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Optimal control approach ininverse radiative transfer problems: the problem on boundary function

Published online by Cambridge University Press:  15 August 2002

Valeri I. Agoshkov  and
Claude Bardos
Valeri I. Agoshkov
Affiliation:
Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia, and CMLA, ENS Cachan, France; agoshkov@inm.ras.ru.
Claude Bardos
Affiliation:
CMLA, ENS Cachan, France.
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Abstract

The paper presents some results related to the optimal control approachs applying to inverse radiative transfer problems, to the theory of reflection operators, to the solvability of the inverse problems on boundary function and to algorithms for solution of these problems.

Keywords

Optimal controlinverse probleminverse radiative transfer problemreflection operatorcontrol equation operatorregularization parameteriterative algorithm.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 5 , 2000 , pp. 259 - 278
Copyright
© EDP Sciences, SMAI, 2000

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References

V.A. Ambartsumyan, Scattering and absorption of light in planetary atmospheres. Uchen. Zap. TsAGI 82 (1941), in Russian.
S. Chandrasekhar, Radiative Transfer. New York (1960).
J.-L. Lions, Contrôle optimal des systèmes gouvernés par des équations aux dérivées partielles. Dunod, Paris (1968).
V.I. Lebedev and V.I. Agoshkov, The Poincaré-Steklov Operators and their Applications in Analysis. Dept. of Numerical Math. of the USSR Academy of Sciences, Moscow (1983), in Russian.
V.I. Agoshkov, Generalized solutions of transport equations and their smoothness properties. Nauka, Moscow (1988), in Russian.
Agoshkov, V.I., Reflection operators and domain decomposition methods in transport theory problems. Sov. J. Numer. Anal. Math. Modelling 2 (1987) 325-347. CrossRef
Agoshkov, V.I., On the existence of traces of functions in spaces used in transport theory problems. Dokl. Akad. Nauk SSSR 288 (1986) 265-269, in Russian.
V.S. Vladimirov, Mathematical problems of monenergetic particle transport theory. Trudy Mat. Inst. Steklov 61 (1961), in Russian.
G.I. Marchuk, Design of Nuclear Reactors. Atomizdat, Moscow (1961), in Russian.
V.V. Sobolev, Light Scattering in Planetary Atmospheres. Pergamon Press, Oxford, U.K. (1973).
Marchuk, G.I. and Agoshkov, V.I., Reflection Operators and Contemporary Applications to Radiative Transfer. Appl. Math. Comput. 80 (1995) 1-19.
V.I. Agoshkov, Domain decomposition methods in problems of hydrodynamics. I. Problem plain circulation in ocean. Moscow: Department of Numerical Mathematics, Preprint No. 96 (1985) 12, in Russian.
V.I. Agoshkov, Domain decomposition methods and perturbation methods for solving some time dependent problems of fluid dynamics, in Proc. of First International Interdisciplinary Conference. Olympia -91 (1991).
V.I. Agoshkov, Control theory approaches in: data assimilation processes, inverse problems, and hydrodynamics. Computer Mathematics and its Applications, HMS/CMA 1 (1994) 21-39.
Ill-posed problems in natural Sciences, edited by A.N. Tikhonov. Moscow, Russia - VSP, Netherlands (1992).
A.L. Ivankov, Inverse problems for the nonstationary kinetic transport equation. In [15].
A.I. Prilepko, D.G. Orlovskii and I.A. Vasin, Inverse problems in mathematical physics. In [15].
Yu.E. Anikonov, New methods and results in multidimensional inverse problems for kinetic equations. In [15].
E.C. Titchmarsh, Introduction to the Theory of Fourier Integral. New York (1937).
C. Bardos, Mathematical approach for the inverse problem in radiative media (1986), not published.
Case, K.M., Inverse problem in transport theory. Phys. Fluids 16 (1973) 16-7-1611. CrossRef
L.P. Niznik and V.G. Tarasov, Reverse scattering problem for a transport equation with respect to directions. Preprint, Institute of Mathematics, Academy Sciences of the Ukrainian SSR (1980).
K.K. Hunt and N.J. McCormick, Numerical test of an inverse method for estimating single-scattering parameters from pulsed multiple-scattering experiments. J. Opt. Soc. Amer. A. 2 (1985).
McCormick, N.J., Recent Development in inverse scattering transport method. Trans. Theory Statist. Phys. 13 (1984) 15-28. CrossRef
Bardos, C., Santos, R. and Sentis, R., Diffusion approximation and the computation of critical size. Trans. Amer. Math. Soc. 284 (1986) 617-649. CrossRef
Bardos, C., Caflish, R. and Nicolaenko, B., Different aspect of the Milne problem. Trans. Theory Statist. Phys. 16 (1987) 561-585. CrossRef
V.P. Shutyaev, Integral reflection operators and solvability of inverse transport problem, in Integral equations in applied modelling. Kiev: Inst. of Electrodynamics, Academy of Sciences of Ukraine, Vol. 2 (1986) 243-244, in Russian.
V.I. Agoshkov and C. Bardos, Inverse radiative problems: The problem on boundary function. CMLA, ENS de Cachan, Preprint No. 9801 (1998).
V.I. Agoshkov and C. Bardos, Inverse radiative problems: The problem on the right-hand-side function. CMLA, ENS de Cachan, Preprint No. 9802 (1998).
V.I. Agoshkov and C. Bardos, Optimal control approach in 3D-inverse radiative problem on boundary function (to appear).
V.I. Agoshkov, C. Bardos, E.I. Parmuzin and V.P. Shutyaev, Numerical analysis of iterative algorithms for an inverse boundary transport problem (to appear).
S.I. Kabanikhin and A.L. Karchevsky, Optimization methods of solving inverse problems of geoelectrics. In [15].
Coron, F., Golse, F. and Sulem, C., Classification, A of Well-Posed Kinetic Layer Problems. Comm. Pure Appl. Math. 41 (1988) 409-435. CrossRef
R. Dautray and J.L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, CEA. Masson, Tome 9.
R. Glowinski and J.-L. Lions, Exact and approximate controllability for distributed parameter systems. Acta Numer. (1994) 269-378.