An Optimization Problem for Mass Transportation with Congested Dynamics

Starting from the work by Brenier [Extended Monge-Kantorovich theory, in Optimal Transportation and Applications (Martina Franca 2001), Lecture Notes in Math. 1813, Springer-Verlag, Berlin (2003), pp. 91-121], 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 Benamou and Brenier [A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), pp. 375-393].

[1]  Giuseppe Savaré,et al.  A new class of transport distances between measures , 2008, 0803.1235.

[2]  Yann Brenier,et al.  A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem , 2000, Numerische Mathematik.

[3]  Guillaume Carlier,et al.  Optimal Transportation with Traffic Congestion and Wardrop Equilibria , 2006, SIAM J. Control. Optim..

[4]  C. Villani.  Topics in Optimal Transportation , 2003 .

[5]  C. JIMENEZ.  DYNAMIC FORMULATION OF OPTIMAL TRANSPORT PROBLEMS , 2006 .

[6]  Guy Bouchitté,et al.  Integral representation of convex functionals on a space of measures , 1988 .

[7]  Olivier Pironneau,et al.  A FICTITIOUS DOMAIN BASED GENERAL PDE SOLVER , 2004 .

[8]  L. Evans,et al.  Differential equations methods for the Monge-Kantorovich mass transfer problem , 1999 .

[9]  Giuseppe Buttazzo,et al.  OPTIMAL NETWORKS FOR MASS TRANSPORTATION PROBLEMS , 2004 .

[10]  Eugene Stepanov,et al.  Optimal Urban Networks via Mass Transportation , 2008 .

[11]  L. Ambrosio,et al.  Gradient Flows: In Metric Spaces and in the Space of Probability Measures , 2005 .

[12]  Giuseppe M. Buttazzo,et al.  A Model for the Optimal Planning of an Urban Area , 2005, SIAM J. Math. Anal..

[13]  M. Fortin,et al.  Augmented Lagrangian methods : applications to the numerical solution of boundary-value problems , 1983 .

[14]  Bertrand Maury,et al.  Un Modèle de Mouvements de Foule , 2007 .

[15]  Pierre Seppecher,et al.  Energies with respect to a measure and applications to low dimensional structures , 1997 .

[16]  Giuseppe Buttazzo,et al.  Optimal Pricing Policies for Public Transportation Networks , 2006, SIAM J. Optim..

[17]  Guy Bouchitté,et al.  New lower semicontinuity results for nonconvex functionals defined on measures , 1990 .

[18]  Pierre Seppecher,et al.  Mathématiques/Mathematics Shape optimization solutions via Monge-Kantorovich equation , 1997 .

[19]  Guy Bouchitté,et al.  Integral representation of nonconvex functionals defined on measures , 1992 .

[20]  G. Buttazzo,et al.  Characterization of optimal shapes and masses through Monge-Kantorovich equation , 2001 .

[21]  Yann Brenier,et al.  Extended Monge-Kantorovich Theory , 2003 .