Claude Bardos (LJLL - CNRS, Sorbonne Université & Université Paris Diderot)

Title : About Olivier Lafitte around Waves and Plasma

Abstract : Starting from mathematical recents mathematical in present time on Wave and Plasmas I will emphasise the contributions of Olivier with their influence on our community. Such contributions are devoted to linear or linearised equations. This turns out to be fully relevant both for the understanding of mathematical sciences as for many applications.

Hélène Barucq (INRIA - Université de Pau)

Title : Radiation boundary conditions for truncating the atmosphere of the Sun

Abstract : We consider a time-harmonic wave equation under ideal atmospheric assumptions. Using the Liouville transformation, we obtain a Schrödinger-type equation. We then propose two diffferent approaches for deriving Radiation Boundary Conditions. The first one relies on the construction of the Schwartz kernel of the resolvent while the second one uses the exact Dirichlet-to-Neumann map. Numerical experiments assess and compare the different RBCs. 

This is joint work with Ha Pham (Inria), Florian Faucher (Inria), Damien Fournier (MPS) and Laurent Gizon (MPS).

Hakim Boumaza (Université Sorbonne Paris Nord)

Title : Semiclassical analysis of the periodic Airy-Schrödinger operator

Abstract : In this talk I present, in the semiclassical regime, a complete description of the spectral bands of the periodic Airy-Schrödinger operator which are in the range of the periodic potential. The potential of this Schrödinger operator is periodic, continuous and piecewise linear.  I will aslo present an explicit formula for the integrated density of states of the periodic Airy-Schrodinger operator on the real line. Note that in these results there are explicit constants which characterize the semiclassical regime. This is joint work with Olivier Lafitte.

Jean-Marc Brossier (INP Grenoble)

Title : Coopération en apprentissage supervisé 

Abstract : L’idée d’améliorer une estimation de paramètre, ou un choix d’hypothèse, grâce à une coopération entre plusieurs estimateurs simples est apparue dans différents domaines. Partant de l’exemple du codage correcteur d’erreurs, cet exposé pose la question d’une telle coopération dans le domaine de l’apprentissage statistique.

Marion Darbas (Université Sorbonne Paris Nord)

Title : Fast BEMs for time-harmonic high-frequency wave propagation problems 

Abstract : In this talk, I will present the development of fast methods for solving wave diffraction problems using boundary integral equations. The first part (in collaboration with Stephanie Chaillat and Frédérique Le Louër) will focus on simple diffraction and the construction of analytical preconditioners, with applications in acoustics and linear elasticity. Local approximations of the associated Dirichlet-to-Neumann operator are introduced. In the second part, I will present results of ongoing work (in collaboration with Stephanie Chaillat, Martin Gander and Laurence Halpern) on multiple scattering problems. The approach combines boundary integral equations with domain decomposition techniques. 

Bruno Despres (LJLL, Sorbonne Université)

Title : Quelle serait la bonne notion de dérivabilité pour les réseaux de neurones ? 

Abstract : L'analyse mathématique des réseaux de neurones artificiels est un challenge moderne.  Néanmoins on peut noter que tout ce qui tournerait autour d'un possible cadre fonctionnel adapté est pour l'instant ouvert, ce qui est un frein manifeste pour les applications. Je montrerai pourquoi certains travaux de Murat-Trombetti 2003 semblent particulièrement séduisants pour développer un tel cadre fonctionnel à partir d'une notion de dérivation originale. Des exemples numériques seront interprétés à partir de ce cadre. 

Diane Guignard (University of Ottawa)

Title : Reproducing kernels of Hilbert spaces and application to classification

Abstract : We study the kernel trick under the eyes of the Hilbert space rather than the kernel itself. That is, instead of considering a symmetric positive definite kernel and invoking the Moore–Aronszajn theorem to infer the Hilbert space, we start from the Hilbert space itself. We show that a large family of classical Sobolev spaces are reproducing kernels Hilbert spaces and deduce universal reproducing kernels for Hm+1(ℝ2m) and Hm+1(ℝ2m+1). We then approximate reproducing kernels of (subspaces of) H2(Ω), where Ω ⊂ ℝ2  is bounded, using finite elements. Finally, we apply these kernels to classification problems using support vector machine. This is joint work with Olivier Lafitte.

Laurence Halpern (Université Sorbonne Paris Nord)

Title : Calcul PML pour les équations de Maxwell dans une boîte par la méthode des images. Part I 

Abstract : We give two talks on the analysis of perfectly matched layers (PML) introduced by Berenger in the 90's. It concerns the question of exponential decay in the layers and uniform boundedness in time of the solution. 

Part I,  by L. Halpern, presents the perfectly matched layers and employing the method of Cagniard-De Hoop and Diaz to compute the fundamental solution for the stretched d'Alembert equation on four dimensional space time. This striking formula is valid for any absorption in the layer, and is the core of analysis.

In Part II, by J. Rauch,  a second family of explicit formulas from a method of images for time stretched time dependent Maxwell equations reduces the solution of the boundary value problem for the PML with perfect conductor boundary condition (or periodic boundary conditions) to the solution of the stretched d'Alembert equation in four space.   The explicit formula from Part I completes the analysis. The questions of global boundedness and decay in the layers remains open for the absorbing boundary conditions that perform better in practice and worse in proofs.  This is so for the d'Alembert's equation as well as Maxwell's equations and dimension 1+2 as well as 1+3.

Bernard Helffer (Université de Nantes)

Title :Lower and Upper bounds for the magnetic lowest Dirichlet-to-Neumann eigenvalue  in the strong magnetic limit

Abstract : Inspired by some questions  presented  in a recent ArXiv preprint (version v1) by T. Chakradhar, K. Gittins, G. Habib and N. Peyerimhoff, we analyze their conjecture that the ground state energy of the magnetic Dirichlet-to-Neumann operator tends to infinity as the magnetic field tends to infinity. More precisely, we explore refined conjectures for general domains in R2 or R3 based on the  previous analysis in the case  of the half-plane and  the disk  by Helffer-Nicoleau. This is a work in collaboration with Ayman Kachmar and François Nicoleau. 

Lise-Marie Imbert-Gérard (University of Arizona)

Title : Quasi-Trefftz methods for a Bessel enthusiast 

Abstract :  Several plasma wave phenomena play a major role in astrophysical and experimental applications, specifically wave absorption, mode conversion, and cyclotron damping. Hierarchies of models describe various aspects of these phenomena, and this talk will focus on various aspects of the so-called cold plasma model. The presentation will start from modeling aspects and some resonance problems of interest to Olivier, leading to partial differential equations (PDEs) with variable coefficients, before turning to numerical methods specifically tailored to handle such PDEs: the so-called quasi-Trefftz methods.

Trefftz methods rely, in broad terms, on the idea of approximating solutions to PDEs via Galerkin-type methods with basis functions solving exactly the PDE locally, making explicit use of information about the ambient medium. Quasi-Trefftz methods are an extension to variable-coefficient models such as the cold plasma model. After an introduction to quasi-Trefftz methods, the talk will explore a new type of quasi-Trefftz basis functions, inspired by Olivier's contagious enthusiasm for Bessel functions. 

Erell Jamelot (Université Paris-Saclay - CEA)

Title : Explicit T -coercivity for the Stokes problem: A coercive finite element discretization 

Abstract : Using the T -coercivity theory as advocated in Chesnel and Ciarlet (2013) [25], we propose a new variational formulation of the Stokes problem which does not involve nonlocal operators. With this new formulation, unstable finite element pairs are stabilized. In addition, the numerical scheme is easy to implement, and a better approximation of the velocity and the pressure is observed numerically when the viscosity is small. 

Tien-Tai NGuyen (Hanoi University of Science)

Title : Nonlinear Rayleigh-Taylor instability and beyond 

Abstract : In this talk, I will summarize my PhD journey under the supervision of Olivier Lafitte in Villetaneuse, beginning with my work on the Rayleigh–Taylor instability. 

Jeffrey Rauch (University of Michigan)

Title : Method of Images and Perfectly Matched Layers for Maxwell in a Box. Part II 

Abstract : We give two talks on the analysis of perfectly matched layers (PML) introduced by Berenger in the 90's. It concerns the question of exponential decay in the layers and uniform boundedness in time of the solution. 

Part I,  by L. Halpern, presents the perfectly matched layers and employing the method of Cagniard-De Hoop and Diaz to compute the fundamental solution for the stretched d'Alembert equation on four dimensional space time. This striking formula is valid for any absorption in the layer, and is the core of analysis.

In Part II, by J. Rauch,  a second family of explicit formulas from a method of images for time stretched time dependent Maxwell equations reduces the solution of the boundary value problem for the PML with perfect conductor boundary condition (or periodic boundary conditions) to the solution of the stretched d'Alembert equation in four space.   The explicit formula from Part I completes the analysis. The questions of global boundedness and decay in the layers remains open for the absorbing boundary conditions that perform better in practice and worse in proofs.  This is so for the d'Alembert's equation as well as Maxwell's equations and dimension 1+2 as well as 1+3.

Olof Runborg (KTH Royal Institute of Technology Stockholm)

Title : Gaussian beam approximations of high frequency waves

Abstract : Gaussian beam superpositions are asymptotically valid high frequency solutions to linear wave equations. In this talk we discuss the construction of the beams and their accuracy in terms of the frequency. In particular, we show a pointwise error estimate at a fold caustic in the setting of Helmholtz equation. The proof uses Airy functions. 

Catherine Sulem (University of Toronto)

Title : Bloch-Floquet band gaps for water waves over a periodic bottom

Abstract : A central object in the analysis of the water wave problem is the Dirichlet-Neumann operator. This study  is devoted to  its spectrum in the context of the water wave system linearized near equilibrium in a domain with a variable bottom, assumed to be a smooth periodic function. We use the analyticity of the Dirichlet-Neumann operator with respect to the bottom variation and combine it with general properties of elliptic systems and spectral theory for self-adjoint operators to develop a Bloch-Floquet theory and describe the structure of its spectrum. We find that under some conditions on the bottom variations, the spectrum is composed of bands separated by gaps, with explicit formulas for their sizes and locations.

This is a joint work with Christophe Lacave and Matthieu Ménard. 

Kevin Zumbrun (Indiana University Bloomington)

Title : Pseudodifferential damping estimates and stability of relaxation shocks: Kreiss symmetrizers meet the Airy equation 

Abstract : A bottleneck in the theory of large-amplitude and multi-d viscous and relaxation shock stability is the development of nonlinear damping estimates controlling higher by lower derivatives. These have traditionally proceeded from time-evolution bounds based on Friedrichs symmetric and Kawashima or Goodman type energy estimates. Here, we propose an alternative program based on frequency-dependent pseudodifferential time-space damping estimates in the spirit of Kreiss. These are seen to be equivalent in the linear case to high-frequency spectral stability, and, just as for the constant-coeffcient analysis of Kreiss, sharp in a pointwise, fixed-frequency, sense. This point of view leads to a number of simplifications and extensions using already-existing analysis. More interesting is the issue of Airy type turning points, analogous to glancing points in the constant-coefficient case, previously studied by Erpenbeck and Lafitte-Williams-Zumbrun in the context of detonation.