Related papers: Optimal transport maps in Monge-Kantorovich proble…
We adapt the problem of continuous congested optimal transport to the Heisenberg group, equipped with a sub-Riemannian metric. Originally introduced in the Euclidean setting by Carlier, Jimenez, and Santambrogio as a path-dependent variant…
Numerical methods for the optimal transport problem is an active area of research. Recent work of Kitagawa and Abedin shows that the solution of a time-dependent equation converges exponentially fast as time goes to infinity to the solution…
We prove quantitative bounds on the stability of optimal transport maps and Kantorovich potentials from a fixed source measure $\rho$ under variations of the target measure $\mu$, when the cost function is the squared Riemannian distance on…
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…
We investigate a new multi-marginal optimal transport problem arising from a dissociation model in the Strong Interaction Limit of Density Functional Theory. In this short note, we introduce such dissociation model, the corresponding…
We review two models of optimal transport, where congestion effects during the transport can be possibly taken into account. The first model is Beckmann's one, where the transport activities are modeled by vector fields with given…
Optimal transport maps define a one-to-one correspondence between probability distributions, and as such have grown popular for machine learning applications. However, these maps are generally defined on empirical observations and cannot be…
In the semi-discrete version of Monge's problem one tries to find a transport map $T$ with minimum cost from an absolutely continuous measure $\mu$ on $\mathbb{R}^d$ to a discrete measure $\nu$ that is supported on a finite set in…
We study the dual formulation of the Monge-Kantorovich optimal transportation problem, in particular under what circumstances it is permitted in an infinite dimensional setting to use cylindrical functions, i.e. functions of the form…
This paper investigates the optimal transport problem within the framework of Linear Quadratic optimal control systems. We establish the well-posedness of the Monge problem and analyze the regularity of the resulting optimal transport map,…
Paper provides a method for solving the reverse Monge-Kantorovich transport problem (TP). It allows to accumulate positive decision-taking experience made by decision-taker in situations that can be presented in the form of TP. The initial…
Some classical mass transportation problems are investigated in a finitely additive setting. Let $\Omega=\prod_{i=1}^n\Omega_i$ and $\mathcal{A}=\otimes_{i=1}^n\mathcal{A}_i$, where $(\Omega_i,\mathcal{A}_i,\mu_i)$ is a ($\sigma$-additive)…
We consider the numerical solution of the optimal transport problem between densities that are supported on sets of unequal dimension. Recent work by McCann and Pass reformulates this problem into a non-local Monge-Amp\`ere type equation.…
In this work we present a numerical method for the Optimal Mass Transportation problem. Optimal Mass Transportation (OT) is an active research field in mathematics.It has recently led to significant theoretical results as well as…
This note establishes that a generalization of $c$-cyclical monotonicity from the Monge-Kantorovich problem with two marginals gives rise to a sufficient condition for optimality also in the multi-marginal version of that problem. To obtain…
We provide a new proof of the known partial regularity result for the optimal transportation map (Brenier map) between two sets. Contrary to the existing regularity theory for the Monge-Amp{\`e}re equation, which is based on the maximum…
This paper develops a comprehensive theory of optimal transport for signed (real) measures on Rd. Extending the classical Brenier theorem, we consider Jordan decompositions of measures with possibly fractal singular parts. Under suitable…
Recently a Dynamic-Monge-Kantorovich formulation of the PDE-based $L^1$-optimal transport problem was presented. The model considers a diffusion equation enforcing the balance of the transported masses with a time-varying conductivity that…
The Monge-Kantorovich mass transfer problem is equivalently formulated as a convex optimization problem for a potential function. In the light of this formulation an interative algorithm is developed for determining the solution. It is a…
The primary purpose of this note is to provide an instructional summary of the state of the art regarding cyclic monotonicity and related notions. We will also present how these notions are tied to optimality in the optimal transport (or…