34,313 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
Geometrically-exact time-integration mesh-free schemes for advection-diffusion problems derived from optimal transportation theory and their connection with particle methods
We develop an Optimal Transportation Meshfree (OTM) particle method for
advection-diffusion in which the concentration or density of the diffusive
species is approximated by Dirac measures. We resort to an incremental
variational principle for purposes of time discretization of the diffusive
step. This principle characterizes the evolution of the density as a
competition between the Wasserstein distance between two consecutive densities
and entropy. Exploiting the structure of the Euler-Lagrange equations, we
approximate the density as a collection of Diracs. The interpolation of the
incremental transport map is effected through mesh-free max-ent interpolation.
Remarkably, the resulting update is geometrically exact with respect to
advection and volume. We present three-dimensional examples of application that
illustrate the scope and robustness of the method.Comment: 19 pages, 8 figure
Optimal Brownian Stopping between radially symmetric marginals in general dimensions
Given an initial (resp., terminal) probability measure (resp., )
on , we characterize those optimal stopping times that
maximize or minimize the functional ,
, where is Brownian motion with initial law
and with final distribution --once stopped at -- equal to .
The existence of such stopping times is guaranteed by Skorohod-type
embeddings of probability measures in "subharmoic order" into Brownian motion.
This problem is equivalent to an optimal mass transport problem with certain
constraints, namely the optimal subharmonic martingale transport. Under the
assumption of radial symmetry on and , we show that the optimal
stopping time is a hitting time of a suitable barrier, hence is non-randomized
and is unique
A Numerical Method to solve Optimal Transport Problems with Coulomb Cost
In this paper, we present a numerical method, based on iterative Bregman
projections, to solve the optimal transport problem with Coulomb cost. This is
related to the strong interaction limit of Density Functional Theory. The first
idea is to introduce an entropic regularization of the Kantorovich formulation
of the Optimal Transport problem. The regularized problem then corresponds to
the projection of a vector on the intersection of the constraints with respect
to the Kullback-Leibler distance. Iterative Bregman projections on each
marginal constraint are explicit which enables us to approximate the optimal
transport plan. We validate the numerical method against analytical test cases
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