[CvGmt News] Seminari Calcolo delle Variazioni
depascal at dm.unipi.it
depascal at dm.unipi.it
Sun Oct 11 20:22:09 CEST 2009
Prof. Filippo Santambrogio (Univ. de Paris Dauphine)
Wednesday 14 october 2009
At 17pm Aula Seminari (Dipartimento di Matematica)
Crowd movements: gradient flow in Wasserstein spaces under density constraints.
A crowd is located in a room and everybody wants to exit, thus moving towards, say, a single small exit. Yet, the density $\rho$ of the crowd cannot exceed a fixed bound ($\rho\leq 1$). The actual velocity that every agent may realize is hence obtained in the following way: take the speed they would like (the unit vector pointing to the door), and project the whole velocity field in $L^2(\rho)$ on the cone of feasible velocities, i.e. those who have a positive divergence on the set where the density already saturates the constraint.
This evolution is actually a gradient flow in the space $W_2$ of probability densities for the functional which associates to any $\rho$ satisfying the density constraint the mean value $\int D d\rho$, where $D$ is the distance function to the exit, and $+\infty$ to any probability violating the same constraint.
This gradient flow approach allows to give existence results for a problem that had previously been studied in a microscopical way (with people represented by small disks, and the density constraint replaced by a non-overlap condition).
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You find this news in the cvgmt preprint server: http://cvgmt.sns.it/news/20091014/
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