
-
Previous Article
Global existence, general decay and blow-up for a nonlinear wave equation with logarithmic source term and fractional boundary dissipation
- DCDS-S Home
- This Issue
-
Next Article
$ \Sigma $-shaped bifurcation curves for classes of elliptic systems
Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.
Readers can access Online First articles via the “Online First” tab for the selected journal.
Physics informed model error for data assimilation
1. | Univ Bretagne - Sud, CNRS UMR 6205, LMBA, F-56000 Vannes, France |
2. | Univ Brest, CNRS UMR 6205, Laboratoire de Mathematiques de Bretagne Atlantique, France |
Data assimilation consists in combining a dynamical model with noisy observations to estimate the latent true state of a system. The dynamical model is generally misspecified and this generates a model error which is usually treated using a random noise. The aim of this paper is to suggest a new treatment for the model error that further takes into account the physics of the system: the physics informed model error. This model error treatment is a noisy stationary solution of the true dynamical model. It is embedded in the ensemble Kalman filter (EnKF), which is a usual method for data assimilation. The proposed strategy is then applied to study the heat diffusion in a bar when the external heat source is unknown. It is compared to usual methods to quantify the model error. The numerical results show that our method is more accurate, in particular when the observations are available at a low temporal resolution.
References:
[1] |
P. Ailliot, A. Cuzol, G. Durrieu, H. Flourent, E. Frénod, J. Guillot, J.-P. Lucas and F. Septier, Synthèse des questions mathématiques soulevées par la mise en oeuvre de jumeaux numériques pour le suivi et le pilotage de systèmes dynamiques en entreprises, hal-03167416. |
[2] |
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, 127 (1999), 2741-2758.
doi: 10.1175/1520-0493(1999)127<2741:AMCIOT>2.0.CO;2. |
[3] |
J. Berner, U. Achatz, L. Batte, L. Bengtsson, A. De La Camara, H. M. Christensen, M. Colangeli, D. R. Coleman, D. Crommelin and S. I. Dolaptchiev,
Stochastic parameterization: Toward a new view of weather and climate models, Bulletin of the American Meteorological Society, 98 (2017), 565-588.
doi: 10.1175/BAMS-D-15-00268.1. |
[4] |
J. Brajard, A. Carrassi, M. Bocquet and L. Bertino, Combining data assimilation and machine learning to infer unresolved scale parametrization, Philos. Trans. Roy. Soc. A, 379 (2021), Paper No. 20200086, 16 pp.
doi: 10.1098/rsta.2020.0086. |
[5] |
G. Desroziers, L. Berre, B. Chapnik and P. Poli,
Diagnosis of observation, background and analysis-error statistics in observation space, Quarterly Journal of the Royal Meteorological Society: A Journal of the Atmospheric Sciences, Applied Meteorology and Physical Oceanography, 131 (2005), 3385-3396.
doi: 10.1256/qj.05.108. |
[6] |
G. Evensen,
Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics, Journal of Geophysical Research: Oceans, 99 (1994), 10143-10162.
doi: 10.1029/94JC00572. |
[7] |
A. Farchi, P. Laloyaux, M. Bonavita and M. Bocquet,
Using machine learning to correct model error in data assimilation and forecast applications, Quarterly Journal of the Royal Meteorological Society, 147 (2021), 3067-3084.
doi: 10.1002/qj.4116. |
[8] |
M. Ghil and P. Malanotte-Rizzoli,
Data assimilation in meteorology and oceanography, Advances in Geophysics, 33 (1991), 141-266.
doi: 10.1016/S0065-2687(08)60442-2. |
[9] |
R. Lguensat, P. Tandeo, P. Ailliot, M. Pulido and R. Fablet,
The analog data assimilation, Monthly Weather Review, 145 (2017), 4093-4107.
doi: 10.1175/MWR-D-16-0441.1. |
[10] |
T. N. Palmer, R. Buizza, F. Doblas-Reyes, T. Jung, M. Leutbecher, G. J. Shutts, M. Steinheimer and A. Weisheimer, Stochastic parametrization and model uncertainty, ECMWF Technical Memoranda. |
[11] |
P. Tandeo, P. Ailliot, M. Bocquet, A. Carrassi, T. Miyoshi, M. Pulido and Y. Zhen,
A review of innovation-based methods to jointly estimate model and observation error covariance matrices in ensemble data assimilation, Monthly Weather Review, 148 (2020), 3973-3994.
|
show all references
References:
[1] |
P. Ailliot, A. Cuzol, G. Durrieu, H. Flourent, E. Frénod, J. Guillot, J.-P. Lucas and F. Septier, Synthèse des questions mathématiques soulevées par la mise en oeuvre de jumeaux numériques pour le suivi et le pilotage de systèmes dynamiques en entreprises, hal-03167416. |
[2] |
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, 127 (1999), 2741-2758.
doi: 10.1175/1520-0493(1999)127<2741:AMCIOT>2.0.CO;2. |
[3] |
J. Berner, U. Achatz, L. Batte, L. Bengtsson, A. De La Camara, H. M. Christensen, M. Colangeli, D. R. Coleman, D. Crommelin and S. I. Dolaptchiev,
Stochastic parameterization: Toward a new view of weather and climate models, Bulletin of the American Meteorological Society, 98 (2017), 565-588.
doi: 10.1175/BAMS-D-15-00268.1. |
[4] |
J. Brajard, A. Carrassi, M. Bocquet and L. Bertino, Combining data assimilation and machine learning to infer unresolved scale parametrization, Philos. Trans. Roy. Soc. A, 379 (2021), Paper No. 20200086, 16 pp.
doi: 10.1098/rsta.2020.0086. |
[5] |
G. Desroziers, L. Berre, B. Chapnik and P. Poli,
Diagnosis of observation, background and analysis-error statistics in observation space, Quarterly Journal of the Royal Meteorological Society: A Journal of the Atmospheric Sciences, Applied Meteorology and Physical Oceanography, 131 (2005), 3385-3396.
doi: 10.1256/qj.05.108. |
[6] |
G. Evensen,
Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics, Journal of Geophysical Research: Oceans, 99 (1994), 10143-10162.
doi: 10.1029/94JC00572. |
[7] |
A. Farchi, P. Laloyaux, M. Bonavita and M. Bocquet,
Using machine learning to correct model error in data assimilation and forecast applications, Quarterly Journal of the Royal Meteorological Society, 147 (2021), 3067-3084.
doi: 10.1002/qj.4116. |
[8] |
M. Ghil and P. Malanotte-Rizzoli,
Data assimilation in meteorology and oceanography, Advances in Geophysics, 33 (1991), 141-266.
doi: 10.1016/S0065-2687(08)60442-2. |
[9] |
R. Lguensat, P. Tandeo, P. Ailliot, M. Pulido and R. Fablet,
The analog data assimilation, Monthly Weather Review, 145 (2017), 4093-4107.
doi: 10.1175/MWR-D-16-0441.1. |
[10] |
T. N. Palmer, R. Buizza, F. Doblas-Reyes, T. Jung, M. Leutbecher, G. J. Shutts, M. Steinheimer and A. Weisheimer, Stochastic parametrization and model uncertainty, ECMWF Technical Memoranda. |
[11] |
P. Tandeo, P. Ailliot, M. Bocquet, A. Carrassi, T. Miyoshi, M. Pulido and Y. Zhen,
A review of innovation-based methods to jointly estimate model and observation error covariance matrices in ensemble data assimilation, Monthly Weather Review, 148 (2020), 3973-3994.
|





generate |
For |
generate |
$\underline{Analysis}:$: for |
generate |
generate |
For |
generate |
$\underline{Analysis}:$: for |
generate |
Parameter | Value |
100 | |
50 | |
30 | |
30 | |
0.05 | |
1 |
Parameter | Value |
100 | |
50 | |
30 | |
30 | |
0.05 | |
1 |
0.016 | |
0.001 | |
0.050 |
0.016 | |
0.001 | |
0.050 |
Algorithm | Global RMSE |
PIME | 0.017 |
0.048 | |
0.025 |
Algorithm | Global RMSE |
PIME | 0.017 |
0.048 | |
0.025 |
[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] |
Laurent Bourgeois. Quantification of the unique continuation property for the heat equation. Mathematical Control and Related Fields, 2017, 7 (3) : 347-367. doi: 10.3934/mcrf.2017012 |
[3] |
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 |
[4] |
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 |
[5] |
Georgios I. Papayiannis. Robust policy selection and harvest risk quantification for natural resources management under model uncertainty. Journal of Dynamics and Games, 2022, 9 (2) : 203-217. doi: 10.3934/jdg.2022004 |
[6] |
Andrew J. Majda, Michal Branicki. Lessons in uncertainty quantification for turbulent dynamical systems. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3133-3221. doi: 10.3934/dcds.2012.32.3133 |
[7] |
Jing Li, Panos Stinis. Mori-Zwanzig reduced models for uncertainty quantification. Journal of Computational Dynamics, 2019, 6 (1) : 39-68. doi: 10.3934/jcd.2019002 |
[8] |
H. T. Banks, Robert Baraldi, Karissa Cross, Kevin Flores, Christina McChesney, Laura Poag, Emma Thorpe. Uncertainty quantification in modeling HIV viral mechanics. Mathematical Biosciences & Engineering, 2015, 12 (5) : 937-964. doi: 10.3934/mbe.2015.12.937 |
[9] |
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 |
[10] |
Richard Archibald, Feng Bao, Yanzhao Cao, He Zhang. A backward SDE method for uncertainty quantification in deep learning. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022062 |
[11] |
Michael Herty, Elisa Iacomini. Uncertainty quantification in hierarchical vehicular flow models. Kinetic and Related Models, 2022, 15 (2) : 239-256. doi: 10.3934/krm.2022006 |
[12] |
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 |
[13] |
Jochen Bröcker. Existence and uniqueness for variational data assimilation in continuous time. Mathematical Control and Related Fields, 2021 doi: 10.3934/mcrf.2021050 |
[14] |
Haruki Umakoshi. A semilinear heat equation with initial data in negative Sobolev spaces. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 745-767. doi: 10.3934/dcdss.2020365 |
[15] |
Yohei Fujishima. Blow-up set for a superlinear heat equation and pointedness of the initial data. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4617-4645. doi: 10.3934/dcds.2014.34.4617 |
[16] |
Mi-Ho Giga, Yoshikazu Giga, Takeshi Ohtsuka, Noriaki Umeda. On behavior of signs for the heat equation and a diffusion method for data separation. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2277-2296. doi: 10.3934/cpaa.2013.12.2277 |
[17] |
Luz de Teresa, Enrique Zuazua. Identification of the class of initial data for the insensitizing control of the heat equation. Communications on Pure and Applied Analysis, 2009, 8 (1) : 457-471. doi: 10.3934/cpaa.2009.8.457 |
[18] |
José Miguel Pasini, Tuhin Sahai. Polynomial chaos based uncertainty quantification in Hamiltonian, multi-time scale, and chaotic systems. Journal of Computational Dynamics, 2014, 1 (2) : 357-375. doi: 10.3934/jcd.2014.1.357 |
[19] |
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 |
[20] |
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 |
2021 Impact Factor: 1.865
Tools
Metrics
Other articles
by authors
[Back to Top]