Continuity and estimates for multimarginal optimal transportation problems with singular costs

一个多边际质量运输问题的显式解

An explicit solution for a multimarginal mass transportation problem

本文针对单位立方体上成本函数为$xyz$且具有一维均匀投影的多边际运输问题构造了显式解，发现原始问题存在多解且集中在非常数局部维度集合上，而对偶问题解唯一。

On Multimarginal Partial Optimal Transport: Equivalent Forms and Computational Complexity

We study the multi-marginal partial optimal transport (POT) problem between $m$ discrete (unbalanced) measures with at most $n$ supports. We first prove that we can obtain two equivalence forms of the multimarginal POT problem in terms of the multimarginal optimal transport problem via novel extensions of cost tensor. The first equivalence form is derived under the assumptions that the total masses of each measure are sufficiently close while the second equivalence form does not require any conditions on these masses but at the price of more sophisticated extended cost tensor. Our proof techni…

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.

Mass Transportation on Sub-Riemannian Manifolds

We study the optimal transport problem in sub-Riemannian manifolds where the cost function is given by the square of the sub-Riemannian distance. Under appropriate assumptions, we generalize Brenier-McCann's Theorem proving existence and uniqueness of the optimal transport map. We show the absolute continuity property of Wassertein geodesics, and we address the regularity issue of the optimal map. In particular, we are able to show its approximate differentiability a.e. in the Heisenberg group (and under some weak assumptions on the measures the differentiability a.e.), which allows to write a…

Hölder continuity for optimal multivalued mappings

Gangbo and McCann showed that optimal transportation between hypersurfaces generally leads to multivalued optimal maps - bivalent when the target surface is strictly convex. In this paper we quantify Hölder continuity of the bivalent map optimizing average distance squared between arbitrary measures supported on Euclidean spheres.

Evolution models for mass transportation problems

We present a survey on several mass transportation problems, in which a given mass dynamically moves from an initial configuration to a final one. The approach we consider is the one introduced by Benamou and Brenier in [5], where a suitable cost functional $F(ρ,v)$, depending on the density $ρ$ and on the velocity $v$ (which fulfill the continuity equation), has to be minimized. Acting on the functional $F$ various forms of mass transportation problems can be modeled, as for instance those presenting congestion effects, occurring in traffic simulations and in crowd motions, or concentration e…