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This paper introduces a numerical algorithm to compute the $L_2$ optimaltransport map between two measures $\mu$ and $\nu$, where $\mu$ derives from adensity $\rho$ defined as a piecewise linear function (supported by atetrahedral mesh), and where $\nu$ is a sum of Dirac masses.  I first give an elementary presentation of some known results on optimaltransport and then observe a relation with another problem (optimal sampling).This relation gives simple arguments to study the objective functions thatcharacterize both problems.  I then propose a practical algorithm to compute the optimal transport mapbetween a piecewise linear density and a sum of Dirac masses in 3D. In thissemi-discrete setting, Aurenhammer et.al [\emph{8th Symposium on ComputationalGeometry conf. proc.}, ACM (1992)] showed that the optimal transport map isdetermined by the weights of a power diagram. The optimal weights are computedby minimizing a convex objective function with a quasi-Newton method. Toevaluate the value and gradient of this objective function, I propose anefficient and robust algorithm, that computes at each iteration theintersection between a power diagram and the tetrahedral mesh that defines themeasure $\mu$.  The numerical algorithm is experimented and evaluated on several datasets,with up to hundred thousands tetrahedra and one million Dirac masses.

A numerical algorithm for $L_2$ semi-discrete optimal transport in 3D