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Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling
Preface.- Primal and Dual Problems.- One-Dimensional Issues.- L^1 and L^infinity Theory.- Minimal Flows.- Wasserstein Spaces.- Numerical Methods.- Functionals over Probabilities.- Gradient Flows.-Expand
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A MACROSCOPIC CROWD MOTION MODEL OF GRADIENT FLOW TYPE
A simple model to handle the flow of people in emergency evacuation situations is considered: at every point x, the velocity U(x) that individuals at x would like to realize is given. Yet, theExpand
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Optimal Transportation with Traffic Congestion and Wardrop Equilibria
TLDR
In the classical Monge-Kantorovich problem, the transportation cost depends only on the amount of mass sent from sources to destinations and not on the paths followed by this mass. Expand
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Absolute continuity and summability of transport densities: simpler proofs and new estimates
The paper presents some short proofs for transport density absolute continuity and Lp estimates. Most of the previously existing results which were proven by geometric arguments are re-proved throughExpand
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Variational Mean Field Games
This paper is a brief presentation of those mean field games with congestion penalization which have a variational structure, starting from the deterministic dynamical framework. The stochasticExpand
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From Knothe's Transport to Brenier's Map and a Continuation Method for Optimal Transport
TLDR
A simple procedure to map two probability measures in Rd is the so-called Knothe-Rosenblatt rearrangement, which consists in rearranging monotonically the marginal distributions of the last coordinate, and then the conditional distributions, iteratively. Expand
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Handling congestion in crowd motion modeling
TLDR
We address here the issue of congestion in the modeling of crowd motion, in the non-smooth framework: contacts between people are not anticipated and avoided, they actually occur, and they are explicitly taken into account in the model. Expand
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Dealing with moment measures via entropy and optimal transport
A recent paper by Cordero-Erausquin and Klartag provides a characterization of the measures $\mu$ on $\R^d$ which can be expressed as the moment measures of suitable convex functions $u$, i.e. are ofExpand
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Asymptotical compliance optimization for connected networks
TLDR
We consider the problem of the optimal location of a Dirichlet region in a two-dimensional domain $\Omega$ subject to a force $f$ in order to minimize the compliance of the configuration. Expand
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