Institut de recherche mathématique avancée

Résumé : In recent years, conservative-characteristic methods have been extensively used to numerically solve different hyperbolic PDEs. These methods use a combinaton of a finite volume method to approximate the cell-averages in mesh cells using conservative form of equations, and an arbitrary method to approximate the point values in edges/faces of the mesh using the non-conservative form of equations. Therefore, the overall method remains conservative (for the averages), retaining a lot of flexibility in how to deal with the point values. In this talk, we will describe 2 of such methods: CABARET and Active Flux, including some applications for both schemes.

CABARET, first introduced by V. Goloviznin and later refined by S. Karabasov [1], is a second-order explicit conservative-characteristic method. Its special feature is the extrapolation of Riemann invariants along the linearized characteristics to evolve the point values. We will discuss this method in detail and introduce various applications for problems in oceanology [2], fluid-structure interaction [3], transonic flows [4] and thermoacoustic instability [5].

Active Flux method, first introduced by T. Eymann and P. Roe [6], has been adapted to solve many problems for hyperbolic systems of PDEs on orthogonal and polygonal meshes. There are many versions of this method, and we will concentrate on the work of R. Abgrall and his group. This version of Active Flux is a third-order scheme [7], which works on general polygonal meshes (for 2D) and uses the method of lines to approximate the point values on edges and nodes of the mesh. We will describe the base algorithm for two-dimensional problems on a plane, and also we introduce the generalization of Active Flux method on triangular meshes to hyperbolic problems on a sphere [8].

References:

[1] S. Karabasov and V. Goloviznin “Compact Accurately Boundary-Adjusting High-REsolution Technique for Fluid Dynamics”, Journal of Computational Physics, 228(19), pp. 7426–7451, 2009.

[2] V.M. Goloviznin, P.A. Maiorov, P.A. Maiorov and A.V. Solovjev “Validation of the Low Dissipation Computational Algorithm CABARET-MFSH for Multilayer Hydrostatic Flows with a Free Surface on the Lock-release Experiments”, Journal of Computational Physics, 463, p. 111239, 2023.

[3] N. Afanasiev, V. Goloviznin, P. Maiorov and A. Solovjev “Simulating the dynamics of a fluid with a free surface in a gravitational field by a CABARET method”, Mathematical notes of NEFU, 29(4), pp. 77–94, 2022.

[4] N. Afanasiev and V. Goloviznin, “A Locally Implicit Time-Reversible Sonic Point Processing Algorithm for One-Dimensional Shallow-Water Equations”, Journal of Computational Physics, 434, p. 110220, 2021.

[5] N. A. Afanasiev, V. M. Goloviznin, V. N. Semenov et al. “Direct simulation of thermoacoustic instability in gas generators using the cabaret scheme”, Mathematical Models and Computer Simulations, 13(5), pp. 820–830, 2021.

[6] T.A. Eymann and P.L. Roe. “Active flux schemes”, AIAA, 382(19), 2011.

[7] R. Abgrall, J. Lin and Y. Liu “Active flux for triangular meshes for compressible flows problems”, Beijing Journal of Pure & Applied Mathematics, 2(1), pp. 1–33, 2025.

[8] N. Afanasev and R.Abgrall “Active Flux Method on a Sphere”, Submitted, 2025.

Résumé : Symbolic Regression (SR) aims to automatically discover analytical expressions that describe data. Such methods yield interpretable, compact, and computationally efficient models, often with remarkable extrapolation capabilities. With the rise of deep learning, SR has seen a resurgence of interest—but its potential remains underexplored in physics, where additional constraints and priors can play a crucial role.

In this talk, I will present Φ-SO (Physical Symbolic Optimization), a framework that leverages deep reinforcement learning to recover analytical physical laws directly from data. I will outline several key advances in the field achieved within this framework:

(1) the incorporation of dimensional analysis and other functional form priors to guide the search,
(2) the discovery of a single analytical form that fits multiple datasets each governed by different parameters (Class SR),
(3) the recursive detection of additive and multiplicative separabilities to uncover structure within data,
(4) the extension of SR to the discovery of differential equations rather than their solutions, and
(5) planned extensions to our system

Finally, I will discuss concrete applications of Φ-SO to astrophysical problems, in particular studies related to galactic dynamics and dark matter.

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 : 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.