Comptes Rendus
no. 9
Numerical Analysis
Optimal transport, shape optimization and global minimization
[Transport optimal généralisé, optimisation de formes et minimisation globale]

Présenté par : Olivier Pironneau

Bijan Mohammadi1
1 Institut de mathématiques et de modélisation de Montpellier, Université Montpellier II, CC51, 34095 Montpellier, France
Comptes Rendus. Mathématique, Volume 344 (2007) no. 9, pp. 591-596.

Nous présentons la résolution de problème de transport optimal généralisé comme solution d'une minimisation globale basée sur la solution de problèmes à valeurs aux limites. On s'intéresse au cas de second membre à signe variable dans l'équation de Monge–Ampère, avec comme application l'optimisation de formes de surfaces à courbure donnée. Les problèmes ont toujours des minima locaux mais l'optimum global est unique.

We present the numerical solution of general optimal transport problems through global minimization formulated as solution of boundary value problems. The paper is not on optimal transport but aims to show that the optimization problem behind needs global solutions. One gives also interest to the variable sign right-hand-side case with application to shape optimization for surfaces at given curvature. Both the positive and variable sign problems have local minima, but have unique global solution.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2007.03.015
Bijan Mohammadi 1

1 Institut de mathématiques et de modélisation de Montpellier, Université Montpellier II, CC51, 34095 Montpellier, France 
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Bijan Mohammadi. Optimal transport, shape optimization and global minimization. Comptes Rendus. Mathématique, Volume 344 (2007) no. 9, pp. 591-596. doi : 10.1016/j.crma.2007.03.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.03.015/

[1] H. Attouch; R. Cominetti A dynamical approach to convex minimization coupling approximation with the steepest descent method, J. Differential Equations (1996), pp. 128-132

[2] Y. Brenier; J.-D. Benamou A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem, Numer. Math., Volume 84 (2000) no. 3, pp. 375-393

[3] G. Carlier; F. Santambrogio A variational model for urban planning with traffic congestion, ESAIM Control Optim. Calc. Var., Volume 11 (2005), pp. 595-613

[4] E. Dean; R. Glowinski Numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions: an augmented Lagrangian approach, C. R. Acad. Sci. Paris, Ser. I, Volume 339 (2004), pp. 887-892

[5] M. Fortin; R. Glowinski Augmented Lagrangian Methods. Applications to the Numerical Solution of Boundary Value Problems, Studies in Mathematics and its Applications, vol. 15, North-Holland, 1983

[6] B. Ivorra; D. Hertzog; B. Mohammadi; J.F. Santiago Global optimization for the design of fast microfluidic protein folding devices, Int. J. Numer. Meth. Engrg. (2006), pp. 26-36

[7] B. Ivorra; B. Mohammadi; A. Ramos Semi-deterministic global optimization method and application to the control of Burgers equation, JOTA, Volume 135 (2007) no. 1

[8] L.V. Kantorovich On a problem of Monge, Uspekhi Mat. Nauk, Volume 3 (1948), pp. 225-226

[9] G. Loeper; F. Rapetti Numerical solution of the Monge–Ampère equation by a Newton's algorithm, C. R. Acad. Sci. Paris, Ser. I, Volume 340 (2005), pp. 319-324

[10] C. Vilani Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58, Amer. Math. Soc., 2003

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