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Fundamental Control Functions and Error Analysis

  • V. Shutyaev
Conference paper
Part of the NATO Science Series book series (NAIV, volume 26)

Abstract

Consider mathematical model of a physical process that is described by the evolution problem
$$ \left\{ {\begin{array}{*{20}c} {\frac{{\partial \phi }} {{\partial t}} = F(\phi ) + f,t \in (0,T)} \\ {\phi |t = 0 = u,} \\ \end{array} } \right. $$
(1.1)
where ϕ = ϕ(t) is the unknown function belonging for any (t) to the Hilbert space X, uX, f is a nonlinear operator mapping X into X. Let Y = L 2(0, T, X), ‖ · ‖Y = (·,·) 1 2/Y , fY. Let us introduce the functional
$$ S(u) = \frac{\alpha } {2}||u - u_0 ||_X^2 + \frac{1} {2}||C\phi - \phi _{obs} ||_{Y_{obs} }^2 , $$
(1.2)
where α = const ≥ 0, u 0X, Y obs are prescribed functions (observational data), Y obs is a Hilbert space (observational space), C : YY obs a linear continuous operator.

Keywords

Data Assimilation Singular Vector Adjoint Equation Linear Continuous Operator Observational Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Agoshkov, V.I. and Marchuk, G.I. (1993) On solvability and numerical solution of data assimilation problems, Russ. J. Numer. Anal. Math. Modelling 8, 1–16.CrossRefGoogle Scholar
  2. Cacuci, D.G. (1981) Sensitivity theory for nonlinear systems: II. Extensions to additional classes of responses, J. Math. Phys 22, 2803–2812.CrossRefGoogle Scholar
  3. Chavent, G. (1983) Local stability of the output least square parameter estimation technique, Math. Appl. Comp. 2, 3–22.Google Scholar
  4. Dontchev, A.L. (1983) Perturbations, Approximations and Sensitivity Analysis of Optimal Control Systems. (Lecture Notes in Control and Information Sciences; 52), Springer, Berlin.CrossRefGoogle Scholar
  5. Gejadze, I.Yu. and Shutyaev, V.P. (1999) An optimal control problem of initial data restoration, Comp. Math. Math. Phys. 39, 1416–1425.Google Scholar
  6. Kravaris, C. and Seinfeld, J.H. (1985) Identification of parameters in distributed parameter systems by regularization, SIAM J. Control and Optimization 23, 217.CrossRefGoogle Scholar
  7. Kurzhanskii, A.B. and Khapalov, A.Yu. (1991) An observation theory for distributed-parameter systems, J. Math. Syst. Estimat. Control 1, 389–440.Google Scholar
  8. Le Dimet, F.X., Navon, I.M., and Daescu, D.N. (2002) Second-order information in data assimilation, Monthly Weather Review 130, 629–648.CrossRefGoogle Scholar
  9. Le Dimet, F.-X., Ngnepieba P., and Shutyaev, V.P. On error analysis in data assimilation problems. Russ. J. Numer. Anal. Math. Modelling (2002), 17 (1), 71–97.Google Scholar
  10. Lions, J.-L. (1988) Contrôllabilité Exacte Perturbations et Stabilisation de Systèmes Distribués, Masson, Paris.Google Scholar
  11. Marchuk, G.I. (1995) Adjoint Equations and Analysis of Complex Systems, Kluwer, Dordrecht.Google Scholar
  12. Marchuk, G.I., Agoshkov, V.I., and Shutyaev, V.P. (1996) Adjoint Equations and Perturbation Algorithms in Nonlinear Problems, CRC Press Inc., New York.Google Scholar
  13. Morozov, V.A. (1987) Regular Methods for Solving the Ill-Posed Problems, Nauka, Moscow.Google Scholar
  14. Navon, I.M. (1995) Variational data assimilation, optimal parameter estimation and sensitivity analysis for environmental problems, in Atluri, Yagawa, and Cruse (eds.) Computational Mechanics’95. V.1, Springer, New York, pp.740–746.Google Scholar
  15. Ngodock, H.E. (1995) Assimilation de Données et Analyse de Sensibilité: une Application à la Circulation Océanique. Thèse de l’Université Joseph Fourier, UJF, Grenoble.Google Scholar
  16. Shutyaev, V.P. (1994) On a class of insensitive control problems, Control and Cybernetics 23, 257–266.Google Scholar
  17. Shutyaev, V.P. (2001) Control Operators and Iterative Algorithms for Variational Data Assimilation Problems, Nauka, Moscow.Google Scholar
  18. Tikhonov, A.N., Leonov, A.S., Yagola, A.G. (1995) Nonlinear Inverse Problems, Nauka, Moscow.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2003

Authors and Affiliations

  • V. Shutyaev
    • 1
  1. 1.Institute of Numerical MathematicsRussian Academy of SciencesMoscowRussia

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