# Séminaire Equations aux dérivées partielles

### organisé par l'équipe Modélisation et contrôle

• ##### Amina Mecherbet

Autour de l'équation de Transport-Stokes

10 janvier 2023 - 14:00Salle 301

L'équation de Transport-Stokes modélise la sédimentation d'une suspension de particules à faible fraction volumique dans un fluide visqueux. Le système est un couplage entre une équation de Stokes pour le fluide et une équation de transport pour la fonction de densité qui désigne la probabilité de présence des particules dans le fluide.

Dans cet exposé je rappellerai dans un premier temps l'origine de la dérivation d'un tel modèle ainsi que les résultats d'existence et d'unicité connus pour des données initiales de type $L^1\cap L^\infty$.

Je présenterai ensuite des résultats récents obtenus en collaboration avec Franck Sueur concernant certaines propriétés des solutions : existence et unicité pour des données initiales de type $L^1 \cap L^p$, $p\geq3$, analyticité des trajectoires et contrôlabilité du système.

Enfin si le temps le permet, j'évoquerai certaines questions ouvertes liées à la modélisation de la sédimentation d'une gouttelette.
• ##### Emmanuel Franck

Soutenance HDR

17 janvier 2023 - 14:00Salle de conférences IRMA

Soutenance de l'HDR d'Emmanuel Franck, à 14h en salle de conférences de l'IRMA. Sujet de l'HDR : Numerical methods for conservation laws. Application to gas dynamics and plasma physics
• ##### Eloi Martinet

REPORTÉ (trains supprimés) -- Maximisation des valeurs propres du Laplacien avec condition de Neumann

24 janvier 2023 - 14:00Salle de conférences IRMA

On s'intéresse au problème d'optimisation de formes consistant à maximiser les valeurs propres du Laplacien avec conditions de Neumann homogènes. Ces valeurs propres interviennent notamment dans des problèmes acoustiques ou thermiques et sont en particulier liées à la "hot spot conjecture". Contrairement aux valeurs propres de Dirichlet, celles associées au problème de Neumann sont de nature plutôt instables, ce qui rend le problème d'optimisation difficile. On verra comment certaines explorations numériques du problème pour des domaines du plan et de la sphère ont permis de mettre en évidence certaines propriétés des optima.
• ##### Stefania Fresca

Deep learning-based reduced order models for parametrized PDEs

31 janvier 2023 - 14:00Salle de conférences IRMA

The solution of differential problems by means of full order models (FOMs), such as, e.g., the finite element method, entails prohibitive computational costs when it comes to real-time simulations and multi-query routines. The purpose of reduced order modeling is to replace FOMs with reduced order models (ROMs) characterized by much lower complexity but still able to express the physical features of the system under investigation. Conventional ROMs anchored to the assumption of modal linear superimposition, such as proper orthogonal decomposition (POD), may reveal inefficient when dealing with nonlinear time-dependent parametrized PDEs, especially for problems featuring coherent structures propagating over time. To overcome these difficulties, we propose an alternative approach based on deep learning (DL) algorithms, where tools such as convolutional neural networks (CNNs) are used to build an efficient nonlinear surrogate. In the resulting DL-ROM, both the nonlinear trial manifold and the nonlinear reduced dynamics are learned in a non-intrusive way by relying on DL models trained on a set of FOM snapshots, obtained for different parameter values [Fresca et al. (2021a), Fresca et al. (2022)]. Accuracy and efficiency of the DL-ROM technique are assessed in several applications, ranging from cardiac electrophysiology [Fresca et al. (2021b)] to fluid dynamics [Fresca et al. (2021c)], showing that new queries to the DL-ROM can be computed in real-time. Finally, with the aim of moving towards a rigorous justification on the mathematical foundations of DL-ROMs, error bounds are derived for the approximation of nonlinear operators by means of CNNs. The resulting error estimates provide a clear interpretation on the role played by the hyperparameters of dense and convolutional layers. Indeed, by exploiting some recent advances in Approximation Theory, and unvealing the intimate relation between CNNs and the discrete Fourier transform, we are able to characterize the complexity of the neural network in terms of depth, kernel size, stride, and number of input-output channels [Franco et al. (2023)]. References S. Fresca, A. Manzoni, L. Dede’ 2021a A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs. Journal of Scientific Computing, 87(2):1-36. S. Fresca, A. Manzoni 2022 POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition. Computer Methods in Applied Mechanics and Engineering, 388, 114181. S. Fresca, A. Manzoni L. Dede’, A. Quarteroni 2021b POD-enhanced deep learning-based reduced order models for the real-time simulation of cardiac electrophysiology in the left atrium. Frontiers in Physiology, 12, 1431. S. Fresca, A. Manzoni 2021c Real-time simulation of parameter-dependent fluid flows through deep learning-based reduced order models. Fluids, 6(7), 259. N. R. Franco, S. Fresca, A. Manzoni, P. Zunino 2023 Approximation bounds for convolutional neural networks in operator learning. Neural Networks, Accepted.
• ##### Andrea Thomann

REPORTÉ (grève) -- Semi-implicit schemes for all Mach number flows

7 février 2023 - 14:00Salle de séminaires IRMA

When considering multi-physics applications described by hyperbolic models, flow regimes, in comparison to single phase flows described by the Euler equations, are not characterized by one Mach number only. Examples are two fluid flows, where each phase is characterized by its own Mach number depending on the sound speed of the respective medium, or the simulation of elastic materials where, in addition to the standard acoustic Mach number, a shear Mach number depending on the shear modulus, describing the elastic shear stiffness of the material, can be defined.

The characteristic speeds of these models scale with the inverse Mach number inflicting a very restrictive CFL condition on the time step for standard explicit schemes.

Consequently, to avoid vanishing time steps, for near incompressible flows especially, implicit or implicit-explicit time integrators are necessary.

Moreover, the monitoring of sound waves is usually less in the focus of a numerical simulation.

Following the slower material waves and contact waves yields a less restrictive, Mach number independent CFL condition, which is advantageous when these slow dynamics are observed over a long time.

In this talk we address implicit explicit time integration approaches for hyperbolic models involving the above mentioned applications as well as issues and difficulties arising in the construction of the corresponding finite volume scheme.
• ##### Guillaume Ferriere

Théorie de Cauchy et ondes progressives pour l'équation de Gross-Pitaevskii logarithmique

28 février 2023 - 14:00Salle de séminaires IRMA

On s'intéresse dans cet exposé à l'équation de Gross-Pitaevskii logarithmique (logGP), qui n'est autre que l'équation de Schrödinger non-linéaire logarithmique (logNLS) dans le contexte de solutions dont le module tend vers 1 à l'infini. La première partie concerne le problème de Cauchy, pour lequel les techniques classiques pour Gross-Pitaevskii avec non-linéarité polynomiale mais également celles utilisées pour logNLS se sont révélées infructueuses. Pour obtenir une bonne théorie de Cauchy, notre preuve de l'existence d'une solution adapte la méthode par compacité utilisée par Ginibre et Velo pour NLS. L'unicité découle du caractère lipschitzien du flot dans L^2 comme pour logNLS. Dans un deuxième temps, on s'intéresse aux ondes progressives, et en particulier au cas 1d, pour lequel plusieurs conclusions similaires au cas avec non-linéarité polynomiale découlent : au-delà d'une certaine vitesse critique explicite, aucune onde progressive n'existe; en deçà, les ondes progressives non-constantes sont uniques à invariants près. Ce travail a été réalisé en collaboration avec R. Carles.
• ANNULÉ (grève)

7 mars 2023 - 14:00Salle de conférences IRMA

• ##### Nelly Boulos Al Makary

Analysis of the Shallow Water equations with two velocities

14 mars 2023 - 14:00Salle de conférences IRMA

In this work, we are interested in the analysis of the "Shallow water model with two velocities". First, we study the steady state solutions using the Bernouilli's principle for $C^1$ regular solutions and the Rankine-Hugoniot relations through discontinuities. Then, we present the types of solutions, their existence and their uniqueness depending on the boundary conditions. Second, we propose several finite volume approximate Riemann solvers for the resolution of the homogeneous Shallow water model with two velocities. The construction of the schemes is based on a recent analysis of the Riemann problem. We present several test cases to illustrate the behavior and the properties of the schemes. Afterwards, we extend these schemes for the model with topography and we propose a suitable numerical approximation of the source term. We prove that the proposed schemes are well-balanced and ensure the positivity of the water heights. Finally, we study the numerical stability of the stationary solutions.
• ##### Frédéric Valet

Collision of two solitary waves for the Zakharov-Kuznetsov equation

21 mars 2023 - 14:00Salle de conférences IRMA

The Zakharov-Kuznetsov (ZK) equation in dimension 2 is a generalization in plasma physics of the one- dimensional Korteweg de Vries equation (KdV). Both equations admit solitary waves, that are solutions moving in one direction at a constant velocity, vanishing at infinity in space. When two solitary waves collide, two phenomena can occur: either the structure of two solitary waves is conserved without any loss of energy and change of sizes (elastic collision), or the structure is lost or modified (inelastic collision). As a completely integrable equation, KdV only admits elastic collisions. The goal of this talk is to explain the collision phenomenon for two solitary waves having almost the same size for ZK, and to describe the inelasticity of the collision. The talk is based on current works with Didier Pilod.
• ##### Frédéric Valet

Collision of two solitary waves for the Zakharov-Kuznetsov equation

21 mars 2023 - 14:00Salle de conférences IRMA

The Zakharov-Kuznetsov (ZK) equation in dimension 2 is a generalization in plasma physics of the one-dimensional Korteweg de Vries equation (KdV). Both equations admit solitary waves, that are solutions moving in one direction at a constant velocity, vanishing at infinity in space. When two solitary waves collide, two phenomena can occur: either the structure of two solitary waves is conserved without any loss of energy and change of sizes (elastic collision), or the structure is lost or modified (inelastic collision). As a completely integrable equation, KdV only admits elastic collisions. The goal of this talk is to explain the collision phenomenon for two solitary waves having almost the same size for ZK, and to describe the inelasticity of the collision. The talk is based on current works with Didier Pilod.