Intersection of paraboloids and application to Minkowski-type problems

Convergence of a Newton algorithm for semi-discrete optimal transport

“…We finally note that recent progress in computational geometry would allow one to implement Algorithm 1 for the quadratic cost on R 3 , refining [22] or [12]. It should also be possible to deal with optimal transport problems arising from geometric optics, such as the far-field reflector problem [10], whose associated cost satisfies the Ma-Trudinger-Wang condition [24]. where the function f (r) is the mean value of f over the sphere S d−1 (r),…”

“…We refer to this setting as semi-discrete optimal transport. Among the several algorithms proposed to solve semi-discrete optimal transport problems, one currently needs to choose between algorithms that are slow but come with a convergence speed analysis [29,8,21] or algorithms that are much faster in practice but which come with no convergence guarantees [5,27,11,22,10]. Algorithms of the first kind rely on coordinate-wise increments and the number of iterations required to reach the solution up to an error of ε is of order N 3 /ε, where N is the number of Dirac masses in the target measure.…”

“…This leads to the notion of Laguerre tessellation, whose cells are given by When the sets X and Y are contained in R d and the cost is the squared Euclidean distance, the computation of the Laguerre tessellation is a classical problem of computational geometry, for which there exists very efficient softwares, such as CGAL [1] or Geogram [2]. For other cost functions, one has to adapt the algorithms, as was done for the reflector cost on the sphere in [10]. The shape of the Voronoi and Laguerre tessellations is depicted in Figure 1.…”

Numerical resolution of Euler equations through semi-discrete optimal transport

“…The semi-discrete approach to optimal transport has been extended to transport costs other than quadratic, in particular those subject to the Ma-Trudinger-Wang property [31], and applied to inverse problems arising in geometric optics [20].…”

Far-Field Reflector Problem Under Design Constraints

“…19 Furthermore, it can also be transformed into an unconstrained convex optimization problem, and solved e ciently for a discretization of the target distribution of light with up to 20k Dirac masses. 13,5 Additional details about this approach are given in Section 2. Moreover, the solution computed via the optimal transport formulation is a convex patch which satisfies the desired optical properties.…”