no. 11-12
Numerical Analysis
Numerical method for a dynamic optimization problem arising in the modeling of a population of aerosol particles
[Une méthode numérique pour un problème d'optimisation dynamique lié à une population de particules]

Présenté par : Roland Glowinski

Caboussat, Alexandre1 ; Leonard, Allison1
1 Department of Mathematics, University of Houston, 4800 Calhoun Rd, Houston, TX 77204-3008, USA
Comptes Rendus. Mathématique, Tome 346 (2008) no. 11-12, pp. 677-680.

Nous proposons un modèle pour le calcul de l'évolution d'une population de particules. Ce modèle couple un système d'équations différentielles et une séquence de problèmes d'optimisation sous contraintes.

Les conditions de premier ordre de chacun des problèmes d'optimisation et une discrétisation implicite des équations différentielles forment un système d'équations non linéaires qui est traité avec une méthode de point intérieur, couplé à une itération de Newton. Le système linéaire correspondant a une structure par blocs. Une méthode de résolution directe basée sur le complément de Schur prend en compte la structure creuse de la matrice et permet de découpler les différentes particules du système. Des résultats numériques pour une population de particules organiques montrent l'évolution de particules de différentes tailles.

A model coupling differential equations and a sequence of constrained optimization problems is proposed for the simulation of the evolution of a population of particles at equilibrium interacting through a common medium.

The first order optimality conditions of the optimization problems relaxed with barrier functions are coupled with the differential equations into a system of differential-algebraic equations that is discretized in time with an implicit first order scheme. The resulting system of nonlinear algebraic equations is solved at each time step with an interior-point/Newton method. The Newton system is block-structured and solved with Schur complement techniques, in order to take advantage of its sparsity. Application to the dynamics of a population of organic atmospheric aerosol particles is given to illustrate the evolution of particles of different sizes.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.04.016
Caboussat, Alexandre 1 ; Leonard, Allison 1

1 Department of Mathematics, University of Houston, 4800 Calhoun Rd, Houston, TX 77204-3008, USA 
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Caboussat, Alexandre; Leonard, Allison. Numerical method for a dynamic optimization problem arising in the modeling of a population of aerosol particles. Comptes Rendus. Mathématique, Tome 346 (2008) no. 11-12, pp. 677-680. doi : 10.1016/j.crma.2008.04.016. http://www.numdam.org/articles/10.1016/j.crma.2008.04.016/

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