166 research outputs found
A Modica-Mortola approximation for branched transport
The M^\alpha energy which is usually minimized in branched transport problems
among singular 1-dimensional rectifiable vector measures with prescribed
divergence is approximated (and convergence is proved) by means of a sequence
of elliptic energies, defined on more regular vector fields. The procedure
recalls the Modica-Mortola one for approximating the perimeter, and the
double-well potential is replaced by a concave power
Models and applications of Optimal Transport in Economics, Traffic and Urban Planning
Some optimization or equilibrium problems involving somehow the concept of
optimal transport are presented in these notes, mainly devoted to applications
to economic and game theory settings. A variant model of transport, taking into
account traffic congestion effects is the first topic, and it shows various
links with Monge-Kantorovich theory and PDEs. Then, two models for urban
planning are introduced. The last section is devoted to two problems from
economics and their translation in the language of optimal transport
Introduction to Optimal Transport Theory
These notes constitute a sort of Crash Course in Optimal Transport Theory.
The different features of the problem of Monge-Kantorovitch are treated,
starting from convex duality issues. The main properties of space of
probability measures endowed with the distances induced by optimal
transport are detailed. The key tools to put in relation optimal transport and
PDEs are provided
Dealing with moment measures via entropy and optimal transport
A recent paper by Cordero-Erausquin and Klartag provides a characterization
of the measures on which can be expressed as the moment measures
of suitable convex functions , i.e. are of the form (\nabla u)\_\\#e^{- u}
for and finds the corresponding by a
variational method in the class of convex functions. Here we propose a purely
optimal-transport-based method to retrieve the same result. The variational
problem becomes the minimization of an entropy and a transport cost among
densities and the optimizer turns out to be . This
requires to develop some estimates and some semicontinuity results for the
corresponding functionals which are natural in optimal transport. The notion of
displacement convexity plays a crucial role in the characterization and
uniqueness of the minimizers
A note on some Poincar\'e inequalities on convex sets by Optimal Transport methods
We show that a class of Poincar\'e-Wirtinger inequalities on bounded convex
sets can be obtained by means of the dynamical formulation of Optimal
Transport. This is a consequence of a more general result valid for convex
sets, possibly unbounded.Comment: 13 page
Optimal transportation for a quadratic cost with convex constraints and applications
We prove existence of an optimal transport map in the Monge-Kantorovich
problem associated to a cost which is not finite everywhere, but
coincides with if the displacement belongs to a given convex
set and it is otherwise. The result is proven for satisfying
some technical assumptions allowing any convex body in and any convex
polyhedron in , . The tools are inspired by the recent
Champion-DePascale-Juutinen technique. Their idea, based on density points and
avoiding disintegrations and dual formulations, allowed to deal with
problems and, later on, with the Monge problem for arbitrary norms
A Modica-Mortola approximation for the Steiner Problem
In this note we present a way to approximate the Steiner problem by a family
of elliptic energies of Modica-Mortola type, with an additional term relying on
the weighted geodesic distance which takes care of the connexity constraint.Comment: short not
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