– Semi-Discrete<\/strong> Far-field reflector problem and intersection of paraboloids<\/a> Far-Field reflector problem under design constraints<\/a> Intersection of paraboloids and application to Minkowski-type problems<\/a> A comparison of two dual methods for discrete optimal transport<\/a> A numerical algorithm for L2 semi-discrete optimal transport in 3D<\/a> – IPFP<\/strong> Iterative Bregman projections for regularized transportation problems<\/a> Scaling Algorithms for Unbalanced Transport Problems<\/a> Convergence of Entropic Schemes for Optimal Transport and Gradient Flows<\/a> – Finite differences<\/strong> Monotone and consistent discretization of the Monge-Amp\u00e8re operator<\/a>![\"layoutpir0\"](\"https:\/\/project.inria.fr\/mokabajour\/files\/2016\/09\/LayoutPIR0.png\")
<\/a><\/p>\n<\/a><\/p>\n
\nThis algorithm computes the solution of the \\(\\displaystyle L^2 \\) optimal transport problem between a piecewise linear density and a finite number of Dirac masses, both located in the Euclidian plane. It is therefor only usable for the PIR0 problem. The density support has to be convex and is numerically represented by a piecewise linear triangulation. For simple densities such as uniform polygons, defined with their convex envelope, the triangulation will have a small number of triangles, making the algorithm very competitive compared to the other ones.<\/p>\n
\nPedro Manh\u00e3es Machado de Castro, Quentin M\u00e9rigot, Boris Thibert
\nNumerische Mathematik, Springer Verlag, 2015<\/p>\n
\nFar-field reflector problem under design constraints
\nJulien Andr\u00e9, Dominique Attali, Quentin M\u00e9rigot, Boris Thibert
\nInternational Journal of Computational Geometry and Applications, Vol 25, No 2, pp 143-163, 2015<\/p>\n
\nPedro Manh\u00e3es Machado de Castro, Quentin M\u00e9rigot, Boris Thibert
\nComputational geometry (SoCG’14), 308–317, ACM, New York, 2014.<\/p>\n
\nQuentin M\u00e9rigot
\nGeometric science of information, 389–396, Lecture Notes in Computer Science, 8085, Springer, Heidelberg, 2013<\/p>\n
\nBruno L\u00e9vy
\nESAIM: Mathematical Modelling and Numerical Analysis, EDP Sciences, 2015, 49 (6), pp.1693 – 1715<\/p>\n
\nThis algorithm is a discrete algorithm which relies on introducing an entropic regularization to the Kantorovich formulation of the optimal transport problem.
\nA COMPLETER
\nWith this solver, best results are obtained with the same resolution for the source and for the target. It is important to notice that the amount of RAM used by this algorithm is ~source resolution * target resolution. For instance, a simple implementation of the IPFP with two marginals of 40000 diracs would use 13Gb of RAM to store the Gibbs Kernel matrix. An optimized version, using sparse matrices and a refinement approach, has been written and is only available for PIR0. A PIR1 version will soon be available.<\/p>\n
\nJean-David Benamou, Guillaume Carlier, Marco Cuturi, Luca Nenna, Gabriel Peyr\u00e9
\nSIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2015, 2 (37), pp.A1111-A1138<\/p>\n
\nLenaic Chizat, Gabriel Peyr\u00e9, Bernhard Schmitzer, Fran\u00e7ois-Xavier Vialard
\narXiv:1607.05816 [pdf, other]<\/p>\n
\nGuillaume Carlier, Vincent Duval, Gabriel Peyr\u00e9, Bernhard Schmitzer
\narXiv:1512.02783 [pdf, other]<\/p>\n
\nA REMPLIR<\/p>\n
\nJean-David Benamou, Francis Collino, Jean-Marie Mirebeau
\nMath. Comp. 85 (2016), no. 302, 2743–2775.<\/p>\n