Institut de recherche mathématique avancée

Résumé : On one hand, compilers successfully automate many important optimizations. On the other hand, compilers often miss critical optimizations, especially when they are general-purpose. A striking example of this is how compilers typically fail to reason about approximating exact arithmetic with finite precision number representations. Due to such compiler limitations, high-performance code is still commonly optimized by hand and packaged into optimized libraries, which is time-consuming and error-prone.

In the first part of this talk, I will present, at a high level, my ongoing work aimed at replacing manual optimization with an interactive optimization process that combines human expertise with compiler automation. In the second part of this talk, I will dive deeper into one strand of this work, which is aimed at combining program optimization with guaranteed numerical accuracy.

About the speaker : Since one year, I am a CNRS researcher in the CAMUS / ICPS team of ICube, in Strasbourg, France. Before joining CNRS, I was a postdoctoral researcher in the same team for almost two years. I received my PhD from the School of Computing Science at the University of Glasgow, in Scotland, supervised by Michel Steuwer and Phil Trinder. I received my Master from Sorbonne Université in Paris, France.

https://thok.eu/
https://www.ins2i.cnrs.fr/fr/cnrsinfo/thomas-koehler-et-loptimisation-de-programmes

Résumé : Je présenterai quelques questions ouvertes sur les représentations galoisiennes géométriques qui ont été mises en évidence par une approche calculatoire menée en collaboration avec X. Caruso et A. David. Puis je montrerai comment la théorie des modèles locaux pour les champs de modules de $(\Phi,\Gamma)-modules étales permet non seulement d'aborder ces questions mais aussi d'obtenir des présentations explicites des anneaux de déformations potentiellement de Barsotti-Tate (travail en collaboration avec B. Le Hung et S. Morra).

Résumé : We are interested here in the simulation of compressible fluid mechanics equations in a low Mach
number regime. More specifically, we study the numerical approximation of the barotropic Euler
equations using finite volume/finite element methods.
Low Mach number flows are notoriously difficult to simulate with classical finite volume methods,
mainly because their accuracy depends on the mesh shape [2]. Inspired by the MAC scheme [3]
(introduced for the simulation of incompressible fluids), one of the proposed solutions to address
this issue consists of staggering the velocity degrees of freedom at the mesh faces to improve the
approximation of the divergence operator. The challenge of such a placement of unknowns lies in
defining conservation, compared to colocated finite volume methods where it directly results from the
scheme’s formulation.
In[4],the authors proposed conservative staggered schemes based on Crouzeix-Raviart and Rannacher-
Turek finite elements for each velocity component.
Our approach follows this line of research with the following originality : we introduce a staggered
discretization based on the de Rham complex of Nédélec-Raviart-Thomas finite elements [1]. More
precisely, the velocity is in the Raviart-Thomas space, requiring only one degree of freedom per mesh
face in any spatial dimension.
The interest in relying on a discrete de Rham complex is illustrated through an asymptotic analysis
in the Mach number [5] :
i) The complex allows us to demonstrate the existence of a discrete Hodge decomposition, which
helps identify the low Mach limit of the scheme.
ii) Using this formalism, stabilization terms have been constructed to propagate low Mach number
acoustic waves in explicit time integration.
In this presentation, we will introduce both the theoretical tools that ensure accuracy at low Mach
numbers and the procedure for obtaining a conservative finite volume scheme. We will illustrate the
scheme’s properties through numerical simulations in 2d.

[1] A. Ern, J.-L. Guermond. Theory and practice of finite elements, vol. 159. Springer, 2004.

[2] H. Guillard. On the behavior of upwind schemes in the low mach number limit. iv : P0 approxi-
mation on triangular and tetrahedral cells. Computers & fluids, 38(10), 1969–1972, 2009.

[3] F. H. Harlow. Mac numerical calculation of time-dependent viscous incompressible flow of fluid
with free surface. Phys. Fluid, 8, 12, 1965.

[4] R. Herbin, W. Kheriji, J.-C. Latché. On some implicit and semi-implicit staggered schemes for
the shallow water and euler equations. ESAIM : Mathematical Modelling and Numerical Analysis,
48(6), 1807–1857, 2014.

[5] J. Jung, V. Perrier. Steady low mach number flows : identification of the spurious mode and
filtering method. Journal of Computational Physics, 468, 111462, 2022.

Résumé : Dans ce travail, nous étudions un modèle de régression visant à décrire le comportement des valeurs extrêmes d’une variable Y à partir de covariables X. Nous proposons une méthode de réduction de dimension spécialement conçue pour les queues de distribution, permettant de surmonter le fléau de la grande dimension et d’améliorer l’estimation de l’indice des valeurs extrêmes conditionnel.

Résumé : In this talk, I will present the meeting of different dynamical systems: tiling billiards, the wind-tree model and the Eaton lenses. The three of them are motivated by physics. In the beginning of the 2000's, physicists have conceived metamaterials with negative index of refraction. Tilling billiards' trajectories consist of light rays moving in a arrangement of metamaterials with opposite index of refraction. The wind-tree model was introduced by Paul and Tatyana Ehrenfest to study a gaz: a particle is moving in a plane where obstacles are periodically placed, on which the particle bounces. The Eaton lenses are a periodic array of lenses in the plane, in which we consider a light ray that is reflected each time it crosses a lens. After having introduced these dynamical systems, I will consider a mix of them: an arrangement of rectangles in the plane, like in the wind-tree model, but made of metamaterials, like for tiling billiards. I study the trajectories of light in this plane. They are refracted each time they cross a rectangle. I show that these trajectories are trapped in a strip, for almost every parameter. This behavior is similar to the one of the Eaton lenses.