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An Entropic Optimal Transport Numerical Approach to the Reflector Problem

hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
dc.contributor.authorBenamou, Jean-David
hal.structure.identifierEindhoven University of Technology [Eindhoven] [TU/e]
dc.contributor.authorIjzerman, Wilbert
hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
dc.contributor.authorRukhaia, Giorgi
dc.date.accessioned2022-02-11T14:18:40Z
dc.date.available2022-02-11T14:18:40Z
dc.date.issued2020
dc.identifier.issn1073-2772
dc.identifier.urihttps://basepub.dauphine.psl.eu/handle/123456789/22575
dc.language.isoenen
dc.subjectinverse reflector problemen
dc.subjectoptimal transportationen
dc.subjectnon-linear optimizationen
dc.subject.ddc515en
dc.titleAn Entropic Optimal Transport Numerical Approach to the Reflector Problemen
dc.typeArticle accepté pour publication ou publié
dc.description.abstractenThe 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.en
dc.relation.isversionofjnlnameMethods and Applications of Analysis
dc.relation.isversionofjnlvol27en
dc.relation.isversionofjnlissue4en
dc.relation.isversionofjnldate2021-09
dc.relation.isversionofjnlpages311 – 340en
dc.relation.isversionofdoi10.4310/MAA.2020.v27.n4.a1en
dc.relation.isversionofjnlpublisherInternational Pressen
dc.subject.ddclabelAnalyseen
dc.relation.forthcomingnonen
dc.description.ssrncandidatenon
dc.description.halcandidatenonen
dc.description.readershiprechercheen
dc.description.audienceInternationalen
dc.relation.Isversionofjnlpeerreviewednonen
dc.date.updated2022-02-11T14:14:28Z
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