An Entropic Optimal Transport Numerical Approach to the Reflector Problem
| Autor: | Jean-David Benamou, Giorgi Rukhaia, Wilbert Ijzerman |
|---|---|
| Přispěvatelé: | Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales (MOKAPLAN), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Eindhoven University of Technology [Eindhoven] (TU/e), This work was supported by ROMSOC EID H2020 network under the Marie Curie Grant Agreement No. 765374, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria) |
| Rok vydání: | 2020 |
| Předmět: |
Work (thermodynamics)
Optimal transportation Discretization Reflector (antenna) Extension (predicate logic) Resolution (logic) Displacement (vector) Non-linear optimization Simple (abstract algebra) Inverse reflector problem Convergence (routing) Applied mathematics [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] 2010 Mathematics Subject Classification49Qxx 65K10 78A46 Mathematics |
| Zdroj: | Methods and Applications of Analysis Methods and Applications of Analysis, International Press, 2020 Methods and Applications of Analysis, 2020, ⟨10.4310/MAA.2020.v27.n4.a1⟩ |
| ISSN: | 1073-2772 |
| DOI: | 10.5281/zenodo.3755363 |
| Popis: | 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 inparticular 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. working paper or preprint |
| Databáze: | OpenAIRE |
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