Existence of optimal maps in the reflector-type problems
Wilfrid Gangbo,Vladimir Oliker +1 more
TLDR
In this article, the authors consider probability measures μ and ν on a d-dimensional sphere in and cost functions of the form that generalize those arising in geometric optics where they prove that if μ and ǫ vanish on -rectifiable sets, if |l'(t)|>0, and is monotone then there exists a unique optimal map T o that transports μ onto where optimality is measured against c.Abstract:
In this paper, we consider probability measures μ and ν on a d -dimensional sphere in and cost functions of the form that generalize those arising in geometric optics where We prove that if μ and ν vanish on -rectifiable sets, if |l'(t)|>0, and is monotone then there exists a unique optimal map T o that transports μ onto where optimality is measured against c. Furthermore, Our approach is based on direct variational arguments. In the special case when existence of optimal maps on the sphere was obtained earlier in [Glimm and Oliker, J. Math. Sci. 117 (2003) 4096-4108] and [Wang, Calculus of Variations and PDE's 20 (2004) 329-341] under more restrictive assumptions. In these studies, it was assumed that either μ and ν are absolutely continuous with respect to the d -dimensional Haussdorff measure, or they have disjoint supports. Another aspect of interest in this work is that it is in contrast with the work in [Gangbo and McCann, Quart. Appl. Math. 58 (2000) 705-737] where it is proved that when l(t)=t then existence of an optimal map fails when μ and ν are supported by Jordan surfaces.read more
Citations
More filters
Journal ArticleDOI
Density Functional Theory and Optimal Transportation with Coulomb Cost
TL;DR: In this paper, the exact exchange-correlation functional reduces to a very interesting functional of novel form, which depends on an optimal transport map T associated with a given density ρ.
Journal ArticleDOI
The Refractor Problem in Reshaping Light Beams
Journal ArticleDOI
Embedding Sn into Rn+1 with given integral Gauss curvature and optimal mass transport on Sn
TL;DR: Aleksandrov's problem is closely connected with the theory of optimal mass transport on a sphere with cost function and constraints arising naturally from geometric considerations in this article, where a variational solution to the problem of existence and uniqueness of a closed convex hypersurface with prescribed integral Gauss curvature is given.
Journal ArticleDOI
Freeform lens design for a point source and far-field target.
TL;DR: This paper considers the design of a single freeform lens that converts the light from an ideal (zero-étendue) point source into a far-field target and uses a generalized least-squares algorithm that can handle a non-quadratic cost function in the corresponding optimal transport problem.
Journal ArticleDOI
Inverse reflector design for a point source and far-field target
TL;DR: In this paper, the authors present a method for the design of a single freeform reflector that converts the light distribution of a point source to a desired light distribution in the far field using the geometrical-optics law of reflection.
References
More filters
Journal ArticleDOI
The geometry of optimal transportation
TL;DR: In this paper, the existence and uniqueness of optimal maps are discussed. But the uniqueness of the optimal map is not discussed. And the role of the map in finding the optimal solution is left open.
Journal ArticleDOI
Duality theorems for marginal problems
TL;DR: In this paper, an analogue of the classical duality theorem of linear programming is established, imposing only weak conditions on the topology of the spaces Xi and the measurability resp. boundedness of the function h.
Journal ArticleDOI
On the optimal mapping of distributions
M. Knott,C. S. Smith +1 more
TL;DR: In this article, the authors consider the problem of minimizing the expected value of ∣X −Y ∣2 by finding the joint distribution of the random variable (X, Y) with specified marginal distributions for X and Y, and give a sufficient condition for the minimizing joint distribution and supply numerical results for two special cases.
Journal ArticleDOI
On the design of a reflector antenna
TL;DR: The reflector antenna design problem is reduced to that of finding a minimizer or a maximizer of a linear functional subject to a linear constraint, therefore it becomes an linear optimization problem and algorithms in linear programming apply.
Journal ArticleDOI
Optical Design of Single Reflector Systems and the Monge–Kantorovich Mass Transfer Problem
Tilmann Glimm,Vladimir Oliker +1 more
TL;DR: In this paper, it was shown that the problem of designing a reflector that transforms a spherical wave front with a given intensity into an output front illuminating a prespecified region of the far-sphere with prescribed intensity can be solved numerically by tools of linear programming.