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Monotone discretization of the Monge-Ampère equation of optimal transport

Abstract : 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 quasilinear operators. In dimension two, we show how using Selling's formula, a tool originating from low-dimensional lattice geometry, in order to choose the parameters of the discretization yields a closed-form formula for the maximum that appears at the discrete level, allowing the scheme to be solved particularly efficiently. We present the numerical results that we obtained when applying the scheme to the far field refractor problem in nonimaging optics.
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https://hal.archives-ouvertes.fr/hal-03255797
Contributor : Guillaume Bonnet <>
Submitted on : Wednesday, June 9, 2021 - 6:02:03 PM
Last modification on : Tuesday, July 13, 2021 - 3:15:08 AM

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  • HAL Id : hal-03255797, version 1

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Guillaume Bonnet, Jean-Marie Mirebeau. Monotone discretization of the Monge-Ampère equation of optimal transport. 2021. ⟨hal-03255797⟩

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