Related papers: Numerical solution of the Optimal Transportation p…
We address the convergence problem in learning the Optimal Transport (OT) map, where the OT Map refers to a map from one distribution to another while minimizing the transport cost. Semi-dual Neural OT, a widely used approach for learning…
This paper provides a theoretical and numerical approach to show existence, uniqueness, and the numerical determination of metalenses refracting radiation with energy patterns. The theoretical part uses ideas from optimal transport and for…
For the Monge-Amp\`ere equation with a right-hand side bounded away from 0 and infinity, we show that the solution, subject to the natural boundary condition arising in optimal transport, is in $W^{2,1+\varepsilon}$ up to the boundary.
Optimal transport (OT) is a versatile framework for comparing probability measures, with many applications to statistics, machine learning, and applied mathematics. However, OT distances suffer from computational and statistical scalability…
The elliptic Monge-Amp\`ere equation is a fully nonlinear partial differential equation which has been the focus of increasing attention from the scientific computing community. Fast three dimensional solvers are needed, for example in…
We consider the entropic regularization of discretized optimal transport and propose to solve its optimality conditions via a logarithmic Newton iteration. We show a quadratic convergence rate and validate numerically that the method…
Optimal Transport, a theory for optimal allocation of resources, is widely used in various fields such as astrophysics, machine learning, and imaging science. However, many applications impose elementwise constraints on the transport plan…
We propose and analyze a numerical method to solve an elliptic transmission problem in full space. The method consists of a variational formulation involving standard boundary integral operators on the coupling interface and an ultra-weak…
Monge map refers to the optimal transport map between two probability distributions and provides a principled approach to transform one distribution to another. Neural network based optimal transport map solver has gained great attention in…
The discrete optimal transport (OT) problem, which offers an effective computational tool for comparing two discrete probability distributions, has recently attracted much attention and played essential roles in many modern applications.…
In its most general form, the optimal transport problem is an infinite-dimensional optimization problem, yet certain notable instances admit closed-form solutions. We identify the common source of this tractability as \textit{symmetry} and…
We introduce a convergent finite difference method for solving the optimal transportation problem on the sphere. The method applies to both the traditional squared geodesic cost (arising in mesh generation) and a logarithmic cost (arising…
Computing exact Optimal Transport (OT) distances for large-scale datasets is computationally prohibitive. While entropy-regularized alternatives offer speed, they sacrifice precision and frequently suffer from numerical instability in…
Optimal transport (OT) theory can be informally described using the words of the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in hand has to move a large pile of sand lying on a construction site. The goal of the…
In this paper, we present a numerical method, based on iterative Bregman projections, to solve the optimal transport problem with Coulomb cost. This is related to the strong interaction limit of Density Functional Theory. The first idea is…
As an alternative to PINNs, a Deep Ritz framework is proposed to solve fully nonlinear PDEs. A least-squares algorithm is advocated to decouple the nonlinearities from the variational features of several fully nonlinear PDEs. A splitting…
Given a $d$-dimensional continuous (resp. discrete) probability distribution $\mu$ and a discrete distribution $\nu$, the semi-discrete (resp. discrete) Optimal Transport (OT) problem asks for computing a minimum-cost plan to transport mass…
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 develop an $\e$-regularity theory at the boundary for a general class of Monge-Amp\`ere type equations arising in optimal transportation. As a corollary we deduce that optimal transport maps between H\"older densities supported on $C^2$…
This article introduces a representation of dynamic meshes, adapted to some numerical simulations that require controlling the volume of objects with free boundaries, such as incompressible fluid simulation, some astrophysical simulations…