Optimal Transport with Coulomb cost. Approximation and duality

We revisit the duality theorem for multimarginal optimal transportation problems. In particular, we focus on the Coulomb cost. We use a discrete approximation to prove equality of the extremal values and some careful estimates of the approximating sequence to prove existence of maximizers for the dual problem (Kantorovich's potentials). Finally we observe that the same strategy can be applied to a more general class of costs and that a classical results on the topic cannot be applied here

Electrophysiological pattern of dream experience

Dreaming is a common human experience investigated from multiple perspectives over the centuries. Recently, this phenomenon has stimulated scientific interest, becoming a peculiar context to study memory processes and consciousne

A study of the dual problem of the one-dimensional L-infinity optimal transport problem with applications

The Monge-Kantorovich problem for the infinite Wasserstein distance presents several peculiarities. Among them the lack of convexity and then of a direct duality. We study in dimension 1 the dual problem introduced by Barron, Bocea and Jensen. We construct a couple of Kantorovich potentials which is "as less trivial as possible". More precisely, we build a potential which is non constant around any point that the plan which is locally optimal moves at maximal distance. As an application, we show that the set of points which are displaced to maximal distance by a locally optimal transport plan is minimal

Duality theory and optimal transport for sand piles growing in a silos

We prove existence and uniqueness of solutions for a system of PDEs which describes the growth of a sandpile in a silos with flat bottom under the action of a vertical, measure source. The tools we use are a discrete approximation of the source and the duality theory for optimal transport (or Monge-Kantorovich) problems

SkiMap: An Efficient Mapping Framework for Robot Navigation

We present a novel mapping framework for robot navigation which features a multi-level querying system capable to obtain rapidly representations as diverse as a 3D voxel grid, a 2.5D height map and a 2D occupancy grid. These are inherently embedded into a memory and time efficient core data structure organized as a Tree of SkipLists. Compared to the well-known Octree representation, our approach exhibits a better time efficiency, thanks to its simple and highly parallelizable computational structure, and a similar memory footprint when mapping large workspaces. Peculiarly within the realm of mapping for robot navigation, our framework supports realtime erosion and re-integration of measurements upon reception of optimized poses from the sensor tracker, so as to improve continuously the accuracy of the map.Comment: Accepted by International Conference on Robotics and Automation (ICRA) 2017. This is the submitted version. The final published version may be slightly differen