An Entropic Optimal Transport Numerical Approach to the Reflector Problem

Jean-David Benamou (1) , Wilbert L Ijzerman (2) , Giorgi Rukhaia (1)

1 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales
2 TU/e - Eindhoven University of Technology [Eindhoven]

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.

Mots clés: Inverse reflector problem; Optimal transportation; Non-linear optimization

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Jean-David Benamou, Wilbert L Ijzerman, Giorgi Rukhaia. An Entropic Optimal Transport Numerical Approach to the Reflector Problem. Methods and Applications of Analysis, 2020, ⟨10.4310/MAA.2020.v27.n4.a1⟩. ⟨hal-02539799⟩