Monge's Transportation Problem Research Articles

Optimal stopping of stochastic transport minimizing submartingale costs

Given a stochastic state process ( X t ) t (X_t)_t and a real-valued submartingale cost process ( S t ) t (S_t)_t , we characterize optimal stopping times τ \tau that minimize the expectation of S τ S_\tau while realizing given initial and target distributions μ \mu and ν \nu , i.e., X 0 ∼ μ X_0\sim \mu and X τ ∼ ν X_\tau \sim \nu . A dual optimization problem is considered and shown to be attained under suitable conditions. The optimal solution of the dual problem then provides a contact set, which characterizes the location where optimal stopping can occur. The optimal stopping time is uniquely determined as the first hitting time of this contact set provided we assume a natural structural assumption on the pair ( X t , S t ) t (X_t, S_t)_t , which generalizes the twist condition on the cost in optimal transport theory. This paper extends the Brownian motion settings studied in Ghoussoub, Kim, and Palmer [Calc. Var. Partial Differential Equations 58 (2019), Paper No. 113, 31] and Ghoussoub, Kim, and Palmer [A solution to the Monge transport problem for Brownian martingales, 2019] and deals with more general costs.

A solution to the Monge transport problem for Brownian martingales

We provide a solution to the problem of optimal transport by Brownian martingales in general dimensions whenever the transport cost satisfies certain subharmonic properties in the target variable as well as a stochastic version of the standard “twist condition” frequently used in deterministic Monge transport theory. This setting includes, in particular, the case of the distance cost c(x,y)=|x−y|. We prove existence and uniqueness of the solution and characterize it as the first time Brownian motion hits a barrier that is determined by solutions to a corresponding dual problem.

On a Solution to the Monge Transport Problem on the Real Line Arising from the Strictly Concave Case

It is well known that the optimal transport problem on the real line for the classical distance cost may not have a unique solution. In this paper we recover uniqueness by considering the transport...

On Sudakov’s type decomposition of transference plans with norm costs

We consider the original strategy proposed by Sudakov for solving the Monge transportation problem with norm costj j D min Z jT(x) xjD d (x); T : R d ! R d ; = T# ; with , probability measures in R d and absolutely continuous w.r.t. L d . The key idea in this approach is to decompose (via disintegration of measures) the Kantorovich optimal transportation problem into a family of transportation problems in Za R d , where fZaga2A R d are disjoint regions such that the construction of an optimal map Ta : Za! R d is simpler than in the original problem, and then to obtain T by piecing together the maps Ta. When the normj j D is strictly convex (25), the sets Za are a family of 1-dimensional segments determined by the Kantorovich potential called optimal rays, while the existence of the map Ta is straightforward provided one can show that the disintegration ofL d (and thus of ) on such segments is absolutely continuous w.r.t. the 1-dimensional Hausdor measure (12). When the normjj D is not strictly convex, the main problems in this kind of approach are two: rst, to identify a suitable family of regionsfZaga2A on which the transport problem decomposes into simpler ones, and then to prove the existence of optimal maps. In this paper we show how these diculties can be overcome, and that the original idea of Sudakov can be successfully implemented. The results yield a complete characterization of the Kantorovich optimal transportation problem, whose straightforward corollary is the solution of the Monge problem in each set Za and then in R d . The strategy is suciently powerful to be applied to other optimal transportation problems. The analysis requires

Eulerian models and algorithms for unbalanced optimal transport

Benamou and Brenier formulation of Monge transportation problem [J.-D. Benamou and Y. Brenier, Numer. Math. 84 (2000) 375–393.] has proven to be of great interest in image processing to compute warpings and distances between pair of images [S. Agenent, S. Haker and A. Tannenbaum, SIAM J. Math. Anal. 35 (2003) 61–97]. One requirement for the algorithm to work is to interpolate densities of same mass. In most applications to image interpolation, this is a serious limitation. Existing approaches [J.-D. Benamou, ESAIM: M2AN 37 (2003) 851–868; B. Piccoli and F. Rossi, Arch. Rational Mech. Anal. 211 (2014) 335–358; B. Piccoli and F. Rossi, Preprint arXiv:1304.7014 (2014)]. to overcome this caveat are reviewed, and discussed. Due to the mix between transport and L2 interpolation, these models can produce instantaneous motion at finite range. In this paper we propose new methods, parameter-free, for interpolating unbalanced densities. One of our motivations is the application to interpolation of growing tumor images.

Optimal transport with convex obstacle

We consider the Monge transportation problem when the cost is the squared geodesic distance around a convex obstacle. We show that there exists at least one—and in general infinitely many—optimal transport maps.

Monge's transport problem in the Heisenberg group

We prove the existence of a solution to Monge's transport problem between two compactly supported Borel probability measures in the Heisenberg group equipped with its Carnot–Carathéodory distance assuming that the initial measure is absolutely continuous with respect to the Haar measure of the group.

Optimal transportation on non-compact manifolds

In this work, we show how to obtain for non-compact manifolds the results that have already been done for Monge Transport Problem for costs coming from Tonelli Lagrangians on compact manifolds. In particular, the already known results for a cost of the type d r , r > 1, where d is the Riemannian distance of a complete Riemannian manifold, hold without any curvature restriction.

Existence and uniqueness of optimal maps on Alexandrov spaces

The purpose of this paper is to show that in a finite dimensional metric space with Alexandrov's curvature bounded below, Monge's transport problem for the quadratic cost admits a unique solution.

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 absolutely continuous. We explain the relations with Mather's minimal measures.

On the equality between Monge's infimum and Kantorovich's minimum in optimal mass transportation

This paper concerns the Monge's transport problem in a general Polish space. We find optimal conditions to establish the equality between the infimum of Monge's problem and the minimum of the Kantorovich's relaxed version of the problem. A preliminary version of the results of this paper is contained in the Ph.D. thesis [A. Pratelli, Existence of optimal transport maps and regularity of the transport density in mass transportation problems, Ph.D. Thesis, Scuola Normale Superiore, Pisa, Italy, 2003. Available on http://cvgmt.sns.it/].

Continuity of an optimal transport in Monge problem

Given two absolutely continuous probability measures f ± in R 2 , 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 disjoint. We show through several examples that our statement is nearly optimal. Moreover, under the same hypotheses, we also obtain the continuity of the transport density.

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

Given two densities on R n \mathbf {R}^n with the same total mass, the Monge transport problem is to find a Borel map s : R n → R n s:\mathbf {R}^n \to \mathbf {R}^n rearranging the first distribution of mass onto the second, while minimizing the average distance transported. Here distance is measured by a norm with a uniformly smooth and convex unit ball. This paper gives a complete proof of the existence of optimal maps under the technical hypothesis that the distributions of mass be compactly supported. The maps are not generally unique. The approach developed here is new, and based on a geometrical change-of-variables technique offering considerably more flexibility than existing approaches.