Semi-discrete optimal transport and applications to non-imaging optics

In this seminar, we show how optimal transport theory can be used
to build a common framework to solve many optical component design problems.

In a first part, we present optimal transport, its main formulations and the
motivations behind it. We then 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 the damped Newton's
algorithm.  We then look at a particular case where the source measure is
supported on a triangulation in R^3 and show the convergence with linear speed
of the damped Newton's method.  The convergence is a direct consequence of the
regularity and strict motonicity of the Kantorovich functional. We also mention
some applications such as optimal quantization of a probability density over a
surface or remeshing.

In a second part, we describe the relation between optimal transport and optical
component design. In particular, we show how we can recast such problems into a
non-linear system of equations that is a discretization of the so-called
Monge-Ampère equation. This formulation allows us to develop a generic,
parameter-free and efficient algorithm. We finish by showing numerous simulated and
fabricated examples.