PDE and applications seminar | Nancy

In this talk, I will briefly recall the historical Kalman filter and its ensemble form. Then I will show how the latter has been successfully implemented for data assimilation, in particular in numerical weather forecasting. More recently, the Ensemble Kalman Filter has been proposed as a methodology for solving very general inverse problems in high-dimensional contexts. I will present the theory, show some simple applications and point out the numerous open problems that remain.

A dynamically protected cat-qubit is an open quantum system that stabilizes a two-dimensional subspace (called code space) of a quantum harmonic oscillator and shows very promising robustness to noise. Experimental realizations of cat-qubits rely on reservoir engineering, a method of coupling a high-quality cavity with a dissipative cavity. In this talk, after an introduction to the mathematics of open quantum systems, we will present a new generalized LaSalle invariance principle to prove the long-time convergence of a cat-qubit to the code space.

De nombreuses applications en traitement d’images (débruitage, segmentation), en science des données (lissage de nuages de points, associations de formes), en sciences des matériaux (évolution des grains dans les alliages, croissance des cristaux) ou en biologie (modélisation cellulaire) nécessitent l’approximation de l’évolution d’interfaces géométriques telles que l’emblématique mouvement par courbure moyenne.
Dans ce contexte, la méthode des champs de phase est un outil particulièrement efficace pour approcher
l’évolution des surfaces orientées, mais les choses se révèlent beaucoup plus difficiles pour les surfaces non orientées.
Dans cet exposé, nous expliquerons comment approcher de telles évolutions en entraînant des réseaux de neurones dont les structures dérivent des schémas classiques de discrétisation de l’équation d’Allen Cahn.
Des applications numériques aux problèmes de Steiner et de Plateau seront aussi proposées.

This talk addresses the question of the interplay between relaxation and irreversibility through
evolution processes in damage mechanics, by inquiring the following question: can the quasi-static
evolution of an elastic material undergoing a process of plastic deformation be derived as the limit
model of a sequence of quasi-static brittle damage evolutions?
This question is motivated by the static analysis led in [1], where the authors have shown
how the brittle damage model introduced by Francfort and Marigo (see [4]) can lead to a model
of (Hencky) perfect plasticity. Problems of damage mechanics being rather described through
evolution processes, it is natural to extend this analysis to quasi-static evolutions, where the inertia
is neglected. We consider the case where the medium is subjected to time-dependent boundary
conditions, in the one-dimensional setting. The idea is to combine the scaling law introduced in [1]
with the quasi-static brittle damage evolution introduced in [3] by Francfort and Garroni, and try
to understand how the irreversibility of the damage process will be expressed in the limit evolution.
Surprisingly, the interplay between relaxation and irreversibility of the damage is not stable
through time evolutions. Indeed, depending on the choice of the prescribed Dirichlet boundary
condition, the effective quasi-static damage evolution obtained may not be of perfect plasticity
type.
References:
[1] J.-F. Babadjian, F. Iurlano, F. Rindler: Concentration versus oscillation effects in brittle damage, Comm.
Pure Appl. Math. 74 (2021) 1803–1854.
[2] G. Dal Maso, A. DeSimone, M. G. Mora: Quasistatic evolution problems for linearly elastic-perfectly plastic
materials, Arch. Ration. Mech. Anal. 180 (2006) no. 2, 237–291.
[3] G. A. Francfort, A. Garroni: A Variational View of Partial Brittle Damage Evolution, Arch. Rational
Mech. Anal 182 (2006) 125–152.
[4] G. A. Francfort, J.-J. Marigo: Revisiting brittle fracture as an energy minimization problem, J. Mech.
Phys. Solids 46 (1998) 1319–1342.

Dans cet exposé, on discutera un résultat récent obtenu en collaboration avec Caroline Lasser et Didier Robert.

Il s’agit de la construction d’approximations du propagateur associé à un opérateur de Schrödinger semi-classique matriciel.

La méthode utilisée repose sur l’utilisation de paquets d’onde gaussiens et notre résultat justifie les méthodes numériques de « multiple spawning » utilisées en chimie quantique.

We study the existence of solutions to a kinetic system
describing the dynamics of a large number of particles undergoing the
effect of a self-generated field (electrical or gravitational) and the
action of random jumps in velocity according to a $2\sigma$-stable
Poisson process. The evolution of the corresponding system can be seen
as a fractional version of the classical Valsov-Poisson-Fokker-Planck
systems in which the dissipating part is described by a fractional
Laplacian. We address the question of local existence in time of mild
solutions for this system in all natural ranges $0 < \sigma < 1$ thanks
to the use of commutator estimates à la Kato-Ponce. We also investigate
the possibility of propagating the lifespan of these solutions in the
range $\frac{1}{2} < \sigma < 1$ and get global solutions in a natural
weighted $L^2$ space, which is possible thanks to the use of fundamental
solutions combined with an approach due to Bouchut (\emph{J. Funct.
Analysis} Vol 111(1) 1993 pp 239-258.).

Lorsque nous résolvons numériquement un problème de contrôle optimal gouverné par des équations aux dérivées partielles instationnaires, les conditions d’optimalité donnent des systèmes avec un grand nombre d’équations fortement couplées. Il est donc souhaitable de résoudre de tels systèmes en parallèle sur plusieurs processeurs. L’approche classique consiste à décomposer le domaine spatial en plusieurs sous-domaines pour obtenir des problèmes plus petits à résoudre en parallèle. Une autre possibilité intéressante est de décomposer le domaine temporel pour obtenir des méthodes “parallèles en temps”. Dans cet exposé, je présenterai deux méthodes de résolution basées sur une telle décomposition : la première utilise uniquement des communications entre sous-domaines voisins, alors que la deuxième nécessite la résolution d’un système global, mais de taille réduite. Je démontrerai la convergence des deux méthodes lorsque l’EDP est de type diffusif. Je présenterai enfin quelques exemples numériques pour montrer le comportement de ces algorithmes en fonction du nombre de sous-domaines.

In this talk, we present a posteriori estimates for finite element approximations of nonlinear elliptic problems satisfying strong-monotonicity and Lipschitz-continuity properties. These estimates include, and build on, any iterative linearization method that satisfies a few clearly identified assumptions; this encompasses the Picard, Newton, and Zarantonello linearizations. The estimates give a guaranteed upper bound on an augmented energy difference (reliability with constant one), as well as a lower bound (efficiency up to a generic constant). We prove that for the Zarantonello linearization, this generic constant only depends on the space dimension, the mesh shape regularity, and possibly the approximation polynomial degree in four or more space dimensions, making the estimates robust with respect to the strength of the nonlinearity. For the other linearizations, there is only a computable dependence on the local variation of the linearization operators. We also derive similar estimates for the energy difference. Numerical experiments illustrate and validate the theoretical results, for both smooth and singular solutions.