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Article information
2017 , Volume 22, Special issue, p.27-43
Grishina A.A., Penenko A.V.
Chemical kinetics modelling with data assimilation schemes
The paper addresses a comparison of the implicitly modified variational data assimilation 3DVar algorithm with the weak-constrained method 4DVar. Both methods are applied to a chemical kinetics problem based on the Robertson system. To model the measurement data for the Robertson system a direct problem is solved. Noise is accounted in the numerical solution and the real-time data assimilation of one reagent is carried out. The numerical solution for the direct problem is obtained with the help of quasi-steady state approximation, while the direct problem step in data assimilation algorithms is performed using linearly implicit Euler method. In the paper an implicit analogue of 3DVar is considered. It differs from the conventional method so that the penalty functional contains the norm of control function instead of the norm of the discrepancy between analysis and forecast. The norm of control function determines this discrepancy. The principal distinction between 3DVar and 4DVar methods lies in the formation of an “assimilation window” which contains different number of measurements as a part of cost functional expression, in the 3DVar method the measurements incoming immediately in the moment of computation are solely used, whereas in the 4DVar method the measurements, which have come earlier, are also taken into account. Identifying concentrations of all species and ongoing processes in time with the help of modified 3DVar algorithm and 4DVar are analyzed both theoretically and numerically. It was shown that the 3DVar algorithm demonstrated better performance for the considered conditions.
[full text]
Keywords: variational data assimilation, 3DVar, 4DVar, chemical kinetics, Robertson system, Quasi Steady-State Approximation
Author(s):
Grishina Anastasiia Alexandrovna
Position: The master of mathematics
Office: Novosibirsk State University, Institute of Computational Mathematics and Mathematical Geophysics SB RAS
Address: 6300090, Russia, Novosibirsk, Pirogova Str., 2
Phone Office: (383) 330-61-52
E-mail: a.a.grishina17@gmail.com
Penenko Alexey Vladimirovich
PhD.
Position: The master of mathematics
Office: Novosibirsk State University, Institute of Computational Mathematics and Mathematical Geophysics SB RAS
Address: 6300090, Russia, Novosibirsk, Pirogova Str., 2
Phone Office: (383) 330-61-52
E-mail: a.penenko@yandex.ru
SPIN-code: 9950-8820 References:
[1] Ipatova, V.M., Shutyaev, V.P. Algoritmy i zadachi assimilyatsii dannykh dlya modeley dinamiki atmosfery i okeana. Nauchno-obrazovatel'nyy kurs [Data assimilation algorithms and problems for atmospheric dynamics and and ocean models]. Moscow: Mir; 2013: 29. Available at: https://mipt.ru/education/chair/mathematics/upload/99f/algsaasimilation.pdf (In Russ.)
[2] Navon, I.M. Data assimilation for numerical weather prediction: A Review. Berlin, Germany: Springer; 2009:21–65.
[3] Kalman, R.E. A New Approach to Linear Filtering and Prediction Problems. Transactions of the ASME Journal of Basic Engineering. Ser. D. 1960; (82):35-45.
[4] Bocquet, M., Elbern, H., Eskes, H., Hirtl, M., Žabkar, R., Carmichael, G. R., Flemming, F., Inness, A., Pagowski, M., Pérez Camaño, J.L., Saide, P.E., San Jose, R., Sofiev, M., Vira, J., Baklanov, A., Carnevale, C., Grell, G., Seigneur, C. Data assimilation in atmospheric chemistry models: current status and future prospects for coupled chemistry meteorology models. Atmospheric Chemistry Physics. 2015; (15):5325–5358.
[5] Sasaki, Y. A fundamental study of the numerical prediction based on the variational principle. Journal of the Meteorological Society of Japan. 1955; (33):30–43.
[6] Krasyuk, T.V. Data assimilation: competition among methods and the problem of assimilation for satellite assimilation. Trudy GU "GMTs RF". 2012: 43–60. (In Russ.)
[7] Penenko, V.V., Obraztsov, N.N. A variational initialization method for the fields of meteorological elements. Meteorologiya i Gidrologiya. 1976; (11):3–16. (In Russ.)
[8] Marchuk, G.I., Penenko, V.V. Application of perturbation theory to problems of simulation of atmospheric processes. In: Monsoon dynamics (Joint IUTAM/IUGG International Symposium on Monsoon Dynamics, Delhi, 1977.) Eds. J.Lighthill & R.P.Pearce. Cambridge University press; 1981: 639–655.
[9] Le Dimet, F.-X., Talagrand, O. Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects. Tellus A. 1986; (38A):97-110.
[10] Marchuk, G.I. Formulation of some inverse problems. Doklady Akademii Nauk SSSR. 1994; 156(3):503–506. (In Russ.)
[11] Singh, K., Sandu, A., Bowman, K., Parrington, M., Jones, D.B., Lee, M. Ozone data assimilation with GEOS-Chem: a comparison between 3D-Var, 4D-Var, and suboptimal Kalman filter approaches. Atmospheric Chemistry and Physics Discussions. 2011; (11):22247–22300.
[12] WMO, 2015: Seamless Prediction of the Earth System: from Minutes to Months, (G Brunet, S Jones, PM Ruti Eds.), (WMO-No. 1156), ISBN 978-92-63-11156-2, Geneva.
[13] Penenko, V.V. Variational methods of data assimilation and inverse problems for studying the atmosphere, ocean, and environment. Sibirskii Zhurnal Vychislitel'noi Matematiki. 2009; 12(4):421–434. (In Russ.)
[14] Penenko, A.V., Penenko, V.V. Direct data assimilation method for convection-diffusion models based on splitting scheme. Computational technologies. 2014; 19(4):69-83. (In Russ.)
[15] Penenko, A. V. , Penenko, V. V., Tsvetova, E.A. Sequential data assimilation algorithms in air quality monitoring models based on a weak-constraint variational principle. Numerical Analysis and Applications. 2016; 9(4):312-325.
[16] Elbern, H., Strunk, A., Schmidt, H., Talagrand, O. Emission rate and chemical state estimation by 4 -dimensional variational inversion. Atmospheric Chemistry and Physics. 2007; (7):3749– 3769.
[17] Tikhonov, A.N. About ill-posed problems solving and regularization method. Doklady AN SSSR. 1963; 151(3):501–504. (In Russ.)
[18] Tikhonov, A.N., Goncharskiy, A.V., Stepanov, V.V., Yagola, A.G. Chislennye metody resheniya nekorrektnykh zadach [Numerical methods for solving ill-posed problems]. Moscow: Nauka; 1990: 230. (In Russ.)
[19] Agoshkov, V.I. Metody optimal'nogo upravleniya i sopryazhennykh uravneniy v zadachakh matematicheskoy fiziki [Optimal control and adjoint equation methods in problems of mathematical physics]. Moscow: IVM RAN; 2003:108–158. (In Russ.)
[20] Hayrer E., Wanner G. Reshenie obyknovennykh differentsial'nykh uravneniy. Zhestkie i differentsial'no-algebraicheskie zadachi [Solving ordinary differential equations. Stiff and differential-algebraic problems]. Moscow: Mir; 1999:685. (In Russ.)
[21] Pantea, C., Gupta, A., Rawlings, J. B., Craciun, G. The QSSA in chemical kinetics: As taught and as practiced. Eds. Jonoska, N., Saito, M. Discrete and Topological Models in Molecular Biology. Natural Computing Series. Berlin Heidelberg: Springer-Verlag; 2014:419-442.
[22] Jay, L.O., Sandu, A., Potra, F.A., Carmichael, G.R. Improved QSSA methods for atmospheric chemistry integration.
SIAM Journal on Scientific Computing. 1997; 18(1):182—202.
[23] GNU Scientific Library Reference Manual Edition 2.2.1, for GSL Version 2.2.1 Available at: https://www.gnu.org/software/gsl/manual/html_node/ (accessed 25.10.2016).
[24] Sabinin, D.A. Fiziologiya razvitiya rasteniy [Physiology of plants growth]. Moscow: Izdatel'stvo AN SSSR; 1963: 15. (In Russ.)
Bibliography link:
Grishina A.A., Penenko A.V. Chemical kinetics modelling with data assimilation schemes // Computational technologies. 2017. V. 22. XVII All-Russian Conference of Young Scientists on Mathematical Modeling and Information Technology. P. 27-43
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