Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-28T14:57:30.406Z Has data issue: false hasContentIssue false
  • English
  • Français

Solvability and numerical algorithms for a class ofvariational dataassimilation problems

Published online by Cambridge University Press:  15 August 2002

Guri Marchuk  and
Victor Shutyaev
Guri Marchuk
Affiliation:
Institute of Numerical Mathematics, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 GSP-1 Russia; shutyaev@inm.ras.ru.
Victor Shutyaev
Affiliation:
Institute of Numerical Mathematics, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 GSP-1 Russia; shutyaev@inm.ras.ru.
Get access

Abstract

A class of variationaldata assimilation problems on reconstructingthe initial-value functions is considered for the models governed byquasilinear evolution equations. The optimality system is reduced to theequation for the control function. The properties of the control equation are studied and thesolvability theorems are proved for linear and quasilinear data assimilationproblems. The iterative algorithms for solving the problem are formulated andjustified.

Keywords

Variational data assimilationquasilinear evolution problemoptimalitysystemcontrol equationsolvabilityiterative algorithms.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 8: A tribute to JL Lions , 2002 , pp. 873 - 883
Copyright
© EDP Sciences, SMAI, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agoshkov, V.I. and Marchuk, G.I., On solvability and numerical solution of data assimilation problems. Russ. J. Numer. Analys. Math. Modelling 8 (1993) 1-16. CrossRef
R. Bellman, Dynamic Programming. Princeton Univ. Press, New Jersey (1957).
Glowinski, R. and Lions, J.-L., Exact and approximate controllability for distributed parameter systems. Acta Numerica 1 (1994) 269-378. CrossRef
Krylov, I.A. and Chernousko, F.L., On a successive approximation method for solving optimal control problems. Zh. Vychisl. Mat. Mat. Fiz. 2 (1962) 1132-1139 (in Russian).
Kurzhanskii, A.B. and Khapalov, A.Yu., An observation theory for distributed-parameter systems. J. Math. Syst. Estimat. Control 1 (1991) 389-440.
O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type. Nauka, Moscow (1967) (in Russian).
Le Dimet, F.X. and Talagrand, O., Variational algorithms for analysis and assimilation of meteorological observations: Theoretical aspects. Tellus 38A (1986) 97-110. CrossRef
J.-L. Lions, Contrôle Optimal des Systèmes Gouvernés par des Équations aux Dérivées Partielles. Dunod, Paris (1968).
J.-L. Lions and E. Magenes, Problémes aux Limites non Homogènes et Applications. Dunod, Paris (1968).
Lions, J.-L., On controllability of distributed system. Proc. Natl. Acad. Sci. USA 94 (1997) 4828-4835. CrossRef
G.I. Marchuk, V.I. Agoshkov and V.P. Shutyaev, Adjoint Equations and Perturbation Algorithms in Nonlinear Problems. CRC Press Inc., New York (1996).
G.I. Marchuk and V.I. Lebedev, Numerical Methods in the Theory of Neutron Transport. Harwood Academic Publishers, New York (1986).
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.
Marchuk, G.I. and Shutyaev, V.P., Iteration methods for solving a data assimilation problem. Russ. J. Numer. Anal. Math. Modelling 9 (1994) 265-279. CrossRef
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.
Marchuk, G.I. and Zalesny, V.B., 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. CrossRef
Parmuzin, E.I. and Shutyaev, V.P., Numerical analysis of iterative methods for solving evolution data assimilation problems. Russ. J. Numer. Anal. Math. Modelling 14 (1999) 265-274. CrossRef
L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mischenko, The Mathematical Theory of Optimal Processes. John Wiley, New York (1962).
Sasaki, Y.K., Some basic formalisms in numerical variational analysis. Mon. Wea. Rev. 98 (1970) 857-883.
Shutyaev, V.P., On a class of insensitive control problems. Control and Cybernetics 23 (1994) 257-266.
Shutyaev, V.P., Some properties of the control operator in a data assimilation problem and algorithms for its solution. Differential Equations 31 (1995) 2035-2041.
Shutyaev, V.P., On data assimilation in a scale of Hilbert spaces. Differential Equations 34 (1998) 383-389.
Tikhonov, A.N., On the solution of ill-posed problems and the regularization method. Dokl. Akad. Nauk SSSR 151 (1963) 501-504.