Related papers: Optimal transport of measures via autonomous vecto…
We study the controllability of a Partial Differential Equation of transport type, that arises in crowd models. We are interested in controlling it with a control being a vector field, representing a perturbation of the velocity, localized…
In this article we show how ideas, methods and results from optimal transportation can be used to study various aspects of the stationary measuresof Iterated Function Systems equipped with a probability distribution. We recover a classical…
We study optimal transport between probability measures supported on the same finite metric space, where the ground cost is a distance induced by a weighted connected graph. Building on recent work showing that the resulting Kantorovich…
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.…
Flow-based methods for sampling and generative modeling use continuous-time dynamical systems to represent a {transport map} that pushes forward a source measure to a target measure. The introduction of a time axis provides considerable…
Optimal transportation distances are valuable for comparing and analyzing probability distributions, but larger-scale computational techniques for the theoretically favorable quadratic case are limited to smooth domains or regularized…
An adaptive, adversarial methodology is developed for the optimal transport problem between two distributions $\mu$ and $\nu$, known only through a finite set of independent samples $(x_i)_{i=1..N}$ and $(y_j)_{j=1..M}$. The methodology…
We study the problem of maximizing a spectral risk measure of a given output function which depends on several underlying variables, whose individual distributions are known but whose joint distribution is not. We establish and exploit an…
We study the problem of minimizing the Wasserstein distance between a probability distribution and an algebraic variety. We consider the setting of finite state spaces and describe the solution depending on the choice of the ground metric…
We study the computational complexity of the optimal transport problem that evaluates the Wasserstein distance between the distributions of two K-dimensional discrete random vectors. The best known algorithms for this problem run in…
We consider the problem of transporting \nota{one probability measure into another through} the flow of a given driftless control-affine system. Under suitable regularity conditions, the controllability of the system by means of open-loop…
In the current book I suggest an off-road path to the subject of optimal transport. I tried to avoid prior knowledge of analysis, PDE theory and functional analysis, as much as possible. Thus I concentrate on discrete and semi-discrete…
The Monge-Kantorovich transportation problem involves optimizing with respect to a given a cost function. Uniqueness is a fundamental open question about which little is known when the cost function is smooth and the landscapes containing…
We develop a theory of optimal transport relative to a distinguished subset, which acts as a reservoir of mass, allowing us to compare measures of different total variation. This relative transportation problem has an optimal solution and…
For a given $1$-Lipschitz map $u\colon\mathbb{R}^n\to\mathbb{R}^m$ we define a partition, up to a set of Lebesgue measure zero, of $\mathbb{R}^n$ into maximal closed convex sets such that restriction of $u$ is an isometry on these sets. We…
An optimal transport path may be viewed as a geodesic in the space of probability measures under a suitable family of metrics. This geodesic may exhibit a tree-shaped branching structure in many applications such as trees, blood vessels,…
In this study, two initial boundary value problems for one dimensional advection-dispersion equation are solved by differential quadrature method based on sine cardinal functions. Pure advection problem modeling transport of conservative…
We investigate the estimation of an optimal transport map between probability measures on an infinite-dimensional space and reveal its minimax optimal rate. Optimal transport theory defines distances within a space of probability measures,…
We introduce a formulation of optimal transport problem for distributions on function spaces, where the stochastic map between functional domains can be partially represented in terms of an (infinite-dimensional) Hilbert-Schmidt operator…
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…