Table of Contents
ISRN Applied Mathematics
Volume 2013, Article ID 635263, 12 pages
http://dx.doi.org/10.1155/2013/635263
Research Article

Iterative Scheme for Solving Optimal Transportation Problems Arising in Reflector Design

1Department of Mathematics, Western Washington University, Bellingham, WA 98225, USA
2Program in Applied Mathematics, University of Arizona, Tucson, AZ 85716, USA

Received 29 July 2013; Accepted 5 September 2013

Academic Editors: M.-H. Hsu, G. Mishuris, L. Rebollo-Neira, and Q. Song

Copyright © 2013 Tilmann Glimm and Nick Henscheid. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the geometric optics problem of finding a system of two reflectors that transform a spherical wavefront into a beam of parallel rays with prescribed intensity distribution. Using techniques from optimal transportation theory, it has been shown previously that this problem is equivalent to an infinite-dimensional linear programming (LP) problem. Here we investigate techniques for constructing the two reflectors numerically by considering the finite-dimensional LP problems which arise as approximations to the infinite-dimensional problem. A straightforward discretization has the disadvantage that the number of constraints increases rapidly with the mesh size, so only very coarse meshes are practical. To address this well-known issue we propose an iterative solution scheme. In each step, an LP problem is solved, where information from the previous iteration step is used to reduce the number of necessary constraints. As an illustration, we apply our proposed scheme to solve a problem with synthetic data, demonstrating that the method allows for much finer meshes than a simple discretization. We also give evidence that the scheme converges. There exists a growing literature for the application of optimal transportation theory to other beam shaping problems, and our proposed scheme is easy to adapt for these problems as well.