Iterative Scheme for Solving Optimal Transportation Problems Arising in Reflector Design

Abstract

We consider the geometrical 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 before that this problem is equivalent to an infinite dimensional linear programming (LP) problem. We investigate techniques for constructing the two reflectors numerically. A straightforward discretization of this problem has the disadvantage that the number of constraints increases rapidly with the mesh size. So with this technique 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. Information from the previous iteration step is used to reduce the number of constraints necessary. As a proof of concept we apply our proposed scheme to solve a problem with synthetic data. We give evidence that the scheme converges. We also show that it allows for much finer meshes than a simple discretization scheme. There exists a growing literature for the application of optimal transportation theory to other beam shaping problems. Our proposed scheme is easy to adapt for these problems as well.

Multiscale Strategies for Computing Optimal Transport

“…Our approach draws from a varied set of work that is briefly summarized in Section 2. The proposed approach generalizes and builds on previous and concurrently developed hierarchical methods (Glimm and Henscheid, 2013;Schmitzer and Schnörr, 2013;Schmitzer, 2015;Oberman and Ruan, 2015). The work in this paper adds the following contributions:…”

“…Very recently a number of approaches have been proposed to solve the optimal transport in a multiscale fashion (Glimm and Henscheid, 2013;Schmitzer and Schnörr, 2013;Schmitzer, 2015;Oberman and Ruan, 2015). Glimm and Henscheid (2013) design an iterative scheme to solve a discrete optimal transport problem in reflector design and propose a heuristic for the iterative refinements based on linear programming duality. This iterative scheme can be interpreted as a multiscale decomposition of the transport problem based on geometry of the source and target sets.…”

“…This iterative scheme can be interpreted as a multiscale decomposition of the transport problem based on geometry of the source and target sets. The proposed potential refinement strategy extends the heuristic proposed by Glimm and Henscheid (2013) to guarantee optimal solutions and adds a more efficient computation strategy: Their approach requires to check all possible variables at the next scale. In Section 3.4.1 we introduce a variation of the approach by Glimm and Henscheid (2013) by adding a branch and bound strategy to avoid checking all variables, and an iterative procedure that guarantees optimal solutions.…”

“…• The description of a general multiscale framework for discrete optimal transport that can be applied in conjunction with a wide range of optimal transport algorithms. • A set of propagation and refinement heuristics, including approaches that are similar and/or refine existing ones (Glimm and Henscheid, 2013;Oberman and Ruan, 2015;Schmitzer, 2015) as well as novel ones. In particular we propose a novel propagation strategy based on capacity restrictions of the network flow problem at each scale.…”

A Stochastic Multi-layer Algorithm for Semi-discrete Optimal Transport with Applications to Texture Synthesis and Style Transfer

“…For which concerns optimal transport, numerous multiscale strategies have already been proposed which essentially consists in working on simplified versions of the measures µ and ν. Several authors [21,5,42] have proposed to discretize the measures on a grid, and then use a grid refinement strategy to target the true OT map. They propose different arguments to demonstrate the stability of the refined transport plans (which exploits directly or indirectly the sparsity of the true OT plan when µ and ν are absolutely continuous).…”