# Laboratoire Jacques-Louis Lions

### Présentation

Le laboratoire, créé en 1969, porte le nom de son fondateur Jacques-Louis Lions. il s'agit maintenant d'une unité de recherche conjointe à l’Université Pierre et Marie Curie, à l’université Paris Diderot et au Centre National de la Recherche Scientifique.

Le Laboratoire Jacques-Louis Lions constitue le plus grand laboratoire de France et l'un des principaux au monde pour la formation et la recherche en mathématiques appliquées.

Il accueille l'activités de deux masters deuxième année ce qui représente un centaine d'étudiants. Ses activités recouvrent l’analyse, la modélisation et le calcul scientifique haute performance de phénomènes représentés par des équations aux dérivées partielles.

Fort d’environ 100 enseignants-chercheurs, chercheurs, ingénieurs, personnels administratifs permanents ou émérites, et d’autant de doctorants ou post-doctorants, il collabore avec le monde économique et avec d'autres domaines scientifiques à travers un large spectre d'applications : dynamique des fluides; physique, mécanique et chimie théoriques; contrôle, optimisation et finance; médecine et biologie; traitement du signal et des données.

### Thèmes de recherche

- Equations aux dérivées partielles et équations différentielles
- Contrôle, optimisation, calcul des variations
- Calcul scientifique, simulations numériques
- Applications des mathématiques

### [hal-01435054] Discretization error cancellation in electronic structure calculation: toward a quantitative study

Date: 20 nov 2017 - 15:35

Desc: It is often claimed that error cancellation plays an essential role in quantum chemistry and first-principle simulation for condensed matter physics and materials science. Indeed, while the energy of a large, or even medium-size, molecular system cannot be estimated numerically within chemical accuracy (typically 1 kcal/mol or 1 mHa), it is considered that the energy difference between two configurations of the same system can be computed in practice within the desired accuracy. The purpose of this paper is to provide a quantitative study of discretization error cancellation. The latter is the error component due to the fact that the model used in the calculation (e.g. Kohn-Sham LDA) must be discretized in a finite basis set to be solved by a computer. We first report comprehensive numerical simulations performed with Abinit [1,2] on two simple chemical systems, the hydrogen molecule on the one hand, and a system consisting of two oxygen atoms and four hydrogen atoms on the other hand. We observe that errors on energy differences are indeed significantly smaller than errors on energies, but that these two quantities asymptotically converge at the same rate when the energy cutoff goes to infinity. We then analyze a simple one-dimensional periodic Schrödinger equation with Dirac potentials, for which analytic solutions are available. This allows us to explain the discretization error cancellation phenomenon on this test case with quantitative mathematical arguments.

### [hal-01333627] Local exact controllability of the 2D-Schrödinger-Poisson system

Date: 18 juin 2016 - 00:26

Desc: In this article, we investigate the exact controllability of the 2D-Schrödinger-Poisson system, which couples a Schrödinger equation on a bounded domain of R 2 with a Poisson equation for the electrical potential. The control acts on the system through a Neumann boundary condition on the potential, locally distributed on the boundary of the space domain. We prove several results, with or without nonlinearity and with dierent boundary conditions on the wave function, of Dirichlet type or of Neumann type.

### [tel-01446511] Null controllability of a Korteweg-de Vries equation and of a coupled parabolic equations system. Stabilisation in finite time by means of non-stationary feedback

Date: 26 jan 2017 - 09:55

Desc: This doctoral thesis focuses on three fields of Control Theory: the control on the edge of the Korteweg-de Vries equation, the control of three heat equations coupled by cubic terms, and the stabilisation in finite time of three classic systems of finite dimension. For the KdV equation, we first demonstrate a Carleman inequality using a well-chosen exponential weight, then we deduce the controllability at zero of the equation. For the system of three heat equations coupled by cubic terms, we show the global controllability at zero even though the linearized system around zero is not controllable. We apply the return method to obtain local controllability: we build control system trajectories going from zero to zero and whose linearised systems are controllable. Then a scale change allows us to obtain a global result. Finally, concerning the three systems of finite dimension, these systems are controllable systems but the linearised systems are not controllable and are not stabilised with means of continuous stationary feedback. We construct an explicit time-dependent feedback leading to a stabilisation in finite time. For this we deal with different parts of systems during different intervals of time.

### [hal-01135304] Analysis of anatomical variability using diffeomorphic iterative centroid in patients with Alzheimer's disease

Date: 6 jan 2016 - 12:17

Desc: This paper presents a new approach for template-based analysis of anatomical variability in populations,in the framework of Large Deformation Diffeomorphic Metric Mappings and mathematical currents. We propose a fast approach in which the template is computed using an diffeomorphic iterative centroid method. Statistical analysis is then performed on the initial momenta that define the deformations between the centroid and each individual subject. We applied the approach to study the variability of the hippocampus in 134 patients with Alzheimer’s disease (AD) and 160 elderly control subjects. We show that this approach can describe the main modes of variability of the two populations and can predict the performance to a memory test in AD patients.

### [hal-01133786] Two-dimensional simulation by regularization of free surface viscoplastic flows with Drucker-Prager yield stress and application to granular collapse

Date: 16 déc 2016 - 15:18

Desc: This work is devoted to numerical modeling and simulation of granular flows relevant to geophysical flows such as avalanches and debris flows. We consider an incompressible viscoplastic fluid, described by a rheology with pressure-dependent yield stress, in a 2D setting with a free surface. We implement a regularization method to deal with the singularity of the rheological law, using a mixed finite element approximation of the momentum and incompressibility equations, and an arbitrary Lagrangian Eulerian (ALE) formulation for the displacement of the domain. The free surface is evolved by taking care of its deposition onto the bottom and of preventing it from folding over itself. Several tests are performed to assess the efficiency of our method. The first test is dedicated to verify its accuracy and cost on a one-dimensional simple shear plug flow. On this configuration we setup rules for the choice of the numerical parameters. The second test aims to compare the results of our numerical method to those predicted by an augmented Lagrangian formulation in the case of the collapse and spreading of a granular column over a horizontal rigid bed. Finally we show the reliability of our method by comparing numerical predictions to data from experiments of granular collapse of both trapezoidal and rectangular columns over horizontal rigid or erodible granular bed made of the same material. We compare the evolution of the free surface, the velocity profiles, and the static-flowing interface. The results show the ability of our method to deal numerically with the front behavior of granular collapses over an erodible bed.