Proceedings of the Thirtieth Annual Symposium on Computational Geometry 2014
DOI: 10.1145/2582112.2582153
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Intersection of paraboloids and application to Minkowski-type problems

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Cited by 3 publications
(5 citation statements)
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“…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),…”
Section: Convergence Of the Damped Newton Algorithmmentioning
confidence: 99%
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“…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),…”
Section: Convergence Of the Damped Newton Algorithmmentioning
confidence: 99%
“…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.…”
Section: Introductionmentioning
confidence: 99%
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“…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].…”
Section: The Kantorovich Functionalmentioning
confidence: 99%
“…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.…”
Section: Related Workmentioning
confidence: 99%