Data assimilation for the time-dependent transport problem

Victor Shutyaev

Banach Center Publications (2000)

  • Volume: 52, Issue: 1, page 213-220
  • ISSN: 0137-6934

Abstract

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In this paper we consider the data assimilation problem for a timedependent transport problem in a slab when the initial condition is not known. The spaces of traces are introduced, the solvability of the original initial-boundary value transport problem is studied. The properties of the control operator are investigated, the solvability of the data assimilation problem is proved. The class of iterative methods for solving the problem is considered, and the convergence conditions are studied. The results are closely connected with some issues raised in [4], [14], [15].

How to cite

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Shutyaev, Victor. "Data assimilation for the time-dependent transport problem." Banach Center Publications 52.1 (2000): 213-220. <http://eudml.org/doc/209059>.

@article{Shutyaev2000,
abstract = {In this paper we consider the data assimilation problem for a timedependent transport problem in a slab when the initial condition is not known. The spaces of traces are introduced, the solvability of the original initial-boundary value transport problem is studied. The properties of the control operator are investigated, the solvability of the data assimilation problem is proved. The class of iterative methods for solving the problem is considered, and the convergence conditions are studied. The results are closely connected with some issues raised in [4], [14], [15].},
author = {Shutyaev, Victor},
journal = {Banach Center Publications},
keywords = {initial-boundary value transport problem; slab; spaces of traces; control operator; iterative methods; convergence conditions},
language = {eng},
number = {1},
pages = {213-220},
title = {Data assimilation for the time-dependent transport problem},
url = {http://eudml.org/doc/209059},
volume = {52},
year = {2000},
}

TY - JOUR
AU - Shutyaev, Victor
TI - Data assimilation for the time-dependent transport problem
JO - Banach Center Publications
PY - 2000
VL - 52
IS - 1
SP - 213
EP - 220
AB - In this paper we consider the data assimilation problem for a timedependent transport problem in a slab when the initial condition is not known. The spaces of traces are introduced, the solvability of the original initial-boundary value transport problem is studied. The properties of the control operator are investigated, the solvability of the data assimilation problem is proved. The class of iterative methods for solving the problem is considered, and the convergence conditions are studied. The results are closely connected with some issues raised in [4], [14], [15].
LA - eng
KW - initial-boundary value transport problem; slab; spaces of traces; control operator; iterative methods; convergence conditions
UR - http://eudml.org/doc/209059
ER -

References

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  1. [1] V. I. Agoshkov, Boundary Value Problems for Transport Equations, Birkhäuser, Boston, 1998. Zbl0914.35001
  2. [2] V. I. Agoshkov, Necessary and sufficient conditions for solvability of some first-order hyperbolic problems, preprint No.248, Dept. Numer. Math., USSR Academy of Science, 1990 (in Russian). 
  3. [3] V. I. Agoshkov, On existence of traces of functions in spaces used in transport theory, Soviet Doklady 288 (1986), 265-269. Zbl0636.46038
  4. [4] V. I. Agoshkov and G. I. Marchuk, On the solvability and numerical solution of data assimilation problems, Russ. J. Numer. Anal. Math. Modelling 8 (1993), 1-16. Zbl0818.65056
  5. [5] C. Bardos, Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d'approximation; application à l'équation de transport, Ann. Scient. Ec. Norm. Sup. 4 (1970), 185-233. Zbl0202.36903
  6. [6] M. Cessenat, Théorèmes de trace L p pour les espaces de fonctions de la neutronique, C. R. Acad. Sci. Paris 299 (1984), 831-834. Zbl0568.46030
  7. [7] T. A. Germogenova, Local properties of solutions of the transport equation, Sov. Doklady 187 (1969), 18-21. Zbl0208.13004
  8. [8] T. A. Germogenova, Local Properties of Solutions of Transport Equations, Nauka, Moscow, 1986. 
  9. [9] W. Greenberg, C. Van der Mee, and V. Protopopescu, Boundary Value Problems in Abstract Kinetic Theory, Birkhäuser, Basel, 1987. Zbl0624.35003
  10. [10] J. L. Lions, Sur le Contrôle Optimal de Systèmes Gouvernés par des Equations aux Dérivées Partielles, Dunod, Paris, 1968. 
  11. [11] G. I. Marchuk and V.I. Lebedev, Numerical Methods in the Theory of Neutron Transport, Harwood Academic Publisher, New York, 1986. Zbl0234.65102
  12. [12] S. Mischler, Equation de Vlasov avec régularité Sobolev du champ: théorèmes de trace et applications, preprint No.13, Université de Versailles, 1997. 
  13. [13] V. P. Shutyaev, Necessary and sufficient conditions of solvability of the initial-boundary value transport problem, in: Mathematical Models of Non-Linear Excitation, Transport, Dynamics, Control in Condensed Systems and other Media, Proc. of the Third International Conf. Tver 1998, V. Mironov (ed.), TGTU, Tver, 1998, 180. 
  14. [14] V. P. Shutyaev, On a class of insensitive control problems, Control and Cybernetics 23 (1994), 257-266. Zbl0809.93022
  15. [15] V. P. Shutyaev, Some properties of the control operator in the problem of data assimilation and iterative algorithms, Russ. J. Numer. Anal. Math. Modelling 10 (1995), 357-371. Zbl0840.65040
  16. [16] V. P. Shutyaev, Some regularity properties of the solution of the time-dependent transport-problem in a slab, preprint No.81, Dept. Numer. Math., USSR Academy of Science, 1985 (in Russian). 
  17. [17] S. Ukai, Solutions of the Boltzmann equations, in: Patterns and Waves-Qualitative Analysis of Nonlinear Differential Equations, Stud. Math. Appl. 18, North-Holland, Amsterdam (1986), 37-96. 
  18. [18] V. S. Vladimirov, Mathematical problems of one-velocity transport theory, Proc. of the Steklov Inst. Math. 61, 1961. 

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