Description
Date depot: 2 avril 2025
Titre: Computer algebra and computational real algebraic geometry with a view towards optical system design
Directeur de thèse:
Mohab SAFEY EL DIN (LIP6)
Domaine scientifique: Sciences et technologies de l'information et de la communication
Thématique CNRS : Calcul arithmétique et formel, codage et cryptologie
Resumé: Optical system design boils down to combine lenses, mirrors and other optical components of the system with the goal to optimize some optical and geometric criteria such as aberration, i.e. deviations from ideal image formation due to imperfections in the system, or feasibility criteria of both technical and physical aspects realization. A crucial task in optical system design is the classification of architectures of optical systems according to optical features (number of mirrors,
their relative position and their nature – spherical or not, etc.), which is a key assett to the engineer for building optimal optical systems. A remarkable fact is that most of optical laws are governed by Euclidean geometry. Hence, turning classification of optical systems into equations and constraints lead to systems of polynomial constraints with real coefficients.
The initial design problem is then reduced to the study of the connected components of the solution sets of such polynomial systems (which are called semi-algebraic sets). The algorithmic problems studied in this PhD project are the following: (a) design dedicated algorithms for grabbing sample points in each connected component of semi-algebraic sets, leveraging geoemtric properties of systems coming from optical system design, (b) design algorithms for counting these connected components and (c) design algorithms for describing (with semi-algebraic formula) these connected components.
Using some advanced techniques from computer algebra, one expects some complexity improvements for (b) and (c). The ultimate goal is to obtain efficient implementations of such algorithms that can be used to solve, in practice, those problems coming from optical system design.