111 research outputs found
A rigorous analysis using optimal transport theory for a two-reflector design problem with a point source
We consider the following geometric optics problem: Construct a system of two
reflectors which transforms a spherical wavefront generated by a point source
into a beam of parallel rays. This beam has a prescribed intensity
distribution. We give a rigorous analysis of this problem. The reflectors we
construct are (parts of) the boundaries of convex sets. We prove existence of
solutions for a large class of input data and give a uniqueness result. To the
author's knowledge, this is the first time that a rigorous mathematical
analysis of this problem is given. The approach is based on optimal
transportation theory. It yields a practical algorithm for finding the
reflectors. Namely, the problem is equivalent to a constrained linear
optimization problem.Comment: 5 Figures - pdf files attached to submission, but not shown in
manuscrip
Some examples of rank one convex functions in dimension two
We study the rank one convexity of some functions f(ξ) where ξ is a 2 × 2 matrix. Examples such as |ξ|2α + h(detξ) and | ξ |2α (| ξ |2 − γdet ξ) are investigated. Numerical computations are done on the example of Dacorogna and Marcellini, indicating that this function is quasiconve
Fast transport optimization for Monge costs on the circle
Consider the problem of optimally matching two measures on the circle, or
equivalently two periodic measures on the real line, and suppose the cost of
matching two points satisfies the Monge condition. We introduce a notion of
locally optimal transport plan, motivated by the weak KAM (Aubry-Mather)
theory, and show that all locally optimal transport plans are conjugate to
shifts and that the cost of a locally optimal transport plan is a convex
function of a shift parameter. This theory is applied to a transportation
problem arising in image processing: for two sets of point masses on the
circle, both of which have the same total mass, find an optimal transport plan
with respect to a given cost function satisfying the Monge condition. In the
circular case the sorting strategy fails to provide a unique candidate solution
and a naive approach requires a quadratic number of operations. For the case of
real-valued point masses we present an O(N |log epsilon|) algorithm that
approximates the optimal cost within epsilon; when all masses are integer
multiples of 1/M, the algorithm gives an exact solution in O(N log M)
operations.Comment: Added affiliation for the third author in arXiv metadata; no change
in the source. AMS-LaTeX, 20 pages, 5 figures (pgf/TiKZ and embedded
PostScript). Article accepted to SIAM J. Applied Mat
Hamiltonian ODEs on a space of deficient measures
We continue the study (initiated in [1]) of Borel measures whose time evolution is provided by an interacting Hamiltonian structure. Here, the principal focus is the development and advancement of deficency in the measure caused by displacement of mass to infinity in finite time. We introduce - and study in its own right - a regularization scheme based on a dissipative mechanism which naturally degrades mass according to distance traveled (in phase space). Our principal results are obtained based on some dynamical considerations in the form of a condition which forbids mass to return from infinity
Structural results on convexity relative to cost functions
Mass transportation problems appear in various areas of mathematics, their
solutions involving cost convex potentials. Fenchel duality also represents an
important concept for a wide variety of optimization problems, both from the
theoretical and the computational viewpoints. We drew a parallel to the
classical theory of convex functions by investigating the cost convexity and
its connections with the usual convexity. We give a generalization of Jensen's
inequality for cost convex functions.Comment: 10 page
On the Regularity of Optimal Transportation Potentials on Round Spheres
In this paper the regularity of optimal transportation potentials defined on
round spheres is investigated. Specifically, this research generalises the
calculations done by Loeper, where he showed that the strong (A3) condition of
Trudinger and Wang is satisfied on the round sphere, when the cost-function is
the geodesic distance squared. In order to generalise Loeper's calculation to a
broader class of cost-functions, the (A3) condition is reformulated via a
stereographic projection that maps charts of the sphere into Euclidean space.
This reformulation subsequently allows one to verify the (A3) condition for any
case where the cost-fuction of the associated optimal transportation problem
can be expressed as a function of the geodesic distance between points on a
round sphere. With this, several examples of such cost-functions are then
analysed to see whether or not they satisfy this (A3) condition.Comment: 24 pages, 4 figure
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