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An Entropic Optimal Transport Numerical Approach to the Reflector Problem

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Date
2020
Publisher city
Paris
Collection title
Cahier de recherche CEREMADE, Université Paris-Dauphine
Link to item file
https://hal.archives-ouvertes.fr/hal-02539799
Dewey
Analyse
Sujet
Reflector Problem
URI
https://basepub.dauphine.fr/handle/123456789/20643
Collections
  • CEREMADE : Publications
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Author
Benamou, Jean-David
Ijzerman, W.
Rukhaia, Giorgi
Type
Document de travail / Working paper
Item number of pages
29
Abstract (EN)
The point source far field reflector design problem is one of the main classic optimal transport problems with a non-euclidean displacement cost [Wang, 2004] [Glimm and Oliker, 2003]. This work describes 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 Entropic bias induced by the Entropic OT, as discussed in particular in [Ramdas et al., 2017] [Genevay et al., 2018] [Feydy et al., 2018], is another important tool to achieve reasonable results. The paper reviews the necessary mathematical and numerical tools needed to produce and discuss 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. Sinkhorn canonical extension to continuous potentials is needed to generate continuous reflector approximations. The use of Sinkhorn divergences [Feydy et al., 2018] is useful to mitigate the entropic bias.

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