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An optimization problem for mass transportation with congested dynamics. (English) Zbl 1282.49035

Summary: Starting from the work by Y. Brenier [”Extended Monge-Kantorovich theory”, Optimal Transportation and Applications (Martina Franca 2001), Lecture Notes in Math. 1813, Springer-Verlag, Berlin, pp. 91-121 (2003; Zbl 1064.49036)], where a dynamic formulation of mass transportation problems was given, we consider a more general framework, where different kinds of cost functions are allowed. This seems relevant in some problems presenting congestion effects as, for instance, traffic on a highway, crowds moving in domains with obstacles, and, in general, in all cases where the transportation does not behave as in the classical Monge setting. We show some numerical computations obtained by generalizing to our framework the approximation scheme introduced in J.-D.Benamou and Y. Brenier [”A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem”, Numer. Math., 84, pp. 375-393 (2000; Zbl 0968.76069)].

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
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