2 Transport Problems and Disintegration Maps (original) (raw)

Transport problems and disintegration maps

ESAIM: Control, Optimisation and Calculus of Variations, 2013

By disintegration of transport plans it is introduced the notion of transport class. This allows to consider the Monge problem as a particular case of the Kantorovich transport problem, once a transport class is fixed. The transport problem constrained to a fixed transport class is equivalent to an abstract Monge problem over a Wasserstein space of probability measures. Concerning solvability of this kind of constrained problems, it turns out that in some sense the Monge problem corresponds to a lucky case. References 20

The Monge-Kantorovich problem for distributions and applications

We study the Kantorovich-Rubinstein transhipment problem when the difference between the source and the target is not anymore a balanced measure but belongs to a suitable subspace X(Ω) of first order distribution. A particular subclass X ♯ 0 (Ω) of such distributions will be considered which includes the infinite sums of dipoles k (δ p k − δ n k ) studied in . In spite of this weakened regularity, it is shown that an optimal transport density still exists among nonnegative finite measures. Some geometric properties of the Banach spaces X(Ω) and X ♯ 0 (Ω) can be then deduced.

On Regularity of Transport Density in the Monge--Kantorovich Problem

SIAM Journal on Control and Optimization, 2003

We show that the optimal regularity result for the transport density in the classical Monge-Kantorovich optimal mass transport problem, with the measures having summable densities, is a Sobolev differentiability along transport rays.

Long History of the Monge-Kantorovich Transportation Problem

The Mathematical Intelligencer, 2013

In 2012, the centenary of his birth was marked in St. Petersburg. Short histories were presented describing some of the main parts of his legacy, which continue in importance today: duality in linear programming, the so-called ''Monge-Kantorovich transportation problem,'' and the ''Kantorovich metric.'' Note that 2012 was also the 70th anniversary of the publication of his historic paper on the transport metric. The present article offers a somewhat expanded version of my talk on that occasion.

Constructing optimal maps for Monge's transport problem as a limit of strictly convex costs

2002

The Monge-Kantorovich problem is to move one distribution of mass onto another as efficiently as possible, where Monge's original criterion for efficiency was to minimize the average distance transported. Subsequently studied by many authors, it was not until 1976 that Sudakov showed solutions to be realized in the original sense of Monge, i.e., as mappings from R n to R n . A second proof of this existence result formed the subject of a recent monograph by Evans and Gangbo [7], who avoided Sudakov's measure decomposition results by using a partial differential equations approach. In the present manuscript, we give a third existence proof for optimal mappings, which has some advantages (and disadvantages) relative to existing approaches: it requires no continuity or separation of the mass distributions, yet our explicit construction yields more geometrical control than the abstract method of Sudakov. (Indeed, this control turns out to be essential for addressing a gap which has recently surfaced in Sudakov's approach to the problem in dimensions n ≥ 3; see the remarks at the end of this section.) It is also shorter and more flexible than either, and can be adapted to handle transportation on Riemannian manifolds or around obstacles, as we plan to show in a subsequent work . The problem considered here is the classical one:

A Brief on Optimal Transport

2020

The presentation covers prerequisite results from Topology and Measure Theory. This is then followed by an introduction into couplings and basic definitions for optimal transport. The Kantrorovich problem is then introduced and an existence theorem is presented. Following the setup of optimal transport is a brief overview of the Wasserstein distance and a short proof of how it metrizes the space of probability measures on a COMPACT domain. This presentation is a detailed examination of Villani's "Optimal Transport: Old and New" chapters 1-4 and part of 6.

Continuity of an optimal transport in Monge problem

Journal De Mathematiques Pures Et Appliquees, 2005

Given two absolutely continuous probability measures f± in R2, we consider the classical Monge transport problem, with the Euclidean distance as cost function. We prove the existence of a continuous optimal transport, under the assumptions that (the densities of) f± are continuous and strictly positive in the interior part of their supports, and that such supports are convex, compact, and

A nonlocal Monge-Kantorovich problem

This paper is concerned with a nonlocal version of the classical Monge-Kantorovich mass transport problem in which we replace the Euclidean distance with a discrete distance to measure the transport cost. We fix the length of a step and the distance that measures the cost of the transport depends of the number of steps that is needed to transport the involved mass from its origin to its destination. For this problem we deal with the issues of existence of Monge maps, Kantorovich potentials and optimal transport plans. We find that, even in one dimension, and for absolutely continuous measures respect to the Lebesgue measure, the existence of an optimal transport map depends on the involved masses. We also prove that when the length of the step tends to zero these nonlocal problems give an approximation to the classical Monge-Kantorovich mass transport problem. Finally, we study how the approach developed by Evans and Gangbo for the classical case works in this context. We find an equation for the potentials, obtained as a limit of nonlocal p−Laplacian problems, and we use it to construct optimal transport plans. We also obtain the transport density for the classical Monge-Kantorovich problem by rescaling. To the memory of Fuensanta Andreu, our friend and colleague.