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Optimal transport, shape optimization and global minimization. (English) Zbl 1115.65075

Summary: 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. The paper also interest in 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 a unique global solution.

MSC:

65K10 Numerical optimization and variational techniques
49J20 Existence theories for optimal control problems involving partial differential equations
49Q10 Optimization of shapes other than minimal surfaces
49M25 Discrete approximations in optimal control
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[1] Attouch, H.; Cominetti, R., A dynamical approach to convex minimization coupling approximation with the steepest descent method, J. Differential Equations, 128-132 (1996) · Zbl 0886.49024
[2] Brenier, Y.; Benamou, J.-D., A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84, 3, 375-393 (2000) · Zbl 0968.76069
[3] Carlier, G.; Santambrogio, F., A variational model for urban planning with traffic congestion, ESAIM Control Optim. Calc. Var., 11, 595-613 (2005) · Zbl 1085.49046
[4] Dean, E.; Glowinski, R., 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, 339, 887-892 (2004) · Zbl 1063.65121
[5] Fortin, M.; Glowinski, R., Augmented Lagrangian Methods. Applications to the Numerical Solution of Boundary Value Problems, Studies in Mathematics and its Applications, vol. 15 (1983), North-Holland · Zbl 0525.65045
[6] Ivorra, B.; Hertzog, D.; Mohammadi, B.; Santiago, J. F., Global optimization for the design of fast microfluidic protein folding devices, Int. J. Numer. Meth. Engrg., 26-36 (2006)
[7] Ivorra, B.; Mohammadi, B.; Ramos, A., Semi-deterministic global optimization method and application to the control of Burgers equation, JOTA, 135, 1 (2007) · Zbl 1146.90053
[8] Kantorovich, L. V., On a problem of Monge, Uspekhi Mat. Nauk, 3, 225-226 (1948)
[9] Loeper, G.; Rapetti, F., Numerical solution of the Monge-Ampère equation by a Newton’s algorithm, C. R. Acad. Sci. Paris, Ser. I, 340, 319-324 (2005) · Zbl 1067.65119
[10] Vilani, C., Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58 (2003), Amer. Math. Soc.
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