Solvability and numerical algorithms for a class of variational data assimilation problems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002) , pp. 873-883.

A class of variational data assimilation problems on reconstructing the initial-value functions is considered for the models governed by quasilinear evolution equations. The optimality system is reduced to the equation for the control function. The properties of the control equation are studied and the solvability theorems are proved for linear and quasilinear data assimilation problems. The iterative algorithms for solving the problem are formulated and justified.

DOI : https://doi.org/10.1051/cocv:2002044
Classification : 65K10
Mots clés : variational data assimilation, quasilinear evolution problem, optimality system, control equation, solvability, iterative algorithms 
@article{COCV_2002__8__873_0,
     author = {Marchuk, Guri and Shutyaev, Victor},
     title = {Solvability and numerical algorithms for a class of variational data assimilation problems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {873--883},
     publisher = {EDP-Sciences},
     volume = {8},
     year = {2002},
     doi = {10.1051/cocv:2002044},
     zbl = {1070.65553},
     mrnumber = {1932977},
     language = {en},
     url = {www.numdam.org/item/COCV_2002__8__873_0/}
}
Marchuk, Guri; Shutyaev, Victor. Solvability and numerical algorithms for a class of variational data assimilation problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002) , pp. 873-883. doi : 10.1051/cocv:2002044. http://www.numdam.org/item/COCV_2002__8__873_0/

[1] V.I. Agoshkov and G.I. Marchuk, On solvability and numerical solution of data assimilation problems. Russ. J. Numer. Analys. Math. Modelling 8 (1993) 1-16. | MR 1211783 | Zbl 0818.65056

[2] R. Bellman, Dynamic Programming. Princeton Univ. Press, New Jersey (1957). | MR 90477 | Zbl 0077.13605

[3] R. Glowinski and J.-L. Lions, Exact and approximate controllability for distributed parameter systems. Acta Numerica 1 (1994) 269-378. | MR 1288099 | Zbl 0838.93013

[4] I.A. Krylov and F.L. Chernousko, On a successive approximation method for solving optimal control problems. Zh. Vychisl. Mat. Mat. Fiz. 2 (1962) 1132-1139 (in Russian). | MR 149044 | Zbl 0154.10402

[5] A.B. Kurzhanskii and A.Yu. Khapalov, An observation theory for distributed-parameter systems. J. Math. Syst. Estimat. Control 1 (1991) 389-440. | MR 1151967

[6] O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type. Nauka, Moscow (1967) (in Russian). | Zbl 0164.12302

[7] F.X. Le Dimet and O. Talagrand, Variational algorithms for analysis and assimilation of meteorological observations: Theoretical aspects. Tellus 38A (1986) 97-110.

[8] J.-L. Lions, Contrôle Optimal des Systèmes Gouvernés par des Équations aux Dérivées Partielles. Dunod, Paris (1968). | MR 244606 | Zbl 0179.41801

[9] J.-L. Lions and E. Magenes, Problémes aux Limites non Homogènes et Applications. Dunod, Paris (1968). | MR 247244 | Zbl 0165.10801

[10] J.-L. Lions, On controllability of distributed system. Proc. Natl. Acad. Sci. USA 94 (1997) 4828-4835. | MR 1453828 | Zbl 0876.93044

[11] G.I. Marchuk, V.I. Agoshkov and V.P. Shutyaev, Adjoint Equations and Perturbation Algorithms in Nonlinear Problems. CRC Press Inc., New York (1996). | MR 1413046

[12] G.I. Marchuk and V.I. Lebedev, Numerical Methods in the Theory of Neutron Transport. Harwood Academic Publishers, New York (1986). | MR 838539

[13] G.I. Marchuk and V.V. Penenko, Application of optimization methods to the problem of mathematical simulation of atmospheric processes and environment, in Modelling and Optimization of Complex Systems, Proc. of the IFIP-TC7 Work. Conf. Springer, New York (1978) 240-252. | MR 556210 | Zbl 0419.49006

[14] G.I. Marchuk and V.P. Shutyaev, Iteration methods for solving a data assimilation problem. Russ. J. Numer. Anal. Math. Modelling 9 (1994) 265-279. | MR 1285111 | Zbl 0818.65058

[15] G. Marchuk, V. Shutyaev and V. Zalesny, Approaches to the solution of data assimilation problems, in Optimal Control and Partial Differential Equations. IOS Press, Amsterdam (2001) 489-497. | Zbl 1061.65509

[16] G.I. Marchuk and V.B. Zalesny, A numerical technique for geophysical data assimilation problem using Pontryagin's principle and splitting-up method. Russ. J. Numer. Anal. Math. Modelling 8 (1993) 311-326. | Zbl 0818.65057

[17] E.I. Parmuzin and V.P. Shutyaev, Numerical analysis of iterative methods for solving evolution data assimilation problems. Russ. J. Numer. Anal. Math. Modelling 14 (1999) 265-274. | MR 1693744 | Zbl 0963.65054

[18] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mischenko, The Mathematical Theory of Optimal Processes. John Wiley, New York (1962). | MR 166037 | Zbl 0117.31702

[19] Y.K. Sasaki, Some basic formalisms in numerical variational analysis. Mon. Wea. Rev. 98 (1970) 857-883.

[20] V.P. Shutyaev, On a class of insensitive control problems. Control and Cybernetics 23 (1994) 257-266. | MR 1284518 | Zbl 0809.93022

[21] V.P. Shutyaev, Some properties of the control operator in a data assimilation problem and algorithms for its solution. Differential Equations 31 (1995) 2035-2041. | MR 1431633 | Zbl 1030.49500

[22] V.P. Shutyaev, On data assimilation in a scale of Hilbert spaces. Differential Equations 34 (1998) 383-389. | MR 1668218 | Zbl 0947.65064

[23] A.N. Tikhonov, On the solution of ill-posed problems and the regularization method. Dokl. Akad. Nauk SSSR 151 (1963) 501-504. | MR 162377 | Zbl 0141.11001