Volume 56, Number 3, May-June 2022
|Page(s)||815 - 865|
|Published online||25 April 2022|
Monotone discretization of the Monge–Ampère equation of optimal transport
LMO, Université Paris-Saclay, Orsay, France
2 Inria-Saclay and CMAP, École Polytechnique, Palaiseau, France
3 Université Paris-Saclay, ENS Paris-Saclay, CNRS, Centre Borelli, Gif-sur-Yvette, France
* Corresponding author: email@example.com
Accepted: 14 March 2022
We design a monotone finite difference discretization of the second boundary value problem for the Monge–Ampère equation, whose main application is optimal transport. We prove the existence of solutions to a class of monotone numerical schemes for degenerate elliptic equations whose sets of solutions are stable by addition of a constant, and we show that the scheme that we introduce for the Monge–Ampère equation belongs to this class. We prove the convergence of this scheme, although only in the setting of quadratic optimal transport. The scheme is based on a reformulation of the Monge–Ampère operator as a maximum of semilinear operators. In dimension two, we recommend to use Selling’s formula, a tool originating from low-dimensional lattice geometry, in order to choose the parameters of the discretization. We show that this approach yields a closed-form formula for the maximum that appears in the discretized operator, which allows the scheme to be solved particularly efficiently. We present some numerical results that we obtained by applying the scheme to quadratic optimal transport problems as well as to the far field refractor problem in nonimaging optics.
Mathematics Subject Classification: 65N06 / 65N12 / 35J70 / 35J96
Key words: Monge–Ampère equation / optimal transport / monotone finite difference schemes / convergence analysis
© The authors. Published by EDP Sciences, SMAI 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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