotic dynamics by merging data assimilation, machine learning and expectation-maximization, Foundations of Data Science, 2 (2020), 55-80.
doi: 10.3934/fods.2020004. |
[4] |
J. Brajard, A. Carassi, M. Bocquet and L. Bertino, Combining data assimilation and machine learning to emulate a dynamical model from sparse and noisy observations: A case study with the Lorenz 96 model, J. Comput. Sci., 44 (2020), 11pp.
doi: 10.1016/j.jocs.2020.101171. |
[5] |
A. Carrassi, M. Bocquet, L. Bertino and G. Evensen, Data assimilation in the geosciences: An overview of methods, issues, and perspectives, Wiley Interdisciplinary Reviews: Climate Change, 9 (2018).
doi: 10.1002/wcc.535. |
[6] |
E. Cleary, A. Garbuno-Inigo, S. Lan, T. Schneider and A. M. Stuart, Calibrate, emulate, sample, J. Comput. Phys., 424 (2021), 20pp.
doi: 10.1016/j.jcp.2020.109716. |
[7] |
D. Crisan and K. Li,
Generalised particle filters with Gaussian mixtures, Stochastic Process. Appl., 125 (2015), 2643-2673.
doi: 10.1016/j.spa.2015.01.008. |
[8] |
A. Doucet, N. de Freitas and N. Gordon, Sequential Monte Carlo Methods in Practice, Statistics for Engineering and Information Science, Springer-Verlag, New York, 2001.
doi: 10.1007/978-1-4757-3437-9. |
[9] |
G. Evensen, Data Assimilation. The Ensemble Kalman Filter, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-642-03711-5. |
[10] |
G. Evensen,
The ensemble Kalman filter: Theoretical formulation and practical implementation, Ocean Dynamics, 53 (2003), 343-367.
doi: 10.1007/s10236-003-0036-9. |
[11] |
G. A. Gottwald and S. Reich, Supervised learning from noisy observations: Combining machine-learning techniques with data assimilation, Phys. D, 423 (2021), 15pp.
doi: 10.1016/j.physd.2021.132911. |
[12] |
M. Gu and J. O. Berger,
Parallel partial Gaussian process emulation for computer models with massive output, Ann. Appl. Stat., 10 (2016), 1317-1347.
doi: 10.1214/16-AOAS934. |
[13] |
M. Gu, J. Palomo and J. O. Berger,
RobustGaSP: Robust Gaussian Stochastic Process Emulation in R, The R Journal, 11 (2019), 112-136.
doi: 10.32614/RJ-2019-011. |
[14] |
M. E. Johnson, L. M. Moore and D. Ylvisaker,
Minimax and maximin distance designs, J. Statist. Plann. Inference, 26 (1990), 131-148.
doi: 10.1016/0378-3758(90)90122-B. |
[15] |
K. Law, A. Stuart and K. Zygalakis, Data Assimilation. A Mathematical Introduction, Texts in Applied Mathematics, 62, Springer, Cham, 2015.
doi: 10.1007/978-3-319-20325-6. |
[16] |
J. Liu and M. West, Combined parameter and state estimation in simulation-based filtering, in Sequential Monte Carlo Methods in Practice, Stat. Eng. Inf. Sci., Springer, New York, 2001,197–223.
doi: 10.1007/978-1-4757-3437-9_10. |
[17] |
J. S. Liu and R. Chen,
Sequential Monte Carlo methods for dynamic systems, J. Amer. Statist. Assoc., 93 (1998), 1032-1044.
doi: 10.1080/01621459.1998.10473765. |
[18] |
X. Liu and S. Guillas,
Dimension reduction for Gaussian process emulation: An application to the influence of bathymetry on tsunami heights, SIAM/ASA J. Uncertain. Quantif., 5 (2017), 787-812.
doi: 10.1137/16M1090648. |
[19] |
E. N. Lorenz, Predictability - A problem partly solved, in Proceedings of Seminar on Predictability, Cambridge University Press, Reading, UK, 1996.
doi: 10.1017/CBO9780511617652.004.![]() ![]() |
[20] |
J. Maclean and E. S. V. Vleck,
Particle filters for data assimilation based on reduced-order data models, Q. J. Roy. Meteor. Soc., 147 (2021), 1892-1907.
doi: 10.1002/qj.4001. |
[21] |
M. Morzfeld and D. Hodyss, Gaussian approximations in filters and smoothers for data assimilation, Tellus A, 71 (2019).
doi: 10.1080/16000870.2019.1600344. |
[22] |
S. Nakano, G. Ueno and T. Higuchi,
Merging particle filter for sequential data assimilation, Nonlin. Processes Geophys., 14 (2007), 395-408.
doi: 10.5194/npg-14-395-2007. |
[23] |
D. Orrell and L. A. Smith,
Visualizing bifurcations in high dimensional systems: The spectral bifurcation diagram, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 3015-3027.
doi: 10.1142/S0218127403008387. |
[24] |
J. Poterjoy,
A localized particle filter for high-dimensional nonlinear systems, Monthly Weather Review, 144 (2016), 59-76.
doi: 10.1175/MWR-D-15-0163.1. |
[25] |
R. Potthast, A. Walter and A. Rhodin,
A localized adaptive particle filter within an operational NWP framework, Monthly Weather Review, 147 (2019), 345-362.
doi: 10.1175/MWR-D-18-0028.1. |
[26] |
C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning, Adaptative Computation and Machine Learning, MIT Press, Cambridge, MA, 2006. Available from: http://www.gaussianprocess.org/gpml/chapters. |
[27] |
S. Reich and C. Cotter, Probabilistic Forecasting and Bayesian Data Assimilation, Cambridge University Press, New York, 2015.
doi: 10.1017/CBO9781107706804.![]() ![]() ![]() |
[28] |
J. Sacks, W. J. Welch, T. J. Mitchell and H. P. Wynn,
Design and analysis of computer experiments, Statist. Sci., 4 (1989), 409-423.
doi: 10.1214/ss/1177012413. |
[29] |
N. Santitissadeekorn and C. Jones,
Two-stage filtering for joint state-parameter estimation, Monthly Weather Review, 143 (2015), 2028-2042.
doi: 10.1175/MWR-D-14-00176.1. |
[30] |
T. J. Santner, B. J. Williams and W. I. Notz, The Design and Analysis of Computer Experiments, Springer Series in Statistics, Springer, New York, 2018.
doi: 10.1007/978-1-4939-8847-1. |
[31] |
C. Snyder, Particle filters, the "optimal" proposal and high-dimensional systems, in Proceedings of the ECMWF Seminar on Data Assimilation for Atmosphere and Ocean, 2011, 1–10. Available from: https://www.ecmwf.int/sites/default/files/elibrary/2012/12354-particle-filters-optimal-proposal-and-high-dimensional-systems.pdf. |
[32] |
C. Snyder, T. Bengtsson, P. Bickel and J. Anderson,
Obstacles to high-dimensional particle filtering, Monthly Weather Review, 136 (2008), 4629-4640.
doi: 10.1175/2008MWR2529.1. |
[33] |
P. J. van Leeuwen,
Nonlinear data assimilation in geosciences: An extremely efficient particle filter, Q. J. Roy. Meteor. Soc., 136 (2010), 1991-1999.
doi: 10.1002/qj.699. |
[34] |
P. J. van Leeuwen, H. R. Künsch, L. Nerger, R. Potthast and S. Reich,
Particle filters for high-dimensional geoscience applications: A review, Q. J. Roy. Meteor. Soc., 145 (2019), 2335-2365.
doi: 10.1002/qj.3551. |
[35] |
W. J. Welch, R. J. Buck, J. Sacks, H. P. Wynn, T. J. Mitchell and M. D. Morris,
Screening, predicting, and computer experiments, Technometrics, 34 (1992), 15-25.
doi: 10.2307/1269548. |










RMSE ( |
Var ( |
RMSE ( |
Var ( |
Resampling | |
Fine PF | 0.066 | 0.0035 | 0.34 | 0.15 | 226 |
Coarse PF | 0.79 | 0.0015 | 2.1 | 0.16 | 663 |
EnKF | 0.048 | 0.0018 | 0.32 | 0.12 | - |
Emu-PF ( |
0.13 | 0.00026 | 2.4 | 5.1 | 483 |
RMSE ( |
Var ( |
RMSE ( |
Var ( |
Resampling | |
Fine PF | 0.066 | 0.0035 | 0.34 | 0.15 | 226 |
Coarse PF | 0.79 | 0.0015 | 2.1 | 0.16 | 663 |
EnKF | 0.048 | 0.0018 | 0.32 | 0.12 | - |
Emu-PF ( |
0.13 | 0.00026 | 2.4 | 5.1 | 483 |
RMSE ( |
Var ( |
RMSE ( |
Var ( |
Resampling | |
Fine PF | 0.074 | 0.0043 | 0.7 | 0.61 | 173 |
Coarse PF | 0.49 | 0.0016 | 4.9 | 0.13 | 312 |
EnKF | 0.065 | 0.0027 | 0.78 | 0.66 | - |
Emu-PF ( |
0.38 | 0.00085 | 3.8 | 6.1 | 526 |
Emu-PF (PCA) | 0.27 | 0.00051 | 3.1 | 0.58 | 339 |
RMSE ( |
Var ( |
RMSE ( |
Var ( |
Resampling | |
Fine PF | 0.074 | 0.0043 | 0.7 | 0.61 | 173 |
Coarse PF | 0.49 | 0.0016 | 4.9 | 0.13 | 312 |
EnKF | 0.065 | 0.0027 | 0.78 | 0.66 | - |
Emu-PF ( |
0.38 | 0.00085 | 3.8 | 6.1 | 526 |
Emu-PF (PCA) | 0.27 | 0.00051 | 3.1 | 0.58 | 339 |
RMSE ( |
Var ( |
RMSE ( |
Var ( |
Resampling | |
Fine PF | 0.062 | 0.0032 | 0.28 | 0.13 | 243 |
Coarse PF | 1 | 0.0012 | 3.4 | 0.11 | 739 |
EnKF | 0.045 | 0.0017 | 0.25 | 0.1 | - |
Emu-PF ( |
0.15 | 0.00034 | 2.4 | 5.1 | 590 |
Emu-PF (PCA) | 0.084 | 0.00075 | 1.5 | 0.085 | 334 |
RMSE ( |
Var ( |
RMSE ( |
Var ( |
Resampling | |
Fine PF | 0.062 | 0.0032 | 0.28 | 0.13 | 243 |
Coarse PF | 1 | 0.0012 | 3.4 | 0.11 | 739 |
EnKF | 0.045 | 0.0017 | 0.25 | 0.1 | - |
Emu-PF ( |
0.15 | 0.00034 | 2.4 | 5.1 | 590 |
Emu-PF (PCA) | 0.084 | 0.00075 | 1.5 | 0.085 | 334 |
RMSE ( |
Var ( |
Resampling | |
Fine PF | 0.47 | 0.15 | 1706 |
Coarse PF | 5.1 | 0.16 | 9917 |
EnKF | 1 | 0.096 | - |
Emu-PF (Localized) | 0.83 | 0.31 | 3566 |
RMSE ( |
Var ( |
Resampling | |
Fine PF | 0.47 | 0.15 | 1706 |
Coarse PF | 5.1 | 0.16 | 9917 |
EnKF | 1 | 0.096 | - |
Emu-PF (Localized) | 0.83 | 0.31 | 3566 |
RMSE ( |
Var ( |
RMSE ( |
Var ( |
Resampling | |
Fine OP-PF | 1.2 | 0.0075 | 1.9 | 3.3 | 226 |
Coarse OP-PF | 1.2 | 0.004 | 2.0 | 2.8 | 205 |
EnKF | 1.1 | 0.00042 | 1.5 | 1.8 | - |
Emu-PF ( |
0.5 | 0.0017 | 2.6 | 3.5 | 243 |
Emu-PF ( |
0.75 | 0.0035 | 2.0 | 3.7 | 232 |
Emu-PF (PCA) | 1.1 | 0.061 | 2.0 | 2.8 | 238 |
RMSE ( |
Var ( |
RMSE ( |
Var ( |
Resampling | |
Fine OP-PF | 1.2 | 0.0075 | 1.9 | 3.3 | 226 |
Coarse OP-PF | 1.2 | 0.004 | 2.0 | 2.8 | 205 |
EnKF | 1.1 | 0.00042 | 1.5 | 1.8 | - |
Emu-PF ( |
0.5 | 0.0017 | 2.6 | 3.5 | 243 |
Emu-PF ( |
0.75 | 0.0035 | 2.0 | 3.7 | 232 |
Emu-PF (PCA) | 1.1 | 0.061 | 2.0 | 2.8 | 238 |
[1] |
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 |
[2] |
Ryan Bennink, Ajay Jasra, Kody J. H. Law, Pavel Lougovski. Estimation and uncertainty quantification for the output from quantum simulators. Foundations of Data Science, 2019, 1 (2) : 157-176. doi: 10.3934/fods.2019007 |
[3] |
Weihong Guo, Yifei Lou, Jing Qin, Ming Yan. IPI special issue on "mathematical/statistical approaches in data science" in the Inverse Problem and Imaging. Inverse Problems and Imaging, 2021, 15 (1) : I-I. doi: 10.3934/ipi.2021007 |
[4] |
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 |
[5] |
Jelena Grbić, Jie Wu, Kelin Xia, Guo-Wei Wei. Aspects of topological approaches for data science. Foundations of Data Science, 2022, 4 (2) : 165-216. doi: 10.3934/fods.2022002 |
[6] |
Alexandre J. Chorin, Fei Lu, Robert N. Miller, Matthias Morzfeld, Xuemin Tu. Sampling, feasibility, and priors in data assimilation. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4227-4246. doi: 10.3934/dcds.2016.36.4227 |
[7] |
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 |
[8] |
Jochen Bröcker. Existence and uniqueness for variational data assimilation in continuous time. Mathematical Control and Related Fields, 2021 doi: 10.3934/mcrf.2021050 |
[9] |
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 |
[10] |
Andreas Chirstmann, Qiang Wu, Ding-Xuan Zhou. Preface to the special issue on analysis in machine learning and data science. Communications on Pure and Applied Analysis, 2020, 19 (8) : i-iii. doi: 10.3934/cpaa.2020171 |
[11] |
Xin Guo, Lei Shi. Preface of the special issue on analysis in data science: Methods and applications. Mathematical Foundations of Computing, 2020, 3 (4) : i-ii. doi: 10.3934/mfc.2020026 |
[12] |
Sarai Hedges, Kim Given. Addressing confirmation bias in middle school data science education. Foundations of Data Science, 2022 doi: 10.3934/fods.2021035 |
[13] |
Issam S. Strub, Julie Percelay, Olli-Pekka Tossavainen, Alexandre M. Bayen. Comparison of two data assimilation algorithms for shallow water flows. Networks and Heterogeneous Media, 2009, 4 (2) : 409-430. doi: 10.3934/nhm.2009.4.409 |
[14] |
Joshua Hudson, Michael Jolly. Numerical efficacy study of data assimilation for the 2D magnetohydrodynamic equations. Journal of Computational Dynamics, 2019, 6 (1) : 131-145. doi: 10.3934/jcd.2019006 |
[15] |
Yuan Pei. Continuous data assimilation for the 3D primitive equations of the ocean. Communications on Pure and Applied Analysis, 2019, 18 (2) : 643-661. doi: 10.3934/cpaa.2019032 |
[16] |
Juan Carlos De los Reyes, Estefanía Loayza-Romero. Total generalized variation regularization in data assimilation for Burgers' equation. Inverse Problems and Imaging, 2019, 13 (4) : 755-786. doi: 10.3934/ipi.2019035 |
[17] |
Adam Larios, Yuan Pei. Approximate continuous data assimilation of the 2D Navier-Stokes equations via the Voigt-regularization with observable data. Evolution Equations and Control Theory, 2020, 9 (3) : 733-751. doi: 10.3934/eect.2020031 |
[18] |
Michele La Rocca, Cira Perna. Designing neural networks for modeling biological data: A statistical perspective. Mathematical Biosciences & Engineering, 2014, 11 (2) : 331-342. doi: 10.3934/mbe.2014.11.331 |
[19] |
Chelsey Legacy, Andrew Zieffler, Benjamin S. Baumer, Valerie Barr, Nicholas J. Horton. Facilitating team-based data science: Lessons learned from the DSC-WAV project. Foundations of Data Science, 2022 doi: 10.3934/fods.2022003 |
[20] |
Karl R. B. Schmitt, Linda Clark, Katherine M. Kinnaird, Ruth E. H. Wertz, Björn Sandstede. Evaluation of EDISON's data science competency framework through a comparative literature analysis. Foundations of Data Science, 2021 doi: 10.3934/fods.2021031 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]