Section: New Results

Various topics

Participants : Virginie Ehrlacher, Tony Lelièvre, Antoine Levitt.

In [18], T. Lelièvre has explored with J. Infante Acevedo (CERMICS) the interest of using the greedy algorithm (also known as the Proper Generalized Decomposition) for the pricing of basket options.

V. Ehrlacher and D. Lombardi have developped a new tensor-based numerical method for the resolution of kinetic equations [40] in a fully Eulerian framework.This theory enables to describe a large system of particles by a distribution function $f(x,v,t)$ that encodes the probability of finding a particle at time $t$, position $x\in {ℝ}^3$ and velocity $v\in {ℝ}^3$. These systems are used to model the behavior of plasma of the transport of electrons in semiconductors for instance. However, simulating such systems involves the resolutions of problems defined on ${ℝ}^3\times {ℝ}^3\times {ℝ}_{+}$, which leads to very high-dimensional systems. The new approach developped in [40] circumvents the curse of dimensionality for these systems, by efficiently adapting the rank of the decomposition of the solution through time. Encouraging preliminary numerical results have been obtained on $3D\times 3D$ systems.

A system of cross-diffusion equations has been proposed in [32] by A. Bakhta and V. Ehrlacher for the modelling of a Physical Vapor Deposition (PVD) process used for the manufacturing of thin film solar cells. This process works as follows: a substrate wafer is introduced in a hot chamber where different chemical species are injected under gaseous form. These different species deposit on the surface of the substrate, so that a thin film layer grows upon the surface of the substrate. Two phenomena have to be taken into account in the modelling: 1) the evolution of the thickness of the thin film layer; 2) the diffusion of the various species inside the bulk. The existence of a weak solution to the system proposed in [32] has been proved, along with the existence of optimal fluxes to be injected in the chamber in order to obtain target concentration profiles at the end of the process. The long-time behavior of solutions has been studied in the case when the injected fluxes are constant. Moreover, numerical results on the simulation of this system have been compared with experimental data given by IRDEP on CIGS (Copper, Indium, Gallium, Selenium) solar cells. The project is a collaboration with IRDEP.

A. Levitt, in collaboration with F. Aviat, L. Lagardère, Y. Maday, J.-P. Piquemal (UPMC), B. Stamm (Aachen), P. Ren (Texas) and J. Ponder (Saint Louis), has proposed a new method for the solution of the equations of polarizable force fields [8]. Previous methods had to solve a linear system to high accuracy in order for the energy to be preserved in simulations. The method presented, based on an explicit differentiation of the energy produced by the truncated iterative method, is able to conserve the energy even with loose convergence criteria, thus allowing stable and fast simulations at degraded accuracy.