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Section: New Results

Computational Optimal Transport

Participants : Bruno Lévy, Erica Schwindt.

We continued working on Optimal Transportation and its applications in fluid simulation and astrophysics  [21], [20]. We developed an efficient and robust algorithm to compute Laguerre diagrams and intersections with tetrahedralized domains, that is, the geometric structure involved in a specific form of optimal transport that we are interested in. In addition, we developed an efficient parallel algorithm to compute Laguerre diagrams, with the possibility of handling periodic boundaries (3-torus), that is to say that the domain is a unit cube with opposite faces that are identified (if one leaves the domain from the left, it enters the domain from the right, etc..., like in the PacMan game). Such a topology is interesting for some simulations in astrophysics, or in material science, that consider a huge domain with homogeneous behavior and replace it with a tiny fraction and periodic boundary conditions (equivalent to a periodic material). We made the algorithms available in the geogram programming library ( http://alice.loria.fr/software/geogram/doc/html/index.html ). In cooperation with Roya Mohayaee (Institut d'Astrophysique de Paris) and Jean-Michel Alimi (Observatoire de Paris), we started applying the method to some inverse problems in astrophysics (Early Universe Reconstruction), that is reconstructing the past history of the universe from a 3D map of the galaxy clusters. Under some simplifying assumptions, the problem is precisely an instance of semi-discrete optimal transport that our algorithm solves efficiently. Our algorithm does the computation on a desktop PC within hours for several tenths of million points. With Quentin Merigot and Hugo Leclerc (U. Paris Sud), we are designing a new algorithm with the aim of scaling up to billions points (as requested by our astrophysicist colleagues).