Methods and Applications of Analysis

Volume 27 (2020)

Number 4

Special Issue for the 60th Birthday of John Urbas: Part I

Guest Editors: Neil Trudinger and Xu-Jia Wang

An entropic optimal transport numerical approach to the reflector problem

Pages: 311 – 340

DOI: https://dx.doi.org/10.4310/MAA.2020.v27.n4.a1

Authors

Jean-David Benamou (INRIA-Paris, MOKAPLAN, Paris, France; and CEREMADE, UMR CNRS 7534, Université Paris Dauphine, Paris, France)

Wilbert L. Ijzerman (Signify Research High Tech Campus 7, Eindhoven, The Netherlands)

Giorgi Rukhaia (INRIA-Paris, MOKAPLAN, Paris, France; and CEREMADE, UMR CNRS 7534, Université Paris Dauphine, Paris, France)

Abstract

The point source far field reflector design problem is a classic example of an optimal transport problem with a non-euclidean displacement cost [Wang, 2004] [Glimm and Oliker, 2003]. This work discusses the use of Entropic Optimal Transport and the associated Sinkhorn algorithm [Cuturi, 2013] to solve it numerically. As the reflector modelling is based on the Kantorovich potentials, several questions arise. First, on the convergence of the discrete entropic approximation and here we follow the recent work of [Berman, 2017] and in particular the imposed discretization requirements therein. Secondly, the correction of the bias induced by the Entropic Optimal Transport using the recent notion of Sinkhorn divergences [Ramdas et al., 2017] [Genevay et al., 2018] [Feydy et al., 2018] is shown to be necessary to achieve reasonable results. The paper does not establish any new theoretical result but discusses the necessary mathematical and numerical tools needed to produce and analyse the obtained numerical results. We find that Sinkhorn algorithm may be adapted, at least in simple academic cases, to the resolution of the far field reflector problem.

Keywords

inverse reflector problem, optimal transportation, non-linear optimization

2010 Mathematics Subject Classification

49Qxx, 65K10, 78A46

Received 9 April 2020

Accepted 6 November 2020

Published 24 September 2021