Email this article Sensitivity of functionals of variational data assimilation problems
- Authors: Shutyaev V.P.1, Le Dimet F.2
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Affiliations:
- Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences
- University Grenoble Alpes
- Issue: Vol 486, No 4 (2019)
- Pages: 421-425
- Section: Mathematics
- URL: https://journals.eco-vector.com/0869-5652/article/view/14442
- DOI: https://doi.org/10.31857/S0869-56524864421-425
- ID: 14442
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Abstract
The problem of variational data assimilation for a nonlinear evolutionary model is formulated as an optimal control problem to find simultaneously unknown parameters and the initial state of the model. The response function is considered as a functional of the optimal solution found as a result of assimilation. The sensitivity of the functional to observational data is studied. The gradient of the functional with respect to observations is associated with the solution of a nonstandard problem involving a system of direct and adjoint equations. On the basis of the Hessian of the original cost function, the solvability of the nonstandard problem is studied. An algorithm for calculating the gradient of the response function with respect to observational data is formulated and justified.
About the authors
V. P. Shutyaev
Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences
Author for correspondence.
Email: victor.shutyaev@mail.ru
Russian Federation, 8, Gubkina street, Moscow, 119991
F.-X. Le Dimet
University Grenoble Alpes
Email: victor.shutyaev@mail.ru
France, 621 Central Avenue, Saint-Martin-d'Heres, 38400
References
- Marchuk G.I. Adjoint Equations and Analysis of Complex Systems. Dordrecht: Kluwer, 1995.
- Lions J.-L. Contrôle Optimal des Systèmes Gouvernés par des Équations aux Dérivées Partielles. Paris: Dunod, 1968.
- Пененко В.В., Образцов Н.Н. Вариационный метод согласования полей метеорологических элементов // Метеорология и гидрология. 1976. № 11. С. 1-11.
- Le Dimet F.-X., Talagrand O. Variational Algorithms for Analysis and Assimilation of Meteorological Observations: Theoretical Aspects // Tellus. 1986. 38A. Р. 97-110.
- Marchuk G.I., Agoshkov V.I., Shutyaev V.P. Adjoint Equations and Perturbation Algorithms in Nonlinear Problems. N.Y.: CRC Press Inc., 1996.
- Le Dimet F.-X., Ngodock H.E., Luong B., Verron J. Sensitivity Analysis in Variational Data Assimilation // J. Meteorol. Soc. Jap. 1997. V. 75. № 1B. Р. 245-255.
- Daescu D.N. On the Sensitivity Equations оf Four-Dimensional Variational (4D-Var) Data Assimilation // Mon.Weather Rev. 2008. V. 136. № 8. Р. 3050-3065.
- Gejadze I., Le Dimet F.-X., Shutyaev V. On Analysis Error Covariances in Variational Data Assimilation // SIAM J. Sci. Computing. 2008. V. 30. № 4. Р. 1847-1874.
- Gejadze I., Le Dimet F.-X., Shutyaev V.P. On Optimal Solution Error Covariances in Variational Data Assimilation Problems // J. Comp. Phys., 2010. V. 229. № 6. Р. 2159-2178.
- Gejadze I. Yu., Shutyaev V.P. On Gauss-Verifiability of Optimal Solutions in Variational Data Assimilation Problems with Nonlinear Dynamics // J. Comput. Phys. 2015. V. 280. Р. 439-456.
- Gejadze I.Yu., Shutyaev V.P., Le Dimet F.-X. Analysis Error Covariance Versus Posterior Covariance in Variational Data Assimilation // Q.J.R. Meteorol. Soc., 2013. V. 139. Р. 1826-1841.
- Smith P. J., Thornhill G.D., Dance S.L., Lawless A.S., Mason D.C., Nichols N.K. Data Assimilation for State and Parameter Estimation: Application to Morphodynamic Modelling // Q.J.R. Meteorol. Soc. 2013. V. 139. Р. 314-327.
- Shutyaev V., Le Dimet F.-X., Shubina E. Sensitivity with Respect to Observations in Variational Data Assimilation // Russ. J. Numer. Anal. Math. Modelling, 2017. V. 32. № 1. Р. 61-71.
- Shutyaev V. P., Le Dimet F.-X., Parmuzin E.I. Sensitivity Analysis with Respect to Observations in Variational Data Assimilation for Parameter Estimation // Nonlin. Processes Geophys. 2018. V. 25. Р. 429-439.
- Шутяев В.П. Операторы управления и итерационные алгоритмы в задачах вариационного усвоения данных. М.: Наука, 2001. 239 с.
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