135 research outputs found
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 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
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 . 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 . We also discuss
efficient numerical methods to solve the embedded corrector problem
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
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