The Artificial Beach; Absorbing Boundary Layers for Water Waves

• Jeffrey Rauch (University of Michigan)

Room: Amphithéâtre Hermite Reports on the construction of absorbing layers for the one dimensional linearized water wave equation. The equation is nonlocal forcing it immediately out of the realm of standard ideas. A key and simple is idea is one way water wave equations related to D'Alembert's method. Joint work with Izbicki, Karni, Carney, Abgrall, and Prigge (in chronological order).

About some operators over a unit disc related to the Laplace equation.

• Jean-Claude Nedelec (CMAP Polytechnique)

Room: Amphithéâtre Hermite We introduce four integral operators closely related to the Laplace equation in three-dimensions on the circular unit disc. Two of them are closed to the simple layer on the disc and the other two are related to the hyper singular operator. Contrary to the case of a closed domain, these operators no longer map fractional Sobolev spaces in a dual fashion but degenerate into different subspaces depending on their extensibility by zero. We establish their variational formulations and the coercivity properties in some Sobolev spaces. They are also linked to the Laplace operator on the disc. These results are a tentative extension to R3 of previous results in R2, contains in a common work with Carlos Jerez-Hanckes that we present in the beginning of the talk. We have introduce explicit and exact variational formulations for some weakly and hyper-singular operators associated to the Log operator overon an open flat slit as well as for their corresponding inverses.

Mathematics of the Faraday cage

• Nick TERFETHEN (University of Oxford)

Room: Amphithéâtre Hermite Everybody has heard of the Faraday cage effect, in which a wire mesh does a good job of blocking electric fields. Surely the mathematics of such a famous and useful phenomenon has been long ago worked out and written up in the textbooks? It seems to be not so. One reason may be that that the effect is not as simple as one might expect: it depends on the wires having finite radius. Nor is it as strong as one might imagine: the shielding improves only linearly as the wire spacing decreases. This talk will present results by Jon Chapman, Dave Hewett and myself including (a) numerical simulations, (b) a theorem proved by conformal mapping, (c) a continuous model derived by multiple scales analysis, (d) a discrete model derived by energy minimization, (e) a connection with the periodic trapezoidal rule for analytic integrands, and (f) a physical explanation.

Mathematicals models for the Weak Interactions in Quantum Field Theory

• Jean-Claude Guillot

Room: Amphithéâtre Hermite We consider Hamiltonians with cutoffs which are self-adjoint operators in appropriate Fock spaces with a unique ground state. A limiting Absorbtion Principle is proved

The resolution of the bounded L2 curvature conjecture in general relativity

• Jérémie Szeftel

Room: Amphithéâtre Hermite In order to control locally a space-time which satisfies the Einstein equations, what are the minimal assumptions one should make on its curvature tensor? The bounded L2 curvature conjecture roughly asserts that one should only need L2 bounds of the curvature tensor on a given space-like hypersurface. This conjecture has its roots in the remarkable developments of the last twenty years centered around the issue of optimal well-posedness for nonlinear wave equations. In this context, a corresponding conjecture for nonlinear wave equations cannot hold, unless the nonlinearity has a very special nonlinear structure. I will present the proof of this conjecture, which sheds light on the specific null structure of the Einstein equations. This talk is intended for a general audience and will require no specific background. This is joint work with Sergiu Klainerman and Igor Rodnianski.

Topology-preserving diffusion of magnetic fields

• Yann Brenier (CMLS École Polytechnique)

Room: Amphithéâtre Hermite In the 1990, Moffatt discussed a dissipative model of Magneto-hydrodynamics (that he called “magnetic relaxation” but could also be called “Darcy” or “Stokes” MHD), in order to get stationary solutions of the Euler equations with prescribed topology. We will discuss the corresponding PDEs and some concepts of generalized solutions related both to P.-L. Lions’ dissipative solutions to the Euler equations and to the recent approach by Ambrosio, Gigli and Savare of the heat equation in general metric spaces.

Periodic topographic open wave guides : theoretical and computational aspects

• Patrick Joly (INRIA)

Room: Amphithéâtre Hermite We consider the propagation of waves in a periodic structure that can be represented as a infinite thick graph. We show that, provided that adequate boundary conditions are satisfied, the introduction of a lineic geometric perurbation of this reference structure can create the apparition of guided waves associated to frequencies inside any band gap of the periodic medium. The proof is based on an asymptotic analysis with respect to the thickness of the graph. We also explain how to compute such waves. The method ins based on specific transparent conditionds for periodic media.

Time Reversed Absorbed Conditions (TRAC) and applications to inverse problems

• Frederic Nataf

Room: Amphithéâtre Hermite We introduce the time reversed absorbing conditions (TRAC) in time reversal methods. They enable to ``recreate the past'' without knowing the source which has emitted the signals that are back-propagated. The method is very insensitive to noise in the data. Applications to coefficients reconstruction and source identification are given.

Multiple scattering by small obstacles - Improved Foldy's model.

• Abderrahmane BENDALI (INSA Toulouse)

Room: Amphithéâtre Hermite Usual Foldy's model is used to approximate a multiple scattering problem involving small scatterers by monopole scatterers. Using the method of matched asymptotic expansions, with P. H. Cocquet and S. Tordeux, we have first proved that the scattered field can be approximated at any order of accuracy by multipoles. This first provides a mathematical justification for the Foldy model and next makes it possible to improve it by adding some self-interaction terms and adequately choosing the centers of phase of the effective monopoles.

Numbers don't count

• Marc Schoenauer (INRIA)

Room: Amphithéâtre Hermite Optimisation is concerned with finding the arguments that result in the largest (or lowest) value of some objective function. However, whereas the range and scale of possible values of the objective function is often arbitrary, the performance of optimisation algorithms very often heavily depends on the chosen coordinates. Comparison-based methods can hence get an edge over value-dependent approaches in many situations. Three examples (time permitting) will illustrate such situations: stochastic black-box optimisation with surrogate models; algorithm selection using Collaborative Filtering; 'Programming by Feedback', or how possibly-dummy users could actually improve their pet robot's behavior.

A two-scale approach for the issue of crack nucleation in fracture mechanics

• Jean-Jacques Marigo (LMS École Polytechnique)

Room: Amphithéâtre Hermite It is well-accepted that Griffith-like models are appropriate for crack propagation at the scale of a structure, but inadequate for the modeling of crack nucleation in brittle materials. Arguably, finer models, where a microscopic (material) length scale plays a fundamental role, are necessary to determine the critical load and crack geometry at the onset. The consistent combined modeling and numerical simulation of crack nucleation and propagation from the material to the structural length-scale is a challenging and largely open issue. In this talk we will consider gradient damage models and cohesive force models which contain both the concepts of critical stress and fracture energy and hence are good candidate to include in a unique setting the issues of nucleation and propagation of cracks. By considering fundamental problems as those of a thermal shock or of a notched body, we will show in particular that the nucleation of cracks predicted by such models is in fact an instability process. Theoretical arguments coupled with numerical tests will support this claim.

Adaptive time stepping and Schwarz Waveform Relaxation (SWR) Method for Compressible Navier–Stokes Equations

• Oana Ciobanu (Paris 13 et Onera)

Room: Amphithéâtre Hermite A space-time domain decomposition algorithm for the compressible Navier–Stokes problem has been designed, with the aim of implementing it in three dimensions, in an industrial code. We improve the SWR method adding an adaptive time stepping inside each time window and compare its performances for different second order explicit/implicit algorithms, on complexe cases.

Turbulent transport : the interplay of Physics, Mathematics and High Performance Computing

• Georges-Henri Cottet

Room: Amphithéâtre Hermite In this talk we show how numerical models and algorithms can be tailored to HPC platforms to address multiscale problems occurring in turbulent transport.

Collective dynamics in living and social systems

• Pierre Degond (Imperial College London)

Room: Amphithéâtre Hermite Collective dynamics refers to the ability of motile agents to achieve large-scale coordination through purely local interactions. Systems exhibiting collective dynamics can be found in the living world (motor proteins, cells, birds, pedestrians) as well as in the social world (opinion, wealth). Collective dynamics challenges the existing theories relating microscopic to macroscopic dynamics. In this talk, we will review some of these mathematical challenges.

A Local Wave Tracking Strategy for Solving High-Frequency Helmholtz Problems

• Mohamed Amara (Université de Pau)

Room: Amphithéâtre Hermite the presentation deals about a procedure for selecting basis function orientation to improve the e?fficiency of solution methodologies that employ local plane-wave approximations. The proposed adaptive approach consists of a local wave tracking strategy. Each plane-wave basis set within considered elements of the mesh partition is individually or collectively rotated to best align one function of the set with the local propagation direction of the field. Systematic determination of the approximated local direction of the field inside the computational domain is formulated as a minimization problem. To illustrate the salient features and evaluate the performance of the proposed wave tracking approach, error estimates as well as numerical results are presented for the case of a least-squares method (LSM) using plane-wave based approach. The numerical results obtained for the case of a two-dimensional rigid scattering problem indicate that (a) convergence was achievable to a prescribed level of accuracy, even upon initial application of the tracking wave strategy outside the pre-asymptotic convergence region, (b) the proposed approach reduced the size of the resulting system by up to two orders of magnitude, depending on the frequency range, with respect to the size of the standard LSM system

On seismic imaging

• Björn Engquist (University of Texas)

Room: Amphithéâtre Hermite There are several computational steps in seismic imaging. We will discuss a few, including algorithms for fast high frequency wave propagation and the application of Wasserstein metric to full waveform inversion.