Related papers: Computing the L1 optimal transport density: a FEM …
We introduce the \emph{transport energy} functional $\mathcal E$ (a variant of the Bouchitt\'e-Buttazzo-Seppecher shape optimization functional) and we prove that its unique minimizer is the optimal transport density $\mu^*$, i.e., the…
We propose a new algorithm to approximate the Earth Mover's distance (EMD). Our main idea is motivated by the theory of optimal transport, in which EMD can be reformulated as a familiar $L_1$ type minimization. We use a regularization which…
We present a numerical method to solve the optimal transport problem with a quadratic cost when the source and target measures are periodic probability densities. This method is based on a numerical resolution of the corresponding…
This paper studies best finitely supported approximations of one-dimensional probability measures with respect to the $L^r$-Kantorovich (or transport) distance, where either the locations or the weights of the approximations' atoms are…
We introduce fast algorithms for generalized unnormalized optimal transport. To handle densities with different total mass, we consider a dynamic model, which mixes the $L^p$ optimal transport with $L^p$ distance. For $p=1$, we derive the…
We present an adaptation of the MA-LBR scheme to the Monge-Amp{\`e}re equation with second boundary value condition, provided the target is a convex set. This yields a fast adaptive method to numerically solve the Optimal Transport problem…
In recent works - both experimental and theoretical - it has been shown how to use computational geometry to efficently construct approximations to the optimal transport map between two given probability measures on Euclidean space, by…
Inspired by the matching of supply to demand in logistical problems, the optimal transport (or Monge--Kantorovich) problem involves the matching of probability distributions defined over a geometric domain such as a surface or manifold. In…
We propose an extension of the computational fluid mechanics approach to the Monge-Kantorovich mass transfer problem, which was developed by Benamou-Brenier. Our extension allows optimal transfer of unnormalized and unequal masses. We…
The aim of this article is to show that the Monge-Kantorovich problem is the limit of a sequence of entropy minimization problems when a fluctuation parameter tends down to zero. We prove the convergence of the entropic values to the…
We propose a biologically inspired dynamic model for the numerical solution of the $L^{1}$-PDE based optimal transportation model.
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…
In this article we study the numerical solution of the $L^1$-Optimal Transport Problem on 2D surfaces embedded in $R^3$, via the DMK formulation introduced in [FaccaCardinPutti:2018]. We extend from the Euclidean into the Riemannian setting…
We prove that if $\Omega\subset \mathbb{R}^{n+1}$ is a (not necessarily strictly) convex, $C^1$ domain, and $\mu$ and $\bar{\mu}$ are probability measures absolutely continuous with respect to surface measure on $\partial \Omega$, with…
We investigate the approximation of the Monge problem (minimizing \int\_$\Omega$ |T (x) -- x| d$\mu$(x) among the vector-valued maps T with prescribed image measure T \# $\mu$) by adding a vanishing Dirichlet energy, namely $\epsilon$…
The problem of optimal mass transport arises in numerous applications including image registration, mesh generation, reflector design, and astrophysics. One approach to solving this problem is via the Monge-Amp\`ere equation. While recent…
The optimal (Monge-Kantorovich) transportation problem is discussed from several points of view. The Lagrangian formulation extends the action of the {\em Lagrangian} $L(v,x,t)$ from the set of orbits in $\R^n$ to a set of measure-valued…
This paper introduces a numerical algorithm to compute the $L_2$ optimal transport map between two measures $\mu$ and $\nu$, where $\mu$ derives from a density $\rho$ defined as a piecewise linear function (supported by a tetrahedral mesh),…
In this paper, we show a new regularity result on the transport density {\sigma} in the classical Monge-Kantorovich optimal mass transport problem between two measures, {\mu} and {\nu}, having some summable densities, f^+ and f^-. More…
In this paper, we formulate the problem of inferring a Finite Mixture Model from discrete data as an optimal transport problem with entropic regularization of parameter $\lambda\geq 0$. Our method unifies hard and soft clustering, the…