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“A Numerical Algorithm for L2 semi-discrete optimal transport in 3D”

Bruno Lévy

ESAIM M2AN (Mathematical Modeling and Numerical Analysis), 2015

Abstract: This paper introduces a numerical algorithm to compute the L_2 optimal transport map between two measures \mu and \nu, where \mu derives from a density \rho defined as a piecewise linear function (supported by a tetrahedral mesh), and where \nu is a sum of Dirac masses. I first give an elementary presentation of some known results on optimal transport and then observe a relation with another problem (optimal sampling). This relation gives simple arguments to study the objective functions that characterize both problems. I then propose a practical algorithm to compute the optimal transport map between a piecewise linear density and a sum of Dirac masses in 3D. In this semi-discrete setting, Aurenhammer et.al showed that the optimal transport map is determined by the weights of a power diagram. The optimal weights are computed by minimizing a convex objective function with a quasi-Newton method. Merigot proposed a hierarchical algorithm that improves the speed of convergence, together with an implementation in 2D. To evaluate the value and gradient of the objective function for the 3D problem, I propose an efficient and robust algorithm, that computes at each iteration the intersection between a power diagram and the tetrahedral mesh that defines the measure \mu. The numerical algorithm is experimented and evaluated on several datasets, with up to hundred thousands tetrahedra and one million Dirac masses.

## BibTex reference

@ARTICLE{L:OT:2015,
TITLE = "A Numerical Algorithm for L2 semi-discrete optimal transport in 3D",
AUTHOR = "Bruno Lévy",
JOURNAL = "ESAIM M2AN (Mathematical Modeling and Numerical Analysis)",
YEAR = "2015",
}

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