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Minimal convex extensions and finite difference discretisation of the quadratic Monge–Kantorovich problem

Part of: Numerical methods in calculus of variations and optimal control Numerical analysis in abstract spaces Partial differential equations, boundary value problems

Published online by Cambridge University Press:  21 September 2018

JEAN-DAVID BENAMOU  and
VINCENT DUVAL
JEAN-DAVID BENAMOU
Affiliation:
INRIA Paris (MOKAPLAN) and CEREMADE, CNRS, Université Paris-Dauphine, PSL Research University, 75016 Paris, France emails: jean-david.benamou@inria.fr; vincent.duval@inria.fr
VINCENT DUVAL
Affiliation:
INRIA Paris (MOKAPLAN) and CEREMADE, CNRS, Université Paris-Dauphine, PSL Research University, 75016 Paris, France emails: jean-david.benamou@inria.fr; vincent.duval@inria.fr
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Abstract

We present an adaptation of the Monge–Ampère (MA) lattice basis reduction scheme to the MA 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 (OT) 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 step size vanishes and show with numerical experiments that it is able to reproduce subtle properties of the OT problem.

Keywords

Optimal transportMonge-Ampère equationfinite-difference scheme

MSC classification

Primary: 65N06: Finite difference methods
Secondary: 65J15: Equations with nonlinear operators (do not use ) 49M15: Newton-type methods
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Special Issue 6: Applied Optimal Transport , December 2019 , pp. 1041 - 1078
Copyright
© Cambridge University Press 2018 

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