C. Le Bris

Existence and Uniqueness of Solutions to Fokker–Planck Type Equations with Irregular Coefficients

We study the existence and the uniqueness of the solution to a class of Fokker–Planck type equations with irregular coefficients, more precisely with coefficients in Sobolev spaces W 1, p . Our arguments are based upon the DiPerna–Lions theory of renormalized solutions to linear transport equations and related equations [5]. The present work extends the results of our previous article [14], where only the simpler case of a Fokker–Planck equation with constant diffusion matrix was addressed. The consequences of the present results on the well-posedness of the associated stochastic differential equations are only outlined here. They will be more thoroughly examined in a forthcoming work [15].

On the thermodynamic limit for Hartree–Fock type models

We continue here our study [10–13] of the thermodynamic limit for various models of Quantum Chemistry, this time focusing on the Hartree–Fock type models. For the reduced Hartree–Fock models, we prove the existence of the thermodynamic limit for the energy per unit volume. We also define a periodic problem associated to the Hartree–Fock model, and prove that it is well-posed.

Simulations of MHD flows with moving interfaces

We report on the numerical simulation of a two-fluid magnetohydrodynamics problem arising in the industrial production of aluminium. The motion of the two non-miscible fluids is modeled through the incompressible Navier-Stokes equations coupled with the Maxwell equations. Stabilized finite elements techniques and an arbitrary Lagrangian-Eulerian formulation (for the motion of the interface separating the two fluids) are used in the numerical simulation. With a view to justifying our strategy, details on the numerical analysis of the problem, with a special emphasis on conservation and stability properties and on the surface tension discretization, as well as results on tests cases are provided. Examples of numerical simulations of the industrial case are eventually presented.

Multilevel domain decomposition for electronic structure calculations

We introduce a new multilevel domain decomposition method (MDD) for electronic structure calculations within semi-empirical and density functional theory (DFT) frameworks. This method iterates between local fine solvers and global coarse solvers, in the spirit of domain decomposition methods. Using this approach, calculations have been successfully performed on several linear polymer chains containing up to 40,000 atoms and 200,000 atomic orbitals. Both the computational cost and the memory requirement scale linearly with the number of atoms. Additional speed-up can easily be obtained by parallelization. We show that this domain decomposition method outperforms the density matrix minimization (DMM) method for poor initial guesses. Our method provides an efficient preconditioner for DMM and other linear scaling methods, variational in nature, such as the orbital minimization (OM) procedure.

A Definition of the Ground State Energy for Systems Composed of Infinitely Many Particles

Abstract We introduce here one possible definition of the energy of an infinite non-periodic system of particles. We investigate the minimization of it, and the link with standard definitions of local ground state for such systems. The specific models under study are issued from Quantum Mechanics (namely from Density-Functional type theories) but our construction and analysis can be adapted to treat other cases.

A coupled system arising in magnetohydrodynamics

Abstract We study an MHD system consisting of the stationary Maxwell equations coupled with the transient Navier-Stokes equations. We prove that a solution exists and is unique for small time and small data. We show that the system may become ill-posed as soon as the fluid velocity becomes too large.

Integral equation methods for molecular scale calculations in the liquid phase

We report here a series of works1–5 devoted to the modelling of the liquid phase for quantum chemistry calculations. The question under consideration is the computation of the electrostatic interaction between charge densities in the presence of a continuum dielectric medium. It consists of solving an elliptic problem of the form -div(∊(x) ∇ V)=ρ where the field ∊(x) exhibits a discontinuity on a closed surface. We show how an enhanced use of integral equations methods has recently led to significant progress in this field: reduction of the computational cost in the standard cases, extension of existing methods to sophisticated cases out of reach so far, development of new possibilities. This work has a wide range of applications in chemistry and biology.

OPTIMAL LASER CONTROL OF MOLECULAR SYSTEMS: METHODOLOGY AND RESULTS

We report on some mathematical and numerical work related to the control of the evolution of molecular systems using laser fields. More precisely, the control of the orientation of molecules is our goal. We treat this as an optimal control problem and optimize the laser field to be used experimentally by using both deterministic and stochastic algorithms. Comparisons between the different strategies are drawn. In particular, when gradients of the cost functional are used, the different ways for their computation are compared and analyzed.

Computing a molecule: A mathematical viewpoint

We give here an overview of the mathematical results known to this day on the models used in Quantum Chemistry for the numerical computations of molecules. We focus on the problems related to the ground state, in the framework of Hartree–Fock type models and Thomas–Fermi type models. More precisely, we outline the most recent results on the following questions: existence and uniqueness of the minimum, and existence of an optimized geometry for the nuclei. We eventually give a list of open problems.

Finite-Temperature Coarse-Graining of One-Dimensional Models: Mathematical Analysis and Computational Approaches

We present a possible approach for the computation of free energies and ensemble averages of one-dimensional coarse-grained models in materials science. The approach is based upon a thermodynamic limit process, and makes use of ergodic theorems and large deviations theory. In addition to providing a possible efficient computational strategy for ensemble averages, the approach allows for assessing the accuracy of approximations commonly used in practice.