PhD proposal (with Charles Dapogny) starting in October 2022 : Distributionally Robust Shape Optimization
I am mostly interested in the development of effective calculations in problems having a geometric flavor, which could be refered
as applied geometry
or numerical geometry
. In particular, I have been working in the field of geometric inference and
on different nonlinear geometric problems, having connections with various domains such as optimal transport, Monge-Ampère equations, inverse
problems in optics, computational geometry or convex integration theory. Below are some of my favorite fields I have contributed to.
Numerical optimal transport / Inverse problems in optics.
I am working on the numerical aspect of optimal transport, in particular in the semi-discrete setting (i.e.
when transporting an absolutely continuous measure to a discrete one).
I am also investigating applications to inverse problems arising in non-imaging optics.
Figure: Parallel light refracted by a lens that concentrates the light into the "hikari" character.
Convex Integration theory / Smooth fractals.
I am working with the Hevea Project on the realization of Nash's isometric embeddings based on the Convex Integration Theory
developped by Gromov in the 70s. I am particularly interested in the simplification of this theory in order to make
it more effective and to numerically solve some nonlinear partial differential equations.
Click here for an application of the flat torus.
Figure: smooth-fractal structure observed on a reduced sphere.
The aim of geometric inference is to get robust estimations of the topological and geometric properties of a
geometric object from an approximation, such as a finite point set. I have in particular been working on the stability of the
(Federer) curvature measures, the regularity of the distance functions to compact sets, Voronoi Covariance Measures
using distances to measures.
Figure: Example of a finite point set from which one wants to infer geometric properties of the underlying object.
More details on my publications web page.
- Anidolix: software to design optical components.
- Hevea: H-principE, Visualisation Et Applications.
- MAGA: Monge Ampère et Géometrie Algorithmique
- CoMeDiC: Convergent Metrics for Digital Calculus
PhD Students and Postdocs
- Anatole Gallouet (2019-...) Inverse problems in nonimaging optics and generated jacobian equations
Co-advised with Quentin Mérigot
- Jean-Baptiste Keck (Postdoc, January 2020 - April 2021) Problèmes inverses en optique avec des sources de lumière étendues Co-advised with Quentin Mérigot (ANR MAGA Grant)
- Mélanie Theillière (2016-2019) Effective Convex Integration Theory
Co-advised with Vincent Borrelli
- Jocelyn Meyron (2015-2018) Transport optimal semi-discret et applications en optique anidolique
Co-advised with Quentin Mérigot
- Julien André (2012-2015) Conception de réflecteurs pour des applications photométriques
Co-advised with Quentin Mérigot and Dominique Attali
- Roland Denis (Postdoc, December 2012- March 2014) Structure en fractale lisse des sphères de Nash-Kuiper
Co-advised with Francis Lazarus within the Hevea Project (UJF Grant).
- Louis Cuel (2011-2014) Discrete Geometric Inference
Co-advised with Jacques-Olivier Lachaud
Laboratoire LJK - Bâtiment IMAG
Université Grenoble Alpes
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