87 research outputs found
Minimal convex extensions and finite difference discretization of the quadratic Monge-Kantorovich problem
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 between two absolutely continuous measures, the second of
which has convex support. The proposed numerical method actually captures a
specific Brenier solution which is minimal in some sense. We prove the
convergence of the method as the grid stepsize vanishes and we show with
numerical experiments that it is able to reproduce subtle properties of the
Optimal Transport problem
A viscosity framework for computing Pogorelov solutions of the Monge-Ampere equation
We consider the Monge-Kantorovich optimal transportation problem between two
measures, one of which is a weighted sum of Diracs. This problem is
traditionally solved using expensive geometric methods. It can also be
reformulated as an elliptic partial differential equation known as the
Monge-Ampere equation. However, existing numerical methods for this non-linear
PDE require the measures to have finite density. We introduce a new formulation
that couples the viscosity and Aleksandrov solution definitions and show that
it is equivalent to the original problem. Moreover, we describe a local
reformulation of the subgradient measure at the Diracs, which makes use of
one-sided directional derivatives. This leads to a consistent, monotone
discretisation of the equation. Computational results demonstrate the
correctness of this scheme when methods designed for conventional viscosity
solutions fail
Monotone and Consistent discretization of the Monge-Ampere operator
We introduce a novel discretization of the Monge-Ampere operator,
simultaneously consistent and degenerate elliptic, hence accurate and robust in
applications. These properties are achieved by exploiting the arithmetic
structure of the discrete domain, assumed to be a two dimensional cartesian
grid. The construction of our scheme is simple, but its analysis relies on
original tools seldom encountered in numerical analysis, such as the geometry
of two dimensional lattices, and an arithmetic structure called the
Stern-Brocot tree. Numerical experiments illustrate the method's efficiency
Generalized incompressible flows, multi-marginal transport and Sinkhorn algorithm
Starting from Brenier's relaxed formulation of the incompressible Euler
equation in terms of geodesics in the group of measure-preserving
diffeomorphisms, we propose a numerical method based on Sinkhorn's algorithm
for the entropic regularization of optimal transport. We also make a detailed
comparison of this entropic regularization with the so-called Bredinger
entropic interpolation problem. Numerical results in dimension one and two
illustrate the feasibility of the method
A Numerical Method to solve Optimal Transport Problems with Coulomb Cost
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 to introduce an entropic regularization of the Kantorovich formulation
of the Optimal Transport problem. The regularized problem then corresponds to
the projection of a vector on the intersection of the constraints with respect
to the Kullback-Leibler distance. Iterative Bregman projections on each
marginal constraint are explicit which enables us to approximate the optimal
transport plan. We validate the numerical method against analytical test cases
Résolution d'un cas test de contrôle optimal pour un système gouverné par l'équation des ondes à l'aide d'une méthode de décomposition de domaine
On utilise une méthode de décomposition de domaine pour résoudre un problème de contrôle optimal pour l'équation des ondes. La décomposition choisie (un sous-domaine par maille de discrétisation) permet d'utiliser facilement la technique de synthèse quadratique
Big Ray Tracing : Multivalued Travel Time Field Computation using Viscosity Solutions of the Eikonal Equation
Projet IDENTAn hybrid method based on a finite difference upwind scheme and ray tracing is presented. It allows an easy and fast computation of multivalued travel time fields for high frequency signals
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