Related papers: Fast iterative solvers for an optimal transport pr…
This chapter describes techniques for the numerical resolution of optimal transport problems. We will consider several discretizations of these problems, and we will put a strong focus on the mathematical analysis of the algorithms to solve…
In this paper, we propose a refinement strategy to the well-known Physics-Informed Neural Networks (PINNs) for solving partial differential equations (PDEs) based on the concept of Optimal Transport (OT). Conventional black-box PINNs…
Motivated by the computational difficulties incurred by popular deep learning algorithms for the generative modeling of temporal densities, we propose a cheap alternative which requires minimal hyperparameter tuning and scales favorably to…
We address the numerical solution via Galerkin type methods of the Monge-Amp\`ere equation with transport boundary conditions arising in optimal mass transport, geometric optics and computational mesh or grid movement techniques. This fully…
We propose a volumetric formulation for computing the Optimal Transport problem defined on surfaces in $\mathbb{R}^3$, found in disciplines like optics, computer graphics, and computational methodologies. Instead of directly tackling the…
In this paper we study the conditioning of optimal control problems constrained by linear parabolic equations with Neumann boundary conditions. While we concentrate on a given end-time target function the results hold also when the target…
In this paper, we introduce a neural network-based method to address the high-dimensional dynamic unbalanced optimal transport (UOT) problem. Dynamic UOT focuses on the optimal transportation between two densities with unequal total mass,…
The optimal transport (OT) problem is a classical optimization problem having the form of linear programming. Machine learning applications put forward new computational challenges in its solution. In particular, the OT problem defines a…
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…
In this work, we propose a generalized alternating nonlinear generalized minimal residual method (GA-NGMRES) to accelerate first-order optimization schemes for PDE-constrained optimization problems governed by transport equations. We apply…
We present a new approach to discretizing shape optimization problems that generalizes standard moving mesh methods to higher-order mesh deformations and that is naturally compatible with higher-order finite element discretizations of…
We prove existence and uniqueness results for solutions to a class of optimal transportation problems with infinitely many marginals, supported on the real line. We also provide a characterization of the solution with an explicit formula.…
We study the discretization, convergence, and numerical implementation of recent reformulations of the quadratic porous medium equation (multidimensional and anisotropic) and Burgers' equation (one-dimensional, with optional viscosity), as…
Nonlinear systems of partial differential equations (PDEs) may permit several distinct solutions. The typical current approach to finding distinct solutions is to start Newton's method with many different initial guesses, hoping to find…
Current state-of-the-art methods for solving discrete optimization problems are usually restricted to convex settings. In this paper, we propose a general approach based on cutting planes for solving nonlinear, possibly nonconvex, binary…
We propose an efficient numerical algorithm for the solution of diffeomorphic image registration problems. We use a variational formulation constrained by a partial differential equation (PDE), where the constraints are a scalar transport…
This paper connects discrete optimal transport to a certain class of multi-objective optimization problems. In both settings, the decision variables can be organized into a matrix. In the multi-objective problem, the notion of Pareto…
A numerical method for coupled 3D-1D problems with discontinuous solutions at the interfaces is derived and discussed. This extends a previous work on the subject where only continuous solutions were considered. Thanks to properly defined…
The fully implicit method is the most commonly used approach to solve black-oil problems in reservoir simulation. The method requires repeated linearization of large nonlinear systems and produces ill-condi\-tioned linear systems. We…
We study the fundamental computational problem of approximating optimal transport (OT) equations using neural differential equations (Neural ODEs). More specifically, we develop a novel framework for approximating unbalanced optimal…