Transport optimal semi-discret et applications en optique anidolique

Keywords :

• géométrie algorithmique
• conception de réflecteurs

In this thesis, we are interested in solving many inverse problems arising in optics. More precisely, we are interested in designing optical components such as mirrors and lenses that satisfy some light conservation constraints meaning that we want to control the reflected (or refracted) light in order match a prescribed intensity. This has applications in car headlight design or caustic design for example. We show that optical component design problems can be recast as optimal transport ones for different cost functions and we explain how this allows to study the existence and the regularity of the solutions of such problems. We also show how, using computational geometry, we can use an efficient numerical method namely the damped Newton's algorithm to solve all these problems. We will end up with a single generic algorithm able to efficiently build an optical component with a prescribed reflected (or refracted) illumination. We show the convergence of the Newton's algorithm to solve the optimal transport problem when the source measure is supported on a finite union of simplices.  We then describe the common relation between eight optical component design problems and show that they can all be seen as discrete Monge-Ampère equations. We also apply the Newton's method to optical component design and show numerous simulated and fabricated examples. Finally, we look at a problem arising in computational optimal transport namely the choice of the initial weights. We develop three simple procedures to find "good" initial weights which can be used as a starting point in computational optimal transport algorithms.

Directors:

• Mme Dominique Attali (Directeur de recherche - CNRS )
• Mr Quentin Mérigot (Professeur - Université Paris-Sud )
• Mr Boris Thibert (Maître de conférence - Université Grenoble Alpes )

Raporteurs:

• Mr Marco Cuturi (Professeur - Université Paris-Saclay )
• Mr Bruno Levy (Directeur de recherche - INRIA Nancy )

Examinators:

• Mr André Lieutier (Ingénieur - Dassault Systèmes )
• Mme Julie Digne (Chargée de recherche - CNRS )
• Mr Edouard Oudet (Professeur - Université Grenoble Alpes )