Geodesic
Solving the Uniform Density Constraint in a Stochastic Downscaling Model
C. Chauvin1 ; S. A. Hirstoaga2 ; P. Kabelikova3 ; A. Rousseau4 ; F. Bernardin5 ; M. Bossy6
1 Grenoble High Magnetic Field Laboratory, 25 ave des Martyrs, 38042 Grenoble, Claire.Chauvin@grenoble.cnrs.fr.
2 CEREMADE, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris, France, sever@ceremade.dauphine.fr
3 Dpt of Applied Mathematics, Faculty of electrical engineering and informatics (FEI), VSB-Technical University of Ostrava, Tr. 17. listopadu 15, CZ 708 33 Ostrava-Poruba, Czech Republic, pavla.kabelikova@vsb.cz
4 INRIA, Team MOISE-LJK, 51 rue des mathématiques, 38041 Grenoble, France, Antoine.Rousseau@inria.fr
5 Centre d'Études Techniques de l'Équipement de Lyon, Laboratoire régional des ponts et chaussées, 8-10 rue Bernard Palissy Z.I. du Brézet, 63017 CLERMONT-FERRAND Cédex 2, Frederic.Bernardin@equipement.gouv.fr
6 INRIA, TOSCA Project, 2004 route des lucioles, 06902 Sophia Antipolis, France, Mireille.Bossy@inria.fr.
ESAIM. Proceedings, Tome 24 (2008), pp. 97-110.

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The present work aims to contribute to the development of a numerical method to compute small scale phenomena in atmospheric models, getting rid of any mesh refinement. In a domain, typically a mesh of a numerical weather prediction model, we simulate some particles that are moved thanks to a Stochastic Lagrangian model adapted from the PDF methods proposed by S.B. Pope. We estimate the Eulerian values of the required fields, thanks to the computation of a local mean value over an ensemble of particles. We are thus using a stochastic particle method. At small scale, our atmospheric model imposes that the mass density ρ is constant in the domain. As a consequence, the particles have to be uniformly distributed at every time step of the particle method. We aim to use D.P. Bertsekas Auction Algorithm in order to move a given cloud of particles to a new position, which is also given in advance, and that realizes the constraint . Naturally, the transport cost will have to be minimum. This is a problem of 3D optimal transport, which is known to be difficult.
DOI : 10.1051/proc:2008032

C. Chauvin 1 ; S. A. Hirstoaga 2 ; P. Kabelikova 3 ; A. Rousseau 4 ; F. Bernardin 5 ; M. Bossy 6

1 Grenoble High Magnetic Field Laboratory, 25 ave des Martyrs, 38042 Grenoble, Claire.Chauvin@grenoble.cnrs.fr.
2 CEREMADE, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris, France, sever@ceremade.dauphine.fr
3 Dpt of Applied Mathematics, Faculty of electrical engineering and informatics (FEI), VSB-Technical University of Ostrava, Tr. 17. listopadu 15, CZ 708 33 Ostrava-Poruba, Czech Republic, pavla.kabelikova@vsb.cz
4 INRIA, Team MOISE-LJK, 51 rue des mathématiques, 38041 Grenoble, France, Antoine.Rousseau@inria.fr
5 Centre d'Études Techniques de l'Équipement de Lyon, Laboratoire régional des ponts et chaussées, 8-10 rue Bernard Palissy Z.I. du Brézet, 63017 CLERMONT-FERRAND Cédex 2, Frederic.Bernardin@equipement.gouv.fr
6 INRIA, TOSCA Project, 2004 route des lucioles, 06902 Sophia Antipolis, France, Mireille.Bossy@inria.fr. 
@article{EP_2008_24_a7,
     author = {C. Chauvin and S. A. Hirstoaga and P. Kabelikova and A. Rousseau and F. Bernardin and M. Bossy},
     title = {Solving the {Uniform} {Density} {Constraint} in a {Stochastic} {Downscaling} {Model}},
     journal = {ESAIM. Proceedings},
     pages = {97--110},
     publisher = {mathdoc},
     volume = {24},
     year = {2008},
     doi = {10.1051/proc:2008032},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/proc:2008032/}
}
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C. Chauvin; S. A. Hirstoaga; P. Kabelikova; A. Rousseau; F. Bernardin; M. Bossy. Solving the Uniform Density Constraint in a Stochastic Downscaling Model. ESAIM. Proceedings, Tome 24 (2008), pp. 97-110. doi : 10.1051/proc:2008032. http://geodesic.mathdoc.fr/articles/10.1051/proc:2008032/

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