In this talk, we will show how optimal transport theory can be a useful tool for solving many applied problems such as optical component design.
We start by presenting optimal transport, its different formulations and the motivations behind it. We describe in more details the so-called semi-discrete setting where the target measure is supported on a point cloud and explain the main numerical method we will use namely a damped Newton's algorithm. We then look at a particular case where the source measure is supported on a finite union of lower-dimensional subsets of R^d. We show the convergence with linear speed of a damped Newton's method to solve this problem. The convergence will be a direct consequence of the regularity and strict monotonicity of the Kantorovich function. We also describe some applications such as optimal quantization of a probability density over a surface or remeshing.
In the second part, we show that optimal transport can be used to recast many different optical component design problems into solving a non-linear system of equations that is a discretization of the so-called Monge-Amp将イre equation. In these problems, we are interested in building optical components (such as mirrors and lenses) that satisfy light energy constraints. We show the relation between these problems and optimal transport using the notion of Visibility cells. We also show numerous numerical results.
Finally, we describe three different initialization strategies which can be used to speed up existing algorithms for solving both discrete and semi-discrete optimal transport. We explain the necessity of having a good initialization and illustrate the methods on many numerical examples.