Local defects are always neutral in the Thomas-Fermi-von Weisz\"acker theory of crystals

The aim of this article is to propose a mathematical model describing the electronic structure of crystals with local defects in the framework of the Thomas-Fermi-von Weizs\"acker (TFW) theory. The approach follows the same lines as that used in {\it E. Canc\`es, A. Deleurence and M. Lewin, Commun. Math. Phys., 281 (2008), pp. 129--177} for the reduced Hartree-Fock model, and is based on thermodynamic limit arguments. We prove in particular that it is not possible to model charged defects within the TFW theory of crystals. We finally derive some additional properties of the TFW ground state electronic density of a crystal with a local defect, in the special case when the host crystal is modelled by a homogeneous medium.Comment: 34 page

A dynamical adaptive tensor method for the Vlasov-Poisson system

A numerical method is proposed to solve the full-Eulerian time-dependent Vlasov-Poisson system in high dimension. The algorithm relies on the construction of a tensor decomposition of the solution whose rank is adapted at each time step. This decomposition is obtained through the use of an efficient modified Progressive Generalized Decomposition (PGD) method, whose convergence is proved. We suggest in addition a symplectic time-discretization splitting scheme that preserves the Hamiltonian properties of the system. This scheme is naturally obtained by considering the tensor structure of the approximation. The efficiency of our approach is illustrated through time-dependent 2D-2D numerical examples

Propagation des incertitudes sur des problèmes de l'obstacle en grande dimension

International audienc

Greedy algorithms for high-dimensional non-symmetric linear problems

In this article, we present a family of numerical approaches to solve high-dimensional linear non-symmetric problems. The principle of these methods is to approximate a function which depends on a large number of variates by a sum of tensor product functions, each term of which is iteratively computed via a greedy algorithm. There exists a good theoretical framework for these methods in the case of (linear and nonlinear) symmetric elliptic problems. However, the convergence results are not valid any more as soon as the problems considered are not symmetric. We present here a review of the main algorithms proposed in the literature to circumvent this difficulty, together with some new approaches. The theoretical convergence results and the practical implementation of these algorithms are discussed. Their behaviors are illustrated through some numerical examples.Comment: 57 pages, 9 figure

Convergence of a greedy algorithm for high-dimensional convex nonlinear problems

In this article, we present a greedy algorithm based on a tensor product decomposition, whose aim is to compute the global minimum of a strongly convex energy functional. We prove the convergence of our method provided that the gradient of the energy is Lipschitz on bounded sets. The main interest of this method is that it can be used for high-dimensional nonlinear convex problems. We illustrate this method on a prototypical example for uncertainty propagation on the obstacle problem.Comment: 36 pages, 9 figures, accepted in Mathematical Models and Methods for Applied Science

Greedy algorithms for high-dimensional eigenvalue problems

In this article, we present two new greedy algorithms for the computation of the lowest eigenvalue (and an associated eigenvector) of a high-dimensional eigenvalue problem, and prove some convergence results for these algorithms and their orthogonalized versions. The performance of our algorithms is illustrated on numerical test cases (including the computation of the buckling modes of a microstructured plate), and compared with that of another greedy algorithm for eigenvalue problems introduced by Ammar and Chinesta.Comment: 33 pages, 5 figure

Statistical methods for critical scenarios in aeronautics

We present numerical results obtained on the CEMRACS project Predictive SMS proposed by Safety Line. The goal of this work was to elaborate a purely statistical method in order to reconstruct the deceleration profile of a plane during landing under normal operating conditions, from a database containing around $1500$ recordings. The aim of Safety Line is to use this model to detect malfunctions of the braking system of the plane from deviations of the measured deceleration profile of the plane to the one predicted by the model. This yields to a multivariate nonparametric regression problem, which we chose to tackle using a Bayesian approach based on the use of gaussian processes. We also compare this approach with other statistical methods.Comment: 14 pages, 5 figure

Global existence of bounded weak solutions to degenerate cross-diffusion equations in moving domain

International audienceThis note focuses on some issues for the analysis of a system of degenerate cross-diffusion partial differential equations (PDEs). This family of models are encountered in a wide variety of contexts, such as population dynamics, biology, chemistry or materials science. The application we have in mind here is the modeling of the evolution of the concentration of chemical species composing a crystalline solid. The functions, that are the solutions of the system of PDEs of interest, represent the local densities of the different components of the material, and thus should be nonnegative, bounded and satisfy some volumic constraints which will be made precise later in the note. These systems are useful for instance for the prediction of the chemical composition of thin solid films grown by Chemical Vapor Deposition (CVD). In this process, a solid wafer is exposed to gaseous precursors, corresponding to the different species entering the composition of the film, which react or decompose on the substrate surface to produce the desireddeposit. This process generally occurs at high temperature and takes several hours, so that the diffusion of the different atomic species within the bulk of the solid has to be taken into account in addition to the evolution of the surface of the film

An embedded corrector problem to approximate the homogenized coefficients of an elliptic equation

We consider a diffusion equation with highly oscillatory coefficients that admits a homogenized limit. As an alternative to standard corrector problems, we introduce here an embedded corrector problem, written as a diffusion equation in the whole space in which the diffusion matrix is uniform outside some ball of radius $R$. Using that problem, we next introduce three approximations of the homogenized coefficients. These approximations, which are variants of the standard approximations obtained using truncated (supercell) corrector problems, are shown to converge when $R \to \infty$. We also discuss efficient numerical methods to solve the embedded corrector problem