Laboratoire de Mathématiques de Besançon

We investigate theoretically the propagation of acoustic waves in a two-dimensional array of cylindrical pillars on the surface of a semi-infinite substrate. Through the computation of the band structure of the periodic array and of the transmission of waves through a finite length array, we show that the phononic crystal can support a number of surface propagating modes in the nonradiative region of the substrate, or sound cone, as limited by the slowest bulk acoustic wave. The modal shape and the polarization of these guided modes are more complex than those of classical surface waves propagating on a homogeneous surface. Significantly, an in-plane polarized wave and a transverse wave with sagittal polarization appear that are not supported by the free surface. In the band structure, guided modes define band gaps that appear at frequencies markedly lower than those expected from the Bragg interference condition. We identify them as originating from local resonances of the individual cylindrical pillars and show their dependence on the geometrical parameters, in particular with the height of the pillars. The transmission of surface acoustic waves across a finite array of pillars shows the signature of the locally resonant band gaps for surface modes and their dependence on the symmetry of the source and its polarization. Numerical simulations are performed by using the finite element method and considering silicon pillars on a silicon substrate.

From a ferromagnetic resonance experiment on a YIG|GGG|YIG crystal, phonons are shown to be able to provide coherent long-range coupling between two magnets, which act as ``microphones'' and ``speakers'' for acoustic waves to induce feedback interference over millimeter distances. In this ``phononic spin valve'' under perpendicular magnetization, the phonons, which are circularly polarized, carry efficiently angular momentum current through a nonmagnetic dielectric. The high acoustic quality of phonon transport in garnets and the strong coupling to the magnetic order may also be useful for quantum communication.

This paper tries to summarize recent results on the controllability ofsystems of (several) parabolic equations. The emphasis is placed on theextension of the Kalman rank condition (for finite dimensional systems ofdifferential equations) to parabolic systems. This question is itself tiedwith the proof of global Carleman estimates for systems and leads to a widefield of open problems.

Abstract. We study a class of quasi-linear Schrödinger equations arising in the theory of superfluid film in plasma physics. Using gauge transforms and a derivation process we solve, under some regularity assumptions, the Cauchy problem. Then, by means of variational methods, we study the existence, the orbital stability and instability of standing waves which minimize some associated energy. Contents

We study the nonlinear Schrödinger equation arising in dipolar Bose--Einstein condensate in the unstable regime. Two cases are studied: the first when the system is free, the second when gradually a trapping potential is added. In both cases we first focus on the existence and stability/instability properties of standing waves. Our approach leads to the search of critical points of a constrained functional which is unbounded from below on the constraint. In the free case, by showing that the constrained functional has a so-called mountain pass geometry, we prove the existence of standing states with least energy, the ground states, and show that any ground state is orbitally unstable. Moreover, when the system is free, we show that small data in the energy space scatter in all regimes, stable and unstable. In the second case, if the trapping potential is small, we prove that two different kinds of standing waves appear: one corresponds to a topological local minimizer of the constrained energy functional and consists in ground states, and the other is again of mountain pass type but now corresponds to excited states. We also prove that any ground state is a topological local minimizer. Despite the problem being mass supercritical and the functional being unbounded from below, the standing waves associated to the set of ground states turn out to be orbitally stable. Actually, from the physical point of view, the introduction of the trapping potential stabilizes the system that is initially unstable. Related to this we observe that it also creates a gap in the ground state energy level of the system. In addition when the trapping potential is active the presence of standing waves with arbitrary small norm does not permit small data scattering. Eventually some asymptotic results are also given.

We consider the existence of normalized solutions in H 1 (ℝ N ) × H 1 (ℝ N ) for systems of nonlinear Schr¨odinger equations, which appear in models for binary mixtures of ultracold quantum gases. Making a solitary wave ansatz, one is led to coupled systems of elliptic equations of the form and we are looking for solutions satisfying where a 1 > 0 and a 2 > 0 are prescribed. In the system, λ 1 and λ 2 are unknown and will appear as Lagrange multipliers. We treat the case of homogeneous nonlinearities, i.e. , with positive constants β , μ i , p i , r i . The exponents are Sobolev subcritical but may be L 2 -supercritical. Our main result deals with the case in which in dimensions 2 ≤ N ≤ 4. We also consider the cases in which all of these numbers are less than 2 + 4 /N or all are bigger than 2 + 4 /N .

We introduce a Nitsche-based finite element discretization of the unilateral contact problem in linear elasticity. It features a weak treatment of the nonlinear contact conditions through a consistent penalty term. Without any additional assumption on the contact set, we can prove theoretically its fully optimal convergence rate in the $H^1(\Omega)$-norm for linear finite elements in two dimensions, which is $O(h^{\frac{1}{2}+\nu})$ when the solution lies in $H^{\frac{3}{2}+\nu}(\Omega)$, $0<\nu\leq 1/2$. An interest of the formulation is that, as opposed to Lagrange multiplier-based methods, no other unknown is introduced and no discrete inf-sup condition needs to be satisfied.

Abstract We study the following nonlinear scalar field equation Here , m &gt; 0 is a given constant and arises as a Lagrange multiplier. In a mass subcritical case but under general assumptions on the nonlinearity f , we show the existence of one nonradial solution for any , and obtain multiple (sometimes infinitely many) nonradial solutions when N = 4 or . In particular, all these solutions are sign-changing.

Abstract Extreme quantile regression provides estimates of conditional quantiles outside the range of the data. Classical quantile regression performs poorly in such cases since data in the tail region are too scarce. Extreme value theory is used for extrapolation beyond the range of observed values and estimation of conditional extreme quantiles. Based on the peaks-over-threshold approach, the conditional distribution above a high threshold is approximated by a generalized Pareto distribution with covariate dependent parameters. We propose a gradient boosting procedure to estimate a conditional generalized Pareto distribution by minimizing its deviance. Cross-validation is used for the choice of tuning parameters such as the number of trees and the tree depths. We discuss diagnostic plots such as variable importance and partial dependence plots, which help to interpret the fitted models. In simulation studies we show that our gradient boosting procedure outperforms classical methods from quantile regression and extreme value theory, especially for high-dimensional predictor spaces and complex parameter response surfaces. An application to statistical post-processing of weather forecasts with precipitation data in the Netherlands is proposed.

Mixmod is a well-established software package for fitting mixture models of multivariate Gaussian or multinomial probability distribution functions to a given dataset with either a clustering, a density estimation or a discriminant analysis purpose. The Rmixmod S4 package provides an interface from the R statistical computing environment to the C++ core library of Mixmod (mixmodLib). In this article, we give an overview of the model-based clustering and classification methods implemented, and we show how the R package Rmixmod can be used for clustering and discriminant analysis.

We investigate the use of invariant polynomials in the construction of data-driven interatomic potentials for material systems. The 'atomic body-ordered permutation-invariant polynomials' comprise a systematic basis and are constructed to preserve the symmetry of the potential energy function with respect to rotations and permutations. In contrast to kernel based and artificial neural network models, the explicit decomposition of the total energy as a sum of atomic body-ordered terms allows to keep the dimensionality of the fit reasonably low, up to just 10 for the 5-body terms. The explainability of the potential is aided by this decomposition, as the low body-order components can be studied and interpreted independently. Moreover, although polynomial basis functions are thought to extrapolate poorly, we show that the low dimensionality combined with careful regularisation actually leads to better transferability than the high dimensional, kernel based Gaussian Approximation Potential.&#13;\n&#13;\n

A general Nitsche method, which encompasses symmetric and non-symmetric variants, is proposed for frictionless unilateral contact problems in elasticity. The optimal convergence of the method is established both for two- and three-dimensional problems and Lagrange affine and quadratic finite element methods. Two- and three-dimensional numerical experiments illustrate the theory.

Prognostics and health management (PHM) techniques for proton exchange membrane fuel cell (PEMFC) systems are of great importance for increasing their reliability and sustainability. PEMFC systems suffer from relatively poor long-term performance and durability, and prediction and prognosis can give early indications about when components should be fixed or replaced. Prognostics modeling needs to take account of a number of phenomena, including degradation mechanisms that are not easily measured. A number of works are currently investigating PHM in fuel cell systems, as well as the problem of estimating remaining useful lifetime. Any reduction in the volume of data required for making predictions is clearly advantageous. In this paper, a univariate prognostic approach based on signal processing, namely discrete wavelet transform (DWT), is proposed. The proposed approach aims at achieving an online prognostic for PEMFC systems. DWT is first introduced, and then, the predictions are built using the power signals of two different PEMFC stacks in two different scenarios, namely static and dynamic operating conditions. Results show that the method is reliable for online prediction of power, with prediction errors less than 3%.

The main goal of this paper is to propose a convergent finite volume method for a reaction–diffusion system with cross-diffusion. First, we sketch an existence proof for a class of cross-diffusion systems. Then the standard two-point finite volume fluxes are used in combination with a nonlinear positivity-preserving approximation of the cross-diffusion coefficients. Existence and uniqueness of the approximate solution are addressed, and it is also shown that the scheme converges to the corresponding weak solution for the studied model. Furthermore, we provide a stability analysis to study pattern-formation phenomena, and we perform two-dimensional numerical examples which exhibit formation of nonuniform spatial patterns. From the simulations it is also found that experimental rates of convergence are slightly below second order. The convergence proof uses two ingredients of interest for various applications, namely the discrete Sobolev embedding inequalities with general boundary conditions and a spacetime L 1 compactness argument that mimics the compactness lemma due to Kruzhkov. The proofs of these results are given in the Appendix.

Max-stable processes play an important role as models for spatial extreme events. Their complex structure as the pointwise maximum over an infinite number of random functions makes their simulation difficult. Algorithms based on finite approximations are often inexact and computationally inefficient. We present a new algorithm for exact simulation of a max-stable process at a finite number of locations. It relies on the idea of simulating only the extremal functions, that is, those functions in the construction of a max-stable process that effectively contribute to the pointwise maximum. We further generalize the algorithm by Dieker & Mikosch (2015) for Brown-Resnick processes and use it for exact simulation via the spectral measure. We study the complexity of both algorithms, prove that our new approach via extremal functions is always more efficient, and provide closed-form expressions for their implementation that cover most popular models for max-stable processes and multivariate extreme value distributions. For simulation on dense grids, an adaptive design of the extremal function algorithm is proposed.

We consider a stationary nonlinear Schröodinger equation with a repulsive delta-function impurity in one space dimension. This equation admits a unique positive solution and this solution is even. We prove that it is a minimizer of the associated energy on the subspace of even functions of $H^1(\R, \C)$, but not on all $H^1(\R, \C)$, and study its orbital stability.

We describe how the Szlenk index has been used in various areas of the geometry of Banach spaces. We cover the following domains of application of this notion: non existence of universal spaces, linear classification of C(K) spaces, descriptive set theory, renorming problems and non linear classification of Banach spaces.

We study a renewal risk model in which the surplus process of the insurance company is modelled by a compound fractional Poisson process. We establish the long-range dependence property of this nonstationary process. Some results for ruin probabilities are presented under various assumptions on the distribution of the claim sizes.

We follow up on our previous work [C. Le Bris, F. Legoll and A. Lozinski,\nChinese Annals of Mathematics 2013] where we have studied a multiscale finite\nelement (MsFEM) type method in the vein of the classical Crouzeix-Raviart\nfinite element method that is specifically adapted for highly oscillatory\nelliptic problems. We adapt the approach to address here a multiscale problem\non a perforated domain. An additional ingredient of our approach is the\nenrichment of the multiscale finite element space using bubble functions. We\nfirst establish a theoretical error estimate. We next show that, on the problem\nwe consider, the approach we propose outperforms all dedicated existing\nvariants of MsFEM we are aware of.\n

This paper addresses the construction of different families of absorbing boundary conditions for the one- and two-dimensional Schrödinger equations with a general variable nonlinear potential. Various semidiscrete time schemes are built for the associated initial boundary value problems. Finally, some numerical simulations give a comparison of the various absorbing boundary conditions and associated schemes to analyze their accuracy and efficiency.