Section: New Results

Semi-discrete Optimal Transport in 3D

Participant : Bruno Lévy.

This work introduces 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  [28] . The optimal weights are computed by minimizing a convex objective function with a quasi-Newton method. To evaluate the value and gradient of this objective function, we 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. Like in the multilevel proposed by Mérigot, we use a hierarchical algorithm, that uses nested point sets to discretize the source measure.

We think this work may lead to interesting discretizations of the physics, that include the conservation laws (conservation of energy, conservation of momentum ...) deep in their definition, as explained by Jean-David Benamou and Yann Brenier in their fluid dynamics formulation of optimal transport  [30] .

This work was published in the journal Mathematical Modeling and Analysis [10] .

Figure 2. Semi-discrete optimal transport from a constant density to a varying one (product of sines).