Optimal mass transportation and Mather theory

Optimal mass transportation and Mather theory

We study the Monge transportation problem when the cost is the action associated to a Lagrangian function on a compact manifold. We show that the transportation can be interpolated by a Lipschitz lamination. We describe several direct variational problems the minimizers of which are these Lipschitz laminations. We prove the existence of an optimal transport map when the transported measure is a...

Optimal Networks for Mass Transportation Problems

In the framework of transport theory, we are interested in the following optimization problem: given the distributions μ of working people and μ− of their working places in an urban area, build a transportation network (such as a railway or an underground system) which minimizes a functional depending on the geometry of the network through a particular cost function. The functional is defined a...

Neuronal Fiber–tracking via Optimal Mass Transportation

Diffusion Magnetic Resonance Imaging (MRI) is used to (noninvasively) study neuronal fibers in the brain white matter. Reconstructing fiber paths from such data (tractography problem) is relevant in particular to study the connectivity between two given cerebral regions. Fiber-tracking models rely on how water molecules diffusion is represented in each MRI voxel. The Diffusion Spectrum Imaging ...

Viscosity Solutions and Aubry-Mather theory

Three Optimization Problems in Mass Transportation Theory

We give a model for the description of an urban transportation network and we consider the related optimization problem which consists in finding the desing of the network which has the best transportation performances. This will be done by introducing, for every admissible network, a suitable metric space with a distance that inserted into the Monge-Kantorovich cost functional provides the cri...