Site du séminaire EDP

A central question in deep learning is how deep neural networks (DNNs) learn features. DNN layers progressively collapse data into a regular low-dimensional geometry. This collective effect of nonlinearity, noise, learning rate, width, depth, and numerous other parameters, has eluded first-principles theories which are built from microscopic neuronal dynamics. We discovered a noise–nonlinearity phase diagram that highlights where shallow or deep layers learn features more effectively. I will describe a macroscopic mechanical theory of feature learning that accurately reproduces this phase diagram, offering a clear intuition for why and how some DNNs are ``lazy'' and some are ``active'', and relating the distribution of feature learning over layers with test accuracy. Joint work with Cheng Shi and Liming Pan.

In this talk, we discuss the Riemann problem for a new hyperbolic model governing two-phase thin film flow. In the first part of the talk, we discuss the modelling of surface tension driven two-phase thin film flow under the influence of an anti-surfactant and obtain a reduced 4×4 triangular hyperbolic model in one-dimension under the assumption that the solute is perfectly soluble and there are negligble capillarity and diffusivity effects. In the second part of the talk, we will discuss about the Riemann problem and an exact and Godunov solver for this reduced model. We will end our talk with some future directions for higher order stable schemes based on generalized Riemann problems also referred as GRP solvers. This is joint work with Christian Rohde (University of Stuttgart) and Yue Wang (IAPCM, Beijing, China).

I will talk about the hydrostatic approximation of the Euler-Boussinesq equations, describing the evolution of an inviscid stratified fluid where the vertical length scale is much smaller than the horizontal one. Even though of importance in oceanography, the justification of the hydrostatic limit in this context has remained an open problem. I will discuss some recent results showing that some instability mechanisms may prevent this limit to hold. This is joint work with R. Bianchini (CNR Rome) and M. Coti Zelati (Imperial College London).

De nombreux travaux récents en traitement du signal audio s'intéressent à la question suivante : "peut-on entendre la forme d'une pièce ?". Bien que cette question s'inspire du célèbre article de Kac "Peut-on entendre la forme d’un tambour ?", les deux problèmes diffèrent considérablement. En effet, Kac examine l’unicité de la forme d’un tambour à partir des fréquences propres de l’opérateur de Laplace-Dirichlet. En pratique, il est impossible d'accéder directement aux fréquences ou fonctions propres à partir de mesures réelles. Nous considérons donc ici le problème inverse, plus réaliste, de la reconstruction de la forme d’une pièce à partir de mesures de réponses impulsionnelles effectuées en un nombre fini de positions de microphones dans la pièce. Deux approches très distinctes seront introduites dans cette présentation. La première se base sur le modèle des sources images, qui ne prend en compte que les réflexions spéculaires de l'impulsion sonore sur les murs (conditions de Neumann). La méthode de reconstruction développée utilise des outils issus de la super-résolution pour estimer les positions des sources images à partir de signaux temporels, puis en déduire les paramètres géométriques de la pièce. La seconde approche repose sur des méthodes d'optimisation de forme, en intégrant des conditions aux limites plus réalistes dans le domaine fréquentiel.

Despite the supreme importance of fluid flow models, the well-posedness of three-dimensional viscous and inviscid flow equations remains unsolved. Promising efforts have recently evolved around the concept of statistical solutions. In this talk, we present stochastic lattice Boltzmann methods for efficiently approximating statistical solutions to the incompressible Navier–Stokes equations in three spatial dimensions. Space-time adaptive kinetic relaxation frequencies are used to find stable and consistent numerical solutions along the inviscid limit toward the Euler equations. With single level Monte Carlo and stochastic Galerkin methods, we approximate responses, e.g., from initial random perturbations of the flow field. The novel combinations of schemes are implemented in the parallel C++ data structure OpenLB and executed on heterogeneous high-performance computing machinery. Based on exploratory computations, we search for scaling of the energy spectra and structure functions in terms of Kolmogorov’s K41 theory. For the first time, we numerically approximate the limit of statistical solutions of the incompressible Navier–Stokes solutions toward weak-strong unique statistical solutions of the incompressible Euler equations in three dimensions. Applications to wall-bounded turbulence and the potential to generate training data for novel generative artificial intelligence algorithms are discussed.

Electronic structure calculations are widely used to predict the physical properties of molecules and materials. They require to solve nonlinear partial differential and eigenvalue equations. These equations are generally numerically very demanding, especially since they are parameterized by the positions of the nuclei in the molecule and must be solved a large number of times when these positions vary. This is the case for example when simulating the dynamics of a molecule. In this talk, I will present a recent work aimed at efficiently calculating approximate solutions of such parameterized PDEs, with the objective of reducing the overall computational time. For this, I will present a non-linear interpolation method between several solutions, based on optimal transport, and using in particular Wasserstein barycenters. I will illustrate this method with simulations carried out on a 1D toy model.

In recent decades, scientific machine learning, utilizing deep learning methodologies, has found widespread application in the fields of scientific computing and computational engineering. That includes learning the neural networks as a solver and learning functions by the neural operators. Neural network solver can be quite promising after a stroke of luck and proper training. However, they generally yield solution with limited accuracy and exhibit potential issues in network generalizability. Besides, unlike the classical numerical linear algebra methods, purely data-driven neural network solvers lack theoretical convergence guarantee. In this talk, we focus on learning different neural operators for getting a preconditioner to accelerate the solution of the parametric Helmholtz equations by the classical Krylov subspace methods. Given the goal is to learning a preconditioner rather than a solver, the required accuracy is not high and so does the training cost. In order to learning an effective neural network preconditioner without re-training, we train various neural operators with different neural network architectures. Furthermore, we investigate the influence of the settings of training datasets and the tuning of hyper-parameters of different neural operators. Finally, we successfully test network generalizability of trained operator from different physical aspects. We also success in applying it to accelerating the solution of a challenging practical human head CT scan dataset on 64 times larger domain size. In short, this operator learning part illustrates that the performance of the trained inference depends on the choosing of network architecture, the setting of training datasets, and the tuning of hyper-parameters of the neural networks. For the neural network solver part, we focus on improving its accuracy by incorporating an idea from classical subspace method. Furthermore, we illustrate the trade-off between higher accuracy and network generalizability, and the difference between learning neural operator and neural network solver. In conclusion, this presentation demonstrates the efficiency of neural operator learning and neural network solver for improving the simulation of the parametric Helmholtz equations.

The Boltzmann equation can be derived rigorously from a system of elastic hard spheres (Lanford’s theorem). In the case of large systems of particles that interact inelastically (sand, snow, interstellar dust), the derivation of the inelastic Boltzmann equation is still open. One major difficulty, already at the microscopic level, comes from the phenomenon of inelastic collapse, when infinitely many collisions take place in finite time.

Assuming that the restitution coefficient r is constant, we obtain general results of convergence and asymptotics concerning the variables of the dynamical system describing a collapsing system of particles. We prove a complete classification of the singularities when a collapse of three particles takes place, obtaining only two possible orders of collisions between the particles: either the particles arrange in a nearly-linear chain (studied in [3]), or they form a triangle, and we show that, after sufficiently many collisions, the particles collide according to a unique order of collisions, which is periodic. Finally, we construct explicit initial configurations leading to a nearly-linear collapse in a stable way, such that the angle between the particles at the time of collapse can be chosen a priori, with an arbitrary precision.

The results are taken from [1] and [2], obtained in collaboration with Juan J. L. Velázquez (Universität Bonn).

[1] Théophile Dolmaire, Juan J. L. Velázquez, “Collapse of inelastic hard spheres in dimension d ≥ 2”, to appear in Journal of Nonlinear Science, preprint arXiv:2402.13803v2 (02/2024).
[2] Théophile Dolmaire, Juan J. L. Velázquez, “Properties of some dynamical systems for three collapsing inelastic particles”, preprint arXiv:2403.16905 (03/2024).
[3] Tong Zhou, Leo P. Kadanoff, “Inelastic collapse of three particles”, Physical Review E, 54:1, 623–628 (07/1996).

In this talk, we will present a new finite difference method, on regular grid, well suited for elliptic problems posed in a domain given by a level-set function. It is inspired by the phi-FEM paradigm which is a fictitious finite element method imposing the boundary conditions thanks to a level-set function describing the domain. We will consider here the Poisson equation with Dirichlet condition.We will prove the optimal convergence of our finite difference method in some Sobolev norms. Moreover, the discrete problem is proven to be well conditioned, i.e. the condition number of the associated matrix is of the same order as that of a standard method on a comparable mesh. We will then give some numerical results that confirm the optimal convergence in the considered Sobolev norms. An other advantage of our approach is that it uses standard libraries such as Numpy and Scipy in Python, and the implementation is very short (less than 100 lines in Python), making it a very low-cost numerical scheme in terms of computation time.

In this talk, I will first present key results from our recent works on the numerical analysis of lattice Boltzmann schemes, which ignore boundary conditions. These schemes can be used to simulate systems of conservation laws. Next, I will theoretically examine boundary conditions in lattice Boltzmann methods, focusing on a simplified two-unknowns model. By mapping lattice Boltzmann schemes to finite difference schemes, we enable rigorous consistency and stability analyses. We propose kinetic boundary conditions for inflow and outflow scenarios, addressing the trade-off between accuracy and stability, which we successfully mitigate. Consistency is analyzed using modified equations, while stability is assessed via GKS (Gustafsson, Kreiss, and Sundström) theory. For coarse meshes, where GKS theory may falter, we employ spectral and pseudo-spectral analyses of the scheme's matrix to explain low-resolution effects. Lastly, I will discuss potential research directions related to boundary conditions in kinetic schemes.

This talk aims at presenting a subcell monolithic DG/FV convex property preserving scheme solving system of conservation laws on 2D unstructured grids. This is known that discontinuous Galerkin (DG) method needs some sort of nonlinear limiting to avoid spurious oscillations or nonlinear instabilities which may lead to the crash of the code. The main idea motivating the present work is to improve the robustness of DG schemes, while preserving as much as possible its high accuracy and very precise subcell resolution. To do so, a convex blending of high-order DG and fist-order finite volume (FV) scheme will be locally performed at the subcell scale where it is needed. To this end, we first prove that it is possible to rewrite DG scheme as a subcell FV scheme on a subgrid provided with some specific numerical fluxes referred to as DG reconstructed fluxes. Then, the monolithic DG/FV scheme will be defined as following: to each face of each subcell will be assigned two fluxes, a 1st-order FV one and a high-order reconstructed one, that will be in the end blended in a convex way. The goal is now to determine, through analysis, optimal blending coefficients to achieve the desire properties (for instance positivity, non-oscillatory, entropy inequalities) while preserving the high accuracy of the scheme. Numerical results on various type problems will be presented to assess the very good performance of the design method. A particular emphasis will be put on entropy consideration. By means of this subcell monolithic framework, we will attempt to address the following questions: what do we mean by entropy stability? What is the cost of such constraints? Is this absolutely needed?

We present a numerical study of solutions to equations appearing in the theory of surface waves, concretely Boussinesq systems (integrable and non-integrable examples) and Serre-Green-Naghdi (SGN) equations. Solitary waves in 1D are constructed, and there stability is studied numerically. The time evolution of localised initial data is explored. Of special interest is the role of the non-cavitation condition. The appearence of dispersive shock waves, zones of rapidly modulated oscillations, in the vicinity of shocks of the corresponding dispersionless systems is studied. In the context of the SGN equations, these questions are also addressed in 2D.

This is work in collaboration with V. Duchene, S. Gavrilyuk and J.-C. Saut.

In this talk, I will present two ways to reduce the computational cost of flood numerical models. The first feature is the subscale parametrization thanks to porosity modeling. It has first been introduced by V. Guinot in th 2000s and is still at the core of LEMON’s scientific production. It allows reducing the cost of simulation thanks to coarser meshes, while the subscale informationis captured with pre-computations on the domain topography. The second part of the talk will be dedicated to large CFL models to handle heterogeneous meshes with explicit time-schemes. The main idea is to consider hydraulic information not only on the neighbouring cells of any flux interface, but to gather it from a pre-defined dependence domain thanks to a convolution process. This provides much larger time steps (hence a reduced computational cost) without any mesh modification or accuracy loss. These two features are part of the team SW2D-LEMON, developed at Inria and Université Montpellier.

Wave propagation simulations are the core of numerous applications and they have reached a high level of fidelity thanks to a continuous improve in numerical modelling and computational resources. When simulating wave propagation in the Earth’s crust, the properties of the propagation domain are subject to large epistemic uncertainties due to the difficulty of conducting geophysical measurements. However, the computational costs of physics-based simulations in three-dimensional (3D) heterogeneous domains prevent uncertainty analyses via a Monte Carlo-like approach. I introduce a surrogate model based on a Multiple Input Fourier Neural Operator (MIFNO), an extension of the popular Fourier Neural Operator [Li et al, 2021]. Fourier Neural Operators rely on the Fast Fourier Transform to learn the frequential representation of Partial Differential Equations (PDEs). Our MIFNO predicts the solution of the 3D elastic wave equation from the properties of the propagation domain and the initial condition. Its main specificities are: - a factorized architecture that limits the number of parameters and improves the scalability - a depth-to-time conversion that predicts 3D time-dependent variables without a 4D surrogate - an implementation depending on the input representation (structured grids and vectors) I will describe the theoretical foundations of the MIFNO architecture, illustrate its prediction ability and quantify the prediction error. I will also show the benefits of transfer learning to fine-tune the MIFNO on a real earthquake and improve its accuracy. This allows us to quantify uncertainties on the solution of the elastic wave equation.

Les lois de bilan scalaires sont des équations de réaction-advection qui apparaissent naturellement dans des contextes physiques / biologiques. Elles sont obtenues par un bilan entre deux instants proches de la variation de la quantité d'intérêt. Dans cette présentation, je me concentrerai sur certaines solutions propagées à vitesse constante, qui connectent deux états d'équilibres distincts. En particulier, je discuterai la stabilité de ces solutions appelées des fronts.

Dans cet exposé, nous utiliserons des outils d'acoustique, de modélisation et d'analyse numérique afin de mieux comprendre le fonctionnement d'une trompette actuellement conservée au sein du Musée de la Musique à Paris. Nous montrerons comment la simulation directe couplée à une méthode d'inversion permettent de reconstruire de façon non destructive, la forme interne de l'instrument de musique, paramètre prépondérant au son émis. À partir de données tomographiques, une première perce (rayon interne de l'instrument) est reconstruite et permet le calcul de la réponse linéaire de l'instrument. Cette dernière est comparée à des données expérimentales de même nature et un problème inverse permet d'affiner la reconstruction. Ces calculs d'acoustique linéaire sous forme mixte en pression et débit se basent sur une discrétisation en espace par une méthode d'éléments finis non standard dont la convergence repose sur des éléments de preuve originaux. A partir de la perce reconstruite et dont le comportement linéaire est validé expérimentalement, une comparaison sonore est souhaitable. La discrétisation en temps repose sur la garantie d'un bilan de puissance au niveau discret, et s'appuie sur un schéma de Störmer-Verlet dans la partie linéaire du tuyau. Ce dernier est prouvé stable pour une source impulsionnelle, y compris lorsque le pas de temps approche sa plus grande valeur admissible, grâce à des éléments de preuve originaux. Enfin, des sons de trompette sont comparés entre ceux d'un musicien jouant une copie de la trompette réalisée à partir du plan issu de la reconstruction de perce, et ceux d'une simulation sonore de la trompette couplée à un modèle rudimentaire non linéaire d'embouchure. Ce travail a fait l'objet d'une collaboration entre la Cité de la Musique- Philharmonie de Paris, le Centre de Recherche et de Restauration des Musées de France, l'Institut Technique Européen des Métiers de la Musique, le fabricant de trompettes Jérôme Wiss et l'équipe MAKUTU de l'Inria Bordeaux.

J’expliquerai comment on peut dériver et justifier des modèles asymptotiques pour l’équation des vagues dans l’hypothèse d’eaux dites profondes.

Dans cette présentation, je parlerai des théorèmes de Brenier en transport optimal.

Ces théorèmes ont la particularité de se traduire particulièrement bien en algorithmes numériques, grâce (entre autres) aux travaux de Benamou, de Mérigot et de Gallouet.

Je montrerai des applications en dynamique des fluides et en cosmologie.

PDE learning is an emerging field at the intersection of machine learning, physics, and mathematics, that aims to discover properties of unknown physical systems from experimental data. Popular techniques exploit the approximation power of deep learning to learn solution operators, which map source terms to solutions of the underlying PDE. Solution operators can then produce surrogate data for data-intensive machine learning approaches such as learning reduced order models for design optimization in engineering and PDE recovery. In most deep learning applications, a large amount of training data is needed, which is often unrealistic in engineering and biology. However, PDE learning is shockingly data-efficient in practice. We provide a theoretical explanation for this behavior by constructing an algorithm that recovers solution operators associated with elliptic PDEs and achieves an exponential convergence rate with respect to the size of the training dataset. The proof technique combines prior knowledge of PDE theory and randomized numerical linear algebra techniques and may lead to practical benefits such as improving dataset and neural network architecture designs.

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Recently, operator learning, or learning mappings between infinite-dimensional function spaces, has garnered significant attention, notably in relation to learning partial differential equations from data. Conceptually clear when outlined on paper, neural operators necessitate discretization in the transition to computer implementations. This step can compromise their integrity, often causing them to deviate from the underlying operators, with practical consequences.

This talk introduces a new take on neural operators, with a novel framework, Representation equivalent Neural Operators, designed to deal with the aforementioned issue. At its core is the concept of operator aliasing, which measures inconsistency between neural operators and their discrete representations. These concepts will be introduced and their practical applications will be discussed, introducing a novel a convolutional based neural operator.

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Des variantes probabilistes des méthodes de réduction de modèle ont émergé récemment pour améliorer les performances des approches existantes, aussi bien bien en terme de stabilité que d'efficacité. Dans cet exposé, nous présentons une méthode des bases réduites probabiliste pour l'approximation d'une famille de fonctions paramétrées. Ce type de méthode repose sur algorithme "greedy" (glouton) probabiliste utilisant un estimateur d'erreur sous la forme d'une espérance d'une variable aléatoire paramétrée. En pratique, des algorithmes de type MC ou bandit peuvent être considérés. Ces algorithmes ont été testés pour l'approximation de famille de fonctions paramétrées pour lesquelles nous avons accès uniquement à des évaluations ponctuelles (bruitées). En particulier, nous avons considéré l'approximation de la "variété" des solutions, d' EDP paramétrées, admettant une représentation probabiliste par le biais de la formule de Feynman-Kac.

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Despite recent advances in Reinforcement Learning (RL), the Markov Decision Processes are not always the best choice to model complex dynamical systems requiring interactions at high frequency. Being able to work with arbitrary time intervals, Continuous Time Reinforcement Learning (CTRL) is more suitable for those problems. Instead of the Bellman equation operating in discrete time, it is the Hamiltonian Jacobi Bellman (HJB) equation that describes value function evolution in CTRL. Even though the value function is a solution of the HJB equation, it may not be its unique solution. To distinguish the value function from other solutions, it is important to look for the viscosity solutions of the HJB equation. The viscosity solutions constitute a special class of solutions that possess uniqueness and stability properties. This paper proposes a novel approach to approximate the value function by training a Physics Informed Neural Network (PINN) through a specific ε-scheduling iterative process constraining the PINN to converge towards the viscosity solution and shows experimental results with classical control tasks.

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Le transfert de chaleur dans le globe oculaire humain, est fortement influencé par divers paramètres physiologiques et externes. En particulier, il affecte de manière critique le comportement des fluides dans l'œil et les processus d'administration de médicaments. Cependant, la modélisation nécessite la connaissance de divers paramètres, dont certains peuvent jouer un rôle essentiel dans le développement de pathologies. Bien que certaines données médicales aient été récemment acquises, seuls quelques paramètres et leur variabilité sont connus, tandis que d'autres ne peuvent être directement mesurés. À cette fin, un modèle 3D pour simuler le transfert de chaleur dans dans l’œil humain est développé. Afin d'identifier les principaux facteurs influençant le comportement du transfert de chaleur, il est nécessaire d'étudier l'influence de ces paramètres à travers un processus de quantification d'incertitude qui implique de nombreuses évaluations des modèles. Cependant, ce processus est coûteux. Par conséquent, l'utilisation d'une approche de réduction de modèle s'avère essentielle pour réduire le coût de calcul. Dans cette présentation, nous présenterons la méthode des bases réduites avec bornes d’erreur comme moyen de réduire le modèle sans compromettre la précision. Ce modèle réduit permettra l'utilisation des indices de Sobol, une approche statistique, pour évaluer l'influence des paramètres du modèle sur les résultats.

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This talk addresses linear and nonlinear elliptic problems on non-periodic, perforated domains. With the domains representing realistic urban geometries, our goal is to model floods in urban areas. For the linear model, we introduce a novel coarse space that is spanned by locally discrete harmonic basis functions that are piecewise polynomial along subdomain boundaries. We combine the coarse approximation with local subdomain solves in a two-level domain decomposition method. For the nonlinear Diffusive Wave model, we present nonlinear preconditioning techniques that allow us to significantly reduce iteration counts when compared to Newton's method.

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An entropy stable, positivity preserving Godunov-type scheme for multidimensional hyperbolic systems of conservation laws on unstructured grids was presented by Gallice et al. in [1]. A specific feature of their Riemann solver is coupling all cells in the vicinity of the current one thanks to a nodal parameter: the velocity of the nodes. Consequently, this Riemann solver is no longer 1D across one edge. Contrarily, it encounters genuine multidimensional effects. In this presentation, we extend their work to handle source terms, with a specific application to the shallow water system. The scheme we obtain is well balanced in 1D and 2D. We show that the numerical scheme appears to be insensitive to the numerical instability known as Carbuncle in supersonic flows. We also investigated the reasons behind the good behaviour of this numerical scheme with respect to this instability. To conclude the presentation, we discuss possible research paths. In particular, we are investigating (with promising results) whether the knowledge of multidimensional effects can improve the numerical results for low-Mach flows. References [1] G. Gallice, A. Chan, R. Loubère, P.-H. Maire. Entropy Stable and Positivity Preserving Godunov-Type Schemes for Multidimensional Hyperbolic Systems on Unstructured Grid. Journal of Computational Physics, Volume 468, 2022, 111493, ISSN 0021-9991, https://doi.org/10.1016/j.jcp.2022.111493.

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The Statistical Wave Field Theory provides the general equations which govern the statistics of a reverberant acoustic field, expressed as a function of the geometric and physical parameters of a room. In order to introduce this theory, I will make a parallel with two well-known physical theories:

- Statistical physics establishes the macroscopic thermodynamic properties of gases from the laws of quantum mechanics governing microscopic particles. In the same spirit, the statistical wave field theory allows us to determine the macroscopic properties of a wave field in an enclosure (in terms of power distribution and statistical dependencies, through space, time and frequencies), from local physical laws: the wave equation in 3 dimensions and its boundary conditions (Neumann and Robin).

- The theory of relativity is twofold: in its special version, space-time is described as a flat 4-dimensional space; in its general version, which is a relativistic theory of gravitation, space-time is described as a curved space (using Riemannian geometry). The statistical wave field theory will also be presented in two parts. In its special version, we will consider rigid walls (Neumann's boundary condition) and we will show that the statistical properties are described in a flat, Euclidean space; in its general version, we will consider non-rigid walls (Robin's boundary condition) and we will show that the statistical properties are described in a curved space (the wave vector space).

In acoustics, the statistical wave field theory allows us to retrieve all the well-known properties of reverberation:

- time-frequency distribution (Polack formula);
- spatial correlation over frequency in the case of a diffuse wave field (Cook formula);
- modal density over frequency (Balian and Bloch formula);
- reverberation time in the case of a diffuse wave field (Eyring equation).

The statistical wave field theory might be a prominent tool in room acoustics, in particular because it should lead to dramatic computational savings. It should also be useful in most audio signal processing applications (including, of course, artificial reverberation and dereverberation), especially those involving spatial data (analysis/synthesis of sound scenes, source separation and localization, spatialization, etc.). Finally, since this theory is entirely based on the wave equation, it could also find applications in a variety of fields, including electromagnetism, optics and nuclear physics.

We present an extension and numerical solution of a multi-phase first order hyperbolic Unified Model of Continuum Mechanics in the Baer-Nunziato type form. It is a hyperbolic formulation of multi-phase flows, by which compressible Newtonian and non-Newtonian, inviscid and viscous fluids as well as elasto-plastic solids can be described.

Past and current research on multi-phase flow modelling mostly focuses on two-phase mathematical models. One of the most relevant, is the one originally proposed by Baer and Nunziato [1]. However, it is known that the model is not closed, i.e. the definition of these interphase terms is not unique and the generalisation of the model with more than two phases is not
clear. For this reason, in this work we intend to illustrate again how a closed multiphase model of the Baer-Nunziato type can be derived from the original theory of the SHTC systems. The SHTC theory of mixtures was first proposed by Romenski in [11, 12] for the case of two fluids and it was generalized to the case of arbitrary number of constituents in [10].

Furthermore, the Eulerian hyperelasticity equations of Godunov and Romenski are used to introduce viscous and elastic forces into this Baer-Nunziato type multi-phase hyperbolic model derived from the SHTC theory. This formulation of hyperelasticity in Eulerian coordinates, rather than the Lagrangian framework more commonly adopted in solid mechanics, is based on
the work of Godunov and Romenski [3, 4, 6, 5, 8], and in [9], Peshkov and Romenski presented the key insight that the Godunov-Romenski model can be applied not only to elasto-plastic solids, but also to fluid flows.

Hence, once the GPR theory is also introduced, we have an hyperbolic formulation of multiphase flows, by which compressible Newtonian and non-Newtonian, inviscid and viscous fluids as well as elasto-plastic solids can be described. The resulting system is large and includes highly nonlinear stiff algebraic source terms as well as non-conservative products. Consequently, the numerical solution of a multi-phase system in the multi-dimensional case, even if on a Cartesian grid, is a great challenge. For this purpose, we propose to employ a robust second-order explicit MUSCL-Hancock method on Cartesian meshes and a path-conservative technique of Castro and Pares for the treatment of non-conservative [7] products, in the context of the diffuse interface approach. Furthermore, the scheme employs a semi-analytical time integration method for the nonlinear stiff source governing the deformation relaxation, which is a rather challenging task, especially in the context of multi-phase flows. This temporal integration approach, which involves a polar decomposition of the stretching and rotation components of the distortion field, has been extended to the complete equations of the Unified Model of Continuum Mechanics in the fluid regime in [2] by Chiocchetti and Dumbser.



References

1. M.R. Baer and J.W. Nunziato. A two-phase mixture theory for the deflagration-to-detonation transition (ddt) in reactive granular materials. International Journal of Multiphase Flow, 6:861–889, 1986.

2. S. Chiocchetti and M. Dumbser. An exactly curl-free staggered semi-implicit finite volume scheme for a first order hyperbolic model of viscous two-phase flows with surface tension. Journal of Scientific Computing, 94:24, 2023.

3. S.K. Godunov. Elements of mechanics of continuous media. 1978.

4. S.K. Godunov, T.Y. Mikhaîlova, and E.I. Romenskî. Systems of thermodynamically coordinated laws of conservation invariant under rotations. Siberian Mathematical Journal, 37(4):690–705, 1996.

5. S.K. Godunov and E.I. Romenski. Nonstationary equations of the nonlinear theory of elasticity in Euler coordinates. Journal of Applied Mechanics and Technical Physics, 13:868–885, 1972.

6. S.K. Godunov and E.I. Romenski. Elements of Continuum Mechanics and Conservation Laws. 2003.

7. Carlos Parés. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM Journal on Numerical Analysis, 44(1):300–321, 2006.

8. I. Peshkov, M. Pavelka, E.I. Romenski, and M. Grmela. Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations. Continuum Mechanics and Thermodynamics, 30(6):1343–1378, 2018.

9. I. Peshkov and E.I. Romenski. A hyperbolic model for viscous Newtonian flows. Continuum Mechanics and Thermodynamics, 28:85–104, 2016.

10. Evgeniy Romenski, Alexander A. Belozerov, and Ilya M. Peshkov. Conservative formulation for compressible multiphase flows. Quarterly of Applied Mathematics, 74(1):113–136, dec 2016.

11. E I Romensky. Hyperbolic systems of thermodynamically compatible conservation laws in continuum mechanics. Mathematical and computer modelling, 28(10):115–130, 1998.

12. Evgeniy I Romensky. Thermodynamics and Hyperbolic Systems of Balance Laws in Continuum Mechanics. In E. F. Toro, editor, Godunov Methods, pages 745–761. Springer US, New York, NY, 2001.

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The remarkable flight capabilities of flapping insects are attributed to their wings, often approximated as flat, rigid plates. However, real wings consist of delicate structures, comprising veins and membranes that can undergo substantial deformation. In this presentation, we offer comprehensive numerical simulations of these deformable wings, focusing on two models: a bumblebee (Bombus ignitus) wing and a blowfly (Calliphora vicina) wing. We employ a mass-spring system that utilizes a functional approach to model the distinct mechanical behaviors of the veins and membranes of the wings. Subsequently, we conduct numerical simulations of tethered flapping insects with flexible wings using a fluid-structure interaction solver. This solver couples the mass-spring model for the flexible wing with a pseudo-spectral code that solves the incompressible Navier-Stokes equations. We apply the no-slip boundary condition through the volume penalization method, describing the time-dependent complex geometry with a mask function. This approach enables us to solve the governing fluid equations on a regular Cartesian grid. Our implementation, designed for massively parallel computers, empowers us to conduct high-resolution computations with up to 500 million grid points. The findings from this study provide insights into the role of wing flexibility in flapping flight. We observed that wing flexibility made only a minor contribution to lift or thrust enhancement. However, a significant reduction in required power suggests that wing flexibility plays a crucial role in conserving the energetic cost of flight.

The time evolution of well isolated quantum systems can be modeled by the bilinear Schrödinger equation, i.e., a standard linear Schrödinger equation, formally x’(t)=A x(t), where the state x(t) of the system at time t is a point in some L^2 space endowed with its Hilbert structure and A is a skew-adjoint linear operator. When submitted to a sufficiently weak external excitation, the dynamics of the system can be formally written x’(t)=A x(t) + u(t) B x(t). That is, the original system is perturbed by a bilinear term u(t) B x(t), where u is the real valued control and B is a fixed (usually unbounded) skew-symmetric operator.

When the unperturbed Schrödinger operator A has a pure point spectrum (that is, the ambient Hilbert space admits a basis made of eigenvectors of the Schrödinger operator) and under reasonable regularity assumptions, this bilinear system is well posed. Moreover, a classical and efficient control strategy to steer the system from one eigenstate of A to (a small neighborhood of) another is to use a periodic control law whose frequency is proportional to the difference of the corresponding eigenvalues.

In this talk, we will expose how this result can be generalized in the case where the eigenvectors of the Schrödinger operator do not span a basis of the ambient space anymore. The proof amounts to an “averaging version” of the celebrated RAGE theorem. This is an ongoing work in collaboration with Nabile Boussaïd and Marco Caponigro.

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Real world applications, e.g. the simulation of turbulent flows around airfoils, require adaptive discretizations to reduce the computational costs and degrees of freedoms. The r-adaptive method involves the re-distribution of the mesh nodes in regions of rapid variation of the solution. In comparison with h-adaptive discretizations, where the mesh is refined and coarsened by changing the number of elements in the tessellation, the r-adaptive method has some advantages, e.g. no hanging nodes appear and the number of elements does not change. On the other hand a r-adaptive method can be only used when the effect of mesh movement is appropriately accounted for the discretization.

Discontinuous Galerkin (DG) methods offer benefits for the discretization of hyperbolic conservation laws on a complex mesh geometry, since no inter-element continuity is required. Furthermore, it is known that on a static mesh DG methods have some useful theoretical properties, e.g. these methods satisfy a cell entropy inequality.

In this talk, the focus is on the construction of moving mesh nodal DG methods that satisfy a discrete entropy inequality. Thereby, a proper methodology to compute the grid point distribution to move the mesh will be not discussed. Numerical experiments as well as results from the simulation of turbulent flows around airfoils will be presented to validate the capabilities of these methods.

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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.

On s’intéresse à un problème scalaire d’homogénéisation périodique faisant intervenir deux matériaux isotropes de conductivités de signes opposés : un matériau classique et un métamatériau négatif. En raison du changement de signe des coefficients apparaissant dans les équations, il n’est pas facile d'obtenir des estimations d'énergie uniformes pour pouvoir appliquer les techniques d'homogénéisation usuelles. En utilisant la méthode de T-coercivité, on prouve le caractère bien posé du problème microscopique de départ et du problème homogénéisé, ainsi qu’un résultat de convergence. Ces résultats sont obtenus sous réserve que le contraste (négatif) entre les deux matériaux soit assez grand ou assez petit en module.

Symbolic Regression is the study of algorithms that automate the search for analytic expressions that fit data. I will introduce the state-of-the-art techniques of the field and give the basic principles of symbolic computational maths.

I will then present our work which was motivated by the fact that although recent advances in deep learning have generated renewed interest in symbolic regression, efforts have not been focused on physics, where we have important additional constraints due to the units associated with our data. I will present Φ-SO, our Physical Symbolic Optimization framework for recovering analytical symbolic expressions from physics data using deep reinforcement learning techniques by learning units constraints (https://arxiv.org/abs/2303.03192).

Adipose cells or adipocytes are the specialized cells composing the adipose tissue in a variety of species.Their role is the storage of energy in the form of a lipid droplet inside their membrane. Based on the amount of lipid they contain, one can consider the distribution of adipocyte per amount of lipid and observe a peculiar feature : the resulting distribution is bimodal, thus having two local maxima. The aim of this talk is to introduce a model built from the work in Soula & al. (2013) that is able to reproduce this bimodal feature using a Lifshitz-Slyozov model. Additionally we present some result on this model and its relation to the Becker-Döring model. We can show that under some assumptions the later converges to the former and by looking at higher order term we can build an extended diffusive Lifshitz-Slyozov model which better describes the dynamics of adipose cells. I will also present some probabilistic insight into this convergence and some numerical simulations.

This talk aims at introducing a new multi-frequency method to reconstruct width defects in waveguides. Different inverse methods already exist. However, those methods are not using some frequencies, called resonant frequencies, where propagation equations
are known to be ill-conditioned. Since waves seem very sensible to defects at these particular frequencies, we exploit them instead. After studying the forward problem at these resonant frequencies, we approach the wavefield and focus on the inverse problem. Given partial wavefield measurements, we reconstruct slowly varying width defects in a stable and precise way and provide numerical validations and comparisons with existing methods.

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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.

This talk is dedicated to a numerical method based on a random choice as proposed in Glimm's scheme. It is applied to the problem of advection of a scalar quantity. The numerical scheme proposed here relies on a fractional step approach for which: the first step is performed using any classical finite-volume scheme, and the second step is a cell-wise update. This second step is a projection based on a random choice. The resulting scheme possesses a very low level of numerical diffusion. In order to assess the capabilities of this approach, several test cases have been investigated including convergence studies with respect to the mesh-size. The algorithm performs very well on one-dimensional and multi-dimensional problems. This algorithm is very easy to implement even for multi-processor computations.

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.

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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.

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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.

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.

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.

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.

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

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.

Many engineering and scientific problems can be described by equations of fluid dynamics, namely, systems of time dependent nonlinear hyperbolic PDEs. The mathematical description of these processes as well as the numerical discretisation of the resulting PDEs will depend on the level of detail required to study them. In this talk I will focus on two directions towards higher fidelity numerical simulations: first, I present a novel structure-preserving arbitrarily high-order method that solves the nonlinear ideal magneto-hydrodynamics equations. Secondly, I will focus on our work using Machine Learning (ML) methods with the aim to improve or speed up numerical simulations, through the development of parameter-free routines as part of a numerical solver or surrogate models, with the goal of creating hybrid simulation pipelines that can improve over time.

In this talk I will survey some recent results on the neural ODE perspective of machine learning, popularised by Weinan E (2017). This point of view is particularly compelling since many central tasks in machine learning find a natural counterpart in control theory. For instance, supervised learning can be seen as a simultaneous control(lability) problem for a nonlinear ODE, in which one seeks to steer a large amount of initial data to equally as many targets by using a single control. Herein, one can quickly see the necessity of using dynamics which are nonlinear — in fact, many of the commonly used dynamics in machine learning practice are “unusual” compared to more classical control settings, and require new analysis. I will focus mainly (but not exclusively) on the optimal control perspective of supervised learning, and present convergence results for the error, and optimal controls, when the final time horizon is large. Implications to the possible generalization beyond data used for constructing the control, the required depth for the corresponding residual neural network (ResNet), and the turnpike property, will be discussed. The talk will be based on works done in collaboration with Carlos Esteve-Yague, Dario Pighin, and Enrique Zuazua.

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This work is motivated by a collaboration with the French Photovoltaic Institute. The aim of the project is to propose a model in order to simulate and optimally control the fabrication process of thin film solar cells. The production of the thin film inside of which occur the photovoltaic phenomena accounting for the efficiency of the whole solar cell is done via a Physical Vapor Deposition (PVD) process. More precisely, a substrate wafer is introduced in a hot chamber where the different chemical species composing the film are injected under a gaseous form. Molecules deposit on the substrate surface, so that a thin film layer grows by epitaxy. In addition, the different components diffuse inside the bulk of the film, so that the local volumic fractions of each chemical species evolve through time. The efficiency of the final solar cell crucially depends on the final chemical composition of the film, which is freezed once the wafer is taken out of the chamber. A major challenge consists in optimizing the fluxes of the different atoms injected inside the chamber during the process for the final local volumic fractions in the layer to be as close as possible to target profiles. Two different phenomena have to be taken into account in order to correctly model the evolution of the composition of the thin film: 1) the cross-diffusion phenomena between the various components occuring inside the bulk; 2) the evolution of the surface. As a consequence, the underlying model reads as a cross-diffusion system defined on a moving boundary domain. The complete optimal control problem of the fluxes injected in the hot chamber is currently out-of-reach in terms of mathematical analysis. The aim of this talk is to theoretically investigate a simpler problem, which is the boundary stabilization of the model used to simulate the PVD process. We show first exponential stabilization and then finite-time stabilization in arbitrary small time of the linearized system around uniform equilibria, provided the underlying cross-diffusion system has an entropic structure with a symmetric mobility matrix. This stabilization is achieved with respect to both the volumic fractions of the different chemical species composing the thin film and the thickness of the latter. The feedback control is derived using the backstepping technique, adapted to the context of a time-dependent domain. In particular, the norm of the backward backstepping transform is carefully estimated with respect to time. Joint work with Jean Cauvin-Vila and Amaury Hayat.

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Exact solutions to systems of conservation laws in multiple spatial dimensions often possess interesting additional properties which are a consequence of the equations and which can be formulated as PDEs. Examples are the evolution equations of vorticity or angular momentum, involutional constraints, stationary states or singular limits. Usually, the same numerical diffusion which stabilizes a Finite Volume method, prevents it from preserving any of those additional properties. I will show strategies how to analyse linear numerical methods on Cartesian grids, and how to modify the numerical diffusion in a truly multi-dimensional fashion in order to obtain vorticity preserving and low Mach number compliant methods. This enables the numerical methods to capture essential properties of the equations without excessive grid refinement.

Highly-oscillatory phenomena present well-known numerical challenges, such as order reduction and energy preservation. The method of high-order averaging allows to separate the "drift" dynamics from the oscillations using formal calculations. This method is used to generates new, modified problems, namely a micro-macro problem or a pulled-back problem, which can be solved with better numerical accuracy. In this talk, I will introduce high-order averaging using a (somewhat recent) closed form approach, based on an ansatz, which facilitates the discussion around such methods. The focus will mostly be on the geometric properties of the method, notably the preservation of a Hamiltonian structure. Perhaps unsurprisingly, the averaging procedure is close to that of normal forms. Therefore if time allows, I will briefly present in which context these methods are equivalent. Of course, discussions around numerical accuracy will permeate the entire talk.

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Lattice Boltzmann schemes rely on the enlargement of the size of the target problem in order to solve PDEs in a highly parallelizable and efficient kinetic-like fashion, split into a collision and a stream phase. This structure, despite the well-known advantages from a computational standpoint, is not suitable to construct a rigorous notion of consistency with respect to the target equations and to provide a precise notion of stability. In order to alleviate these shortages and introduce a rigorous framework, we demonstrate that any lattice Boltzmann scheme can be rewritten as a corresponding multi-step Finite Difference scheme on the conserved variables. This is achieved by devising a suitable formalism based on operators, commutative algebra and polynomials. Therefore, the notion of consistency of the corresponding Finite Difference scheme allows to invoke the Lax-Richtmyer theorem in the case of linear lattice Boltzmann schemes. Moreover, we show that the frequently-used von Neumann-like stability analysis for lattice Boltzmann schemes entirely corresponds to the von Neumann stability analysis of their Finite Difference counterpart. More generally, the usual tools for the analysis of Finite Difference schemes are now readily available to study lattice Boltzmann schemes. Their relevance is verified by means of numerical illustrations.

In this talk, i will present some results for the time discretization MAC schemes for the shallow water equations (SWE). The space discretization is staggered where the height is stored on the cell-center and the velocity on the cell-edges. Several time integration techniques are discussed like explicit, decoupled, pressure correction and second order Heun method. Numerical simulations will be investigated to assess the strong stability, well-balanced and accuracy first for the SWE and then for the SWE with Coriolis source term.

In the context of preserving stationary states, e.g. lake at rest and moving equilibria, a new formulation of the shallow water system, called Flux Globalization has been introduced by Cheng et al. (2019). This approach consists in including the integral of the source term in the global flux and reconstructing the new global flux rather than the conservative variables. The resulting scheme is able to preserve a large family of smooth and discontinuous steady-state moving equilibria. In this work, we focus on an arbitrary high order WENO Finite Volume (FV) generalization of the global flux approach. The most delicate aspect of the algorithm is the appropriate definition of the source flux (integral of the source term) and the quadrature strategy used to match it with the WENO reconstruction of the hyperbolic flux. When this construction is correctly done, one can show that the resulting WENO FV scheme admits exact discrete steady states characterized by constant global fluxes. We also show that, by an appropriate quadrature strategy for the source, we can embed exactly some particular steady states, e.g. the lake at rest for the shallow water equations. It can be shown that an exact approximation of global fluxes leads to a scheme with better convergence properties and improved solutions. The novel method has been tested and validated on classical cases and their perturbation: subcritical, supercritical and transcritical flows.

We study a collisionless kinetic model for plasmas in the neighborhood of a cylindrical metallic Langmuir probe. This model consists in a bi-species Vlasov-Poisson equation in a domain contained between two cylinders with prescribed boundary conditions. The interior cylinder models the probe while the exterior cylinder models the interaction with the plasma core. We prove the existence of a weak-strong solution for this model in the sense that we get a weak solution for the 2 Vlasov equations and a strong solution for the Poisson equation. The first parts of this work are devoted to explain the model and proceed to a detailed study of the Vlasov equations. This study then leads to a reformulation of the Poisson equation as a 1D non-linear and non-local equation and we prove it admits a strong solution using an iterative fixed-point procedure. Eventually we proceed to a qualitative description of the solution under the so-called "generalized Bohm condition" on the incomming fluxes and a numerical investigation of the obtained equation. Due to technical obstacles, we mainly focussed on the "quasi-radial" fluxes for the numerical analysis, which turns out to be enough to validate the model. Curves of the obtained trajectories of particles and curves of the collected current versus the applied voltage are presented.

Nous proposons de décrire les principaux aspects d’un projet de développement d’un outil de simulation de la propagation du Covid 19 dans différents contextes : établissements scolaires (écoles, collèges ou lycées), universités, et entreprises. L’approche proposée est basée sur une équation déterministe d’évolution sur un graphe dynamique dont les sommets sont des personnes ou des groupes de personnes, et dont les arêtes suivent la matrice des contacts évoluant au fil du temps. Nous décrirons certaines propriétés théoriques de versions simplifiées de ce modèle, et préciserons la manière dont il peut être interprété comme une équation de chimiotaxie discrète. Dans un second temps, nous évoquerons des applications effectives de cette approche, en particulier une étude récente effectuée au sein du CHU du Kremlin-Bicêtre impliquant 210 étudiants en médecine, dont les contacts ont été tracés pendant plusieurs mois à l’aide de petits badges portés en permanence par les volontaires. Ces travaux résultent d’une collaboration avec S. Faure (Orsay) et F. Bourdin (ENS-PSL), ainsi qu’avec l’entreprise Kerlink (pour le contact tracing).

Some fluid and kinetic systems of equations in the presence of (potentially multiple) small parameters admit so-called asymptotic regimes, where they reduce to a smaller set of equations, potentially with a different mathematical structure. However, classic numerical approaches, such as finite volume methods, do not naturally degenerate in these asymptotic regimes to consistent discretizations of the limit equations. Furthermore, even though stability conditions usually become more and more restrictive when we approach these asymptotic regimes, meaning smaller and smaller time steps, accuracy can be dramatically reduced and the results frequently unexploitable. Asymptotic preserving schemes are designed to both lift the restrictive stability conditions and remain accurate in the asymptotic regime. We introduce a new class of second-order in time and space numerical schemes, which are uniformly asymptotic preserving schemes. The proposed Implicit-Explicit (ImEx) approach, does not follow the usual path relying on the method of lines, either with multi-step methods or Runge-Kutta methods, or semi-discretized in time equations, but is inspired from the Lax-Wendroff approach with the proper level of implicit treatment of the source term. We are able to rigorously show that both the second-order accuracy and the stability conditions are independent of the fast scales in every asymptotic regime, including the study of boundary conditions. The method is also able to yield very accurate steady solutions in the nonlinear case when the source term depends on space. A thorough numerical assessment of the proposed strategy is provided by investigating smooth solutions, solutions with shocks and solutions leading to a steady state with variable source term in space. Our aim also includes plasma discharges with sheaths, where we have two small parameters related to Debye length and mass ratio, and we present some numerical simulations that assess and illustrate the potential of a method similar to the one we have introduced but applied to the isothermal Euler-Poisson equations.

In this presentation, we consider a population (typically bacteria) structured by both a spatial variable and a phenotypical trait. Our model takes into account the effects of migrations, mutations, growth and competition. When the environment is assumed homogeneous, if the population survives, it spreads to the whole space, and we have a complete picture of the large-time propagation: the solution converges towards a front, which connects a positive steady state to zero, and spreads at a determined speed.

When the environment is heterogeneous, the situation is much more complex. Depending on the profile of heterogeneities, the invasion may be either slowed or completely blocked. In some cases, the population adaption to the local environment is crucial for invasion to occur. We first consider a linear profile of heterogeneities, and then investigate the fully nonlinear case numerically as well as analytically (in a perturbative framework for the latter).

Le transport optimal est un outil naturel pour comparer de manière géométrique des distributions de probabilité. Il trouve des applications à la fois pour l'apprentissage supervisé (pour la classification) et pour l'apprentissage non supervisé (pour entrainer des réseaux de neurones génératifs). Le transport optimal souffre cependant de la "malédiction de la dimension", le nombre d'échantillons nécessaires pouvant croitre exponentiellement vite avec la dimension. Dans cet exposé, j'expliquerai comment tirer parti de techniques de régularisation entropique afin d'approcher de façon rapide le transport optimal et de réduire l'impact de la dimension sur le nombre d'échantillons nécessaires. Plus d'informations et de références peuvent être trouvées sur le site de notre livre "Computational Optimal Transport" https://optimaltransport.github.io/

Les solutions des systèmes hyperboliques contiennent des discontinuités. Ces solutions faibles vérifient non seulement les EDP de départ, mais aussi une inégalité d'entropie qui agit comme un critère de sélection déterminant si une discontinuité est physique ou non. Il est très important d'obtenir une version discrète de ces inégalités d'entropie lorsqu'on approxime numériquement les solutions, sans quoi le schéma est susceptible de converger vers des solutions non physiques ou pire d'être instable. Obtenir une inégalité d'entropie discrète est en général un travail difficile, souvent inatteignable pour des schémas d'ordre élevé. Dans cet exposé, je présenterai une approche où ces inégalités sont obtenues a posteriori en minimisant une fonctionnelle bien choisie. La difficulté principale est de prendre en compte la notion de consistance. Cette méthode permet d'obtenir des "cartes de diffusion numérique" pour des schémas d'ordre quelconque. Elle permet aussi de trouver, par une autre procédure d'optimisation, la pire donnée initiale vis à vis de l'entropie. C'est un travail en collaboration avec Emmanuel Audusse, Vivien Desveaux et Julien Salomon.

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La résolution approchée de problèmes d’évolution par des schémas de différences finies nécessite un traitement spécifique des bords, ceci de façon à tronquer artificiellement le domaine de calcul et/ou à incorporer de façon satisfaisante les conditions de bords réalistes. Ce traitement du bord affecte les propriétés de consistance et de stabilité du schéma global et est susceptible, à ce titre, de nuire de façon parfois rédhibitoire à la qualité de l’approximation. La cause typique est l’apparition de modes discrets parasites au voisinage du bord. Je présenterai comment un développement à plusieurs échelles de la solution numérique permet d’analyser ce phénomène, en concentrant la discussion sur quelques exemples simples pour le transport linéaire avec Dirichlet ou Neumann en sortie.

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It has been proved by Zuazua in 1993 that the internally controlled semilinear 1D wave equation $\partial_{tt}y-\partial_{xx}y + f(y)=v 1_{\omega}$, with Dirichlet boundary conditions, is exactly controllable in $H^1_0(0,1)\cap L^2(0,1)$ with controls $f\in L^2((0,1)\times(0,T))$, for any $T>0$ and any nonempty open subset $\omega$ of $(0,1)$, assuming that $f\in \mathcal{C}^1(\R)$ does not grow faster than $\beta\vert r\vert \ln^{2}\vert r\vert$ at infinity for some $\beta>0$ small enough. The clever proof, based on the Leray-Schauder fixed point theorem, is not constructive.

In this talk,
- we present a constructive proof and algorithm for the exact controllability of semilinear 1D wave equations.
Assuming that $f^\prime$ does not grow faster than $\beta \ln^{2}\vert r\vert$ at infinity for some $\beta>0$ small enough and that $f^\prime$ is uniformly H\"older continuous on $\R$ for some exponent $p\in[0,1]$, we design a least-squares algorithm yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order $1+p$ after a finite number of iterations.
- We extend the proof to the multidimensional case assuming that $f^\prime$ does not grow faster than $\beta \ln^{1/2}\vert r\vert$ at infinity, by using a result of Fu, Yong and Zhang in 2007.
- We show that the method also applies for the (much more intricate situation of) heat equation by considering appropriate cost functional for the controlled pair of the corresponding linearized equation, depending on parametrized Carleman weights. Large enough parameters ensure the convergence of the algorithm. We end the talk by remarking that a zero order fixed point operator derived from a zero order linearization is indeed contracting for any large enough Carleman parameter. This allows notably to greatly simplified the seminal proof of controllability due to Fernandez-Zuazua in 2000.

This talk is based on a series of recent works with Kuntal Bhandari (Clermont-Ferrand), Arthur Bottois (Clermont-Ferrand), Sylvain Ervedoza (Bordeaux), Jérome Lemoine (Clermont-Ferrand), Irène Gayte (Sevilla) and Emmanuel Trélat (Sorbonne Paris).

Solving Poisson equation is ubiquitous in Physics simulation. All numerical methods (Spectral, Finite Differences, Finite Elements, Discontinuous Galerkin) complement themselves with an ad hoc Poisson solver.
In uniform Cartesian grids with periodic boundary conditions the Fast Fourier Transform (FFT) with its complexity in O(N log(N)) –N denotes the number of points– and its spectral accuracy beats all concurrent numerical methods. In the 80’s, the multigrid methods [HACKBUSCH 85] with their complexity in O(N log(N)) (the log(N) factor stands for the number of iterations necessary to reach the accuracy corresponding to the increase of the number of points N) opened the door to efficient numerical methods suited to non periodic boundaries and immersed boundaries.
Their principle (to separate scales to apply Gauss Seidel iterations) inspired the preconditioning of powerful Linear Solvers (e.g. preconditioning of GMRES) establishing the algebraic multigrid methods. These are blind to the underlying grid structure and can be used in any contexts such as the adaptive grids for instance.
In the 90’s, the Fast Multipole Method [GREENGARD 1987] based on the integral solution of the Poisson Equation and on the properties of its Green kernel, appeared as a concurrent method efficiently addressing the adaptive context and the presence of boundaries.

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This work settles in the high-stakes emerging field of Scientific Machine Learning which studies the application of machine learning to scientific computing. More specifically, we consider the use of deep learning to accelerate numerical simulations. We focus on approximating some components of Partial Differential Equation (PDE) based simulation software by a neural network. This idea boils down to constructing a data set, selecting and training a neural network, and embedding it into the original code, resulting in a hybrid numerical simulation. Although this approach may seem trivial at first glance, the context of numerical simulations comes with several challenges stemming from an accuracy-performances trade-off. To tackle these challenges, we thoroughly study each step of the deep learning methodology while considering the aforementioned constraints. By doing so, we emphasize interplays between numerical simulations and machine learning that can benefit each of these fields. We identify the main steps of the deep learning methodology as the construction of the training data set, the choice of the hyperparameters of the neural network, its training, and the implementation of the neural network for the final use case (here, numerical simulations). In this talk, we will go through the contributions related to the third step (training) and the last step (design of a hybrid simulation code). For the third step, we formally define an analogy between stochastic resolution of PDEs and the optimization process at play when training a neural network. This analogy leads to a PDE-based framework for training neural networks that opens up possibilities for improving existing optimization algorithms. Finally, we apply these contributions to a computational fluid dynamics simulation coupled with a multi-species chemical equilibrium library. The obtained deep-learning-based hybrid code achieves an acceleration factor of 18.7 with controlled to no degradation from the prediction of the original simulation code.

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De nombreux problèmes pratiques en traitement du signal et d’images, comme les problèmes de déconvolution, consistent à essayer de reconstruire à partir d’observations altérées, bruitées et dépendant linéairement d'un signal source, ce même signal. Cette tâche étant mal posée, une famille de techniques consiste à reformuler la reconstruction à travers la résolution d’un problème d’optimisation. On cherche alors en général à reconstruire un signal dont les observations ne s’éloignent pas trop des données. Parfois, il apparaît également que le signal d'intérêt est parcimonieux dans un certain sens : par exemple s’il est composé de sources ponctuelles. Il est alors possible d’incorporer cet a priori dans le problème d’optimisation grâce à une régularisation favorisant ce genre de solutions. Dans cet exposé, nous verrons quelques garanties théoriques de reconstruction dans l’étude de tels problèmes faisant intervenir la norme de la variation totale définie sur l’espace des mesures de Radon. Nous verrons que cette régularisation favorise l’émergence de solutions parcimonieuses composées de sommes de masses de Dirac. Une originalité de cette approche réside dans son absence de discrétisation du domaine sur lequel sont définis les signaux considérés. Ceci permet notamment d’améliorer la simplicité des solutions en évitant les soucis liés aux grilles mal adaptées. Même si l’espace de recherche est de dimension infinie, nous verrons qu’il est possible de construire des méthodes de résolution, avec des garanties de convergence, basées sur l’algorithme de Frank-Wolfe. Enfin ces outils seront illustrés sur un problème de localisation de protéines sur des structures filamentaires en microscopie par fluorescence.

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A class of cell centered Finite Volume schemes has been introduced to discretize the equations of Lagrangian hydrodynamics on moving grid [Loubère, Maire, Rebourcet, 2016]. In this framework, the numerical fluxes are evaluated by means of an approximate Riemann solver located at the grid nodes, which provides the nodal velocity required to move the grid in a compatible manner. In this presentation, we describe the generalization of this type of discretization to hyperbolic systems of conservation laws written in Eulerian representation. The evaluation of the numerical fluxes relies on a nodal solver resulting from a node-based conservation condition. The construction of this nodal solver utilizes the Lagrange-to-Euler transformation introduced by Gallice [Gallice, 2003] and revisited in [Chan, Gallice, Loubère, Maire, 2021] to build positive and entropic Eulerian Riemann solvers from their Lagrangian counterparts. The application of this formalism to the case of gas dynamics provides a multidimensional Finite Volume scheme which is positive and entropic under an explicit condition on the time step. Moreover, this study allows us to rigorously recover the original scheme described in [Shen, Yan, Yuan, 2014] for the Euler equations while correcting its defects. An associated Finite Volume simulation code has been built in multi-dimensions for unstructured meshes. Parallelization has been accomplished using the MPI library embedded in PETSc. A large set of 2D/3D numerical experiments show that the proposed solver is less sensitive to spurious instabilities such as the infamous carbuncle, compared to the classical one. To further improve accuracy, the current scheme has been extended to second-order in time and space. The numerical assessment of this new method by means of representative test cases is very promising in terms of robustness.

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Dans cet exposé, on présente le développement de deux schémas volumes finis explicites d'ordre élevé pour des systèmes de lois de conservation avec terme source qui peuvent dégénérer vers des équations de diffusion. La construction se fait un choisissant le schéma limite ou en limitant la diffusion numérique. L'extension à l'ordre élevé s'effectue avec des reconstructions polynomiales et la méthode MOOD comme principe de limitation. On présente différents résultats sur maillage 2D non structuré. Ceci un travail en collaboration avec Christophe Chalons et Rodolphe Turpault.

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The equations of motion for compressible (barotropic) fluids have the structure of a simple conservative dynamical system when expressed in Lagrangian variables. This can be exposed interpreting the Lagrangian flow as a curve of vector-valued L2 functions, and the internal energy of the fluid as a functional on the same space. Particle methods are a natural discretization strategy in this setting, since in this case the flow is discretized using piecewise constant functions on a given partition of the domain, but they require some form of regularization to define the internal energy of the fluid. In this talk I will describe a particle method in which the internal energy is replaced by its Moreau-Yosida regularization in the L2 space, which can be efficiently computed as a semi-discrete optimal transport problem. I will also show how the convexity of the energy in the Eulerian variables can be exploited in the non-convex Lagrangian setting to prove quantitative convergence estimates towards smooth solution of this problem, and how this result generalizes to dissipative porous media flow.

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Je présenterai et analyserai dans cet exposé un modèle mathématique pour les écoulements compressibles sous une contrainte de densité maximale. Il s’agit de modéliser pour des mélanges biphasiques des phénomènes de saturation (congestion) correspondant à la disparition d’une des deux phases du mélange. Étant donnée une contrainte de densité maximale fixée, les solutions couplent une dynamique compressible dans les zones où la densité est inférieure à cette densité maximale, avec une dynamique incompressible dans les zones où la valeur critique est atteinte, i.e. dans les zones saturées. L'exposé portera plus particulièrement sur la discrétisation et la simulation numérique de ces équations au moyen de schémas mixtes volumes finis / éléments finis.

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We present a general new compact scheme method to achieve very high-order approximation named structural method. The main idea is to definitively separated the physics (the physical equations, including the boundary conditions) to the discretization (the structural equations). The first part is dedicated to the design of such schemes, where we show that the discretization equations are obtained from the kernel of a matrix. The second part is dedicated to one-dimensional problems in space such as convection diffusion, Burger, Euler system and the construction of fourth- or sixth-order schemes in space. The last part concerns a new class of very high order scheme in time (4th,6th), unconditionally stable and demonstrate the high efficiency and nice spectral properties of our schemes. Simulations of non-stationary problems: heat equation, wave equation, burger, Euler and Schrödinger equation, highlight the advantages of structural time-scheme regarded to traditional 4th or 6th order RK method.

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In this communication, we present a conservative cell-centered Lagrangian finite volume scheme for the solution of the hyper-elasticity model equations on unstructured multidimensional grids. The method is derived from the Eucclhyd scheme discussed in [3,1,2], and is second-order accurate in space and is combined with the a posteriori Multidimensional Optimal Order Detection (MOOD) limiting strategy to ensure robustness and stability at shock waves. Second-order of accuracy in time is achieved via the ADER (Arbitrary high order schemes using DERivatives) approach. This method has been tested in an hydrodynamics context in [4] and the present work aims at extending it to the case of hyper-elasticity models. Such models are derived in a first part in a Lagrangian framework. The dedicated Lagrangian numerical scheme is derived in terms of nodal solver, GCL compliance, subcell forces and compatible discretization. The Lagrangian numerical method has been implemented in 3D under MPI parallelisation framework allowing to handle genuinely large meshes. A relative large set of numerical test cases is presented to assess the ability of the method to achieve effective second order of accuracy on smooth flows, maintaining an essentially non-oscillatory behavior, general robustness ensuring at least physical admissibility where appropriate. Pure elastic neo-Hookean and non-linear materials are considered for our benchmark test problems in 2D and 3D. These test cases feature material bending, impact, compression, non-linear deformation and further bouncing/detaching motions. This is joint work with Walter Boscheri and Pierre-Henri Maire. [1] P.-H. Maire. A unified sub-cell force-based discretization for cell-centered Lagrangian hydrodynamics on polygonal grids. International Journal for Numerical Methods in Fluids, 65:1281–1294, 2011. [2] P.-H. Maire. A high-order one-step sub-cell force-based discretization for cell-centered lagrangian hydrodynamics on polygonal grids. Computers and Fluids, 46(1):341–347, 2011. [3] P.-H. Maire, R. Abgrall, J. Breil, and J. Ovadia. A cell-centered Lagrangian scheme for two- dimensional compressible flow problems. SIAM Journal on Scientific Computing, 29:1781–1824, 2007. [4] R. Loubère W. Boscheri, M. Dumbser and P.-H. Maire. A second-order cell-centered lagrangian ADER-MOOD finite volume scheme on multidimensional unstructured meshes for hydrodynamics. Journal of Computational Physics, 358:103 – 129, 2018.

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Plusieurs facteurs font que les écoulements sanguins veineux sont beaucoup moins étudiés que leurs équivalents artériels. Par exemple, l'arborescence veineuse présente de grande variation d'un individu à l'autre : prépondérance de certaines structures, asymétrie, absence d'autres structures. Cela rend une étude générique très complexe.
Cependant nous verrons dans cette présentation qu'en nous limitant aux écoulements sanguins cérébraux, nous arrivons à dégager empiriquement des groupes d'individus. De plus la restriction au compartiment intra-crânien nous permet d'utiliser l'hypothèse d'incompressibilité des vaisseaux sanguins veineux.
Ainsi nous verrons et comparerons plusieurs modèles pour ces écoulements mais aussi leurs possibles extensions. Nous aborderons finalement la possibilité d'utiliser des modèles réduits et les applications qui y sont associées.

Kinetic relaxation schemes are efficient numerical methods to solve an hyperbolic system. The method consists in solving a kinetic model with n_v velocities corresponding to n_v kinetic variables. However, kinetic schemes can be difficult to analyze. To do that, we propose to study the equivalent equation on n_v variables. The study of this equivalent equation allows then to obtain a simple stability analysis. We will also talk about the construction of adapted boundary conditions.

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On s'intéresse à un problème de contrôle approché de l'équation de la chaleur par des "formes" : des contrôles internes, qui en espace sont des fonctions caractéristiques d'ensembles de mesures uniformément bornées. Pour faire cela, on voit la recherche de contrôles comme la recherche de contrôles optimaux sous contraintes pour un certain coût bien choisi. En peut appliquant la dualité de Fenchel-Rockafellar, qui associe à un problème d'optimisation (dit primal) un problème dit dual, et le principe "de la baignoire", qui concerne l'optimisation sous contraintes d'un produit scalaire, on trouve le "bon problème de contrôle optimal à résoudre" en travaillant sur le problème dual. Une fois trouvé le "bon problème dual", la solution du problème dual permet de construire le contrôle optimal (sous de bonnes hypothèses). On peut alors étudier le coût de contrôlabilité en fonction du temps final, ainsi que les liens de ce problème de contrôle par des formes avec un problème de contrôle en temps minimal.

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Je présenterai d'abord les idées générales de la théorie des jeux à champ moyen, comme elle a été introduite il y a une quinzaine d'années par J.-M. Lasry et P.-L. Lions, en me concentrant en particulier sur le cas où les agents cherchent à éviter la congestion due à une densité trop élevée, ce qui se retrouve dans plusieurs modèles de trafic routier ou piétonnier. Sous certaines hypothèses sur le structure du coût, le jeu est un jeu à potentiel, où l'équilibre peut être trouvé en minimisant une énergie globale, typiquement convexe, mais d'autres modèles tout aussi raisonnables n'ont pas cette structure variationnelle. Le but principal de l'exposé sera d'expliquer les enjeux de la théorie et les liens entre EDP, calcul des variations, et théorie des jeux, avec une attention particulière au besoin de régularité des solutions. Je mentionnerai aussi les stratégies les plus courantes pour prouver des résultats d'existence (point fixe de Kakutani, méthodes variationnelles) et pour l'approximation numérique dans certains cas.

Nous étudions des systèmes couplés d'équations non linéaires de plus bas niveau de Landau, pour lesquels nous prouvons des résultats d'existence globale avec des bornes polynomiales sur la croissance possible des normes de Sobolev des solutions. Nous présentons également des trajectoires explicites non bornées qui montrent que ces bornes sont optimales. Dans un second temps, nous montrons l'existence de multi-solitons, puis nous obtenons un résultat d'unicité. Il s'agit d'un travail commun avec Valentin Schwinte (Université de Lorraine).

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Dans cette présentation, j'introduirai une nouvelle technique de réduction de modèle basée sur l'apprentissage machine pour les modèles cinétiques. Celle-ci sera développée sur le modèle de Vlasov-Poisson. En effet, il décrit l'évolution d'une distribution de particules chargées dans un champ électromagnétique. Ce champ pouvant être auto-consistant i.e. dépendant de la distribution susmentionnée, la dynamique peut être fortement non linéaire. Il existe des méthodes classiques inspirées de l'ACP mais elles sont peu efficaces dans le cas auto-consistant. Notre méthode, utilisant des réseaux de neurones, permet de réduire efficacement cette dynamique en un modèle réduit de petite dimension.

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La prise en compte de la force de Coriolis liée à la rotation de la Terre dans les équations de Saint-Venant permet de modéliser des phénomènes d'écoulement de fluide à grande échelle. L'ajout de cette force induit l'apparition de nouveaux états stationnaires à préserver. Dans ce contexte, plusieurs travaux ont proposé des schémas numériques capables de préserver ceux au repos, i.e à vitesse nulle. Dans cette présentation, nous nous intéresserons à la construction de solveurs de Riemann approchés pour le modèle 1D, qui préservent tout ou partie des états stationnaires, et qui mènent à des schémas dits "well-balanced" ou "fully well-balanced". D'autre part, l'extension à l'ordre deux d'un schéma d'ordre 1 préservant tous les états stationnaires soulève plusieurs difficultés. Nous proposerons une méthode MUSCL basée sur des détecteurs d'équilibre pour répondre à cette question.

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The Shallow Water system is one of the simplest yet accurate model describing fluid flows in a varying geometry delimited by a bottom topography and a free surface. The study of this model is valuable for numerous applications, including water management, forecasting natural disasters or understanding climate change. We are especially interested in the low Froude regime, where the material speed of fluid particles is much smaller than that of acoustic waves. This multi-scale dynamic has proven challenging to deal with from the numerical perspective, and implicit-explicit schemes are a good starting point as they allow for the use of scale-independent time steps. We then focus on the asymptotic behavior of such methods, insisting on the importance of preserving nearly-incompressible states. This is indeed a key ingredient for achieving accurate results.

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Durant ces 20 dernières années, les descriptions cinétiques de matériaux granulaires ont reçu beaucoup d'attentions de la part de la communauté mathématique, ainsi que de disciplines plus appliquées, telles que la physique et l'ingénierie. Néanmoins, de nombreux problèmes mathématiques sont toujours ouverts, que ce soit au niveau de la modélisation, de l'analyse, ou des simulations numériques de ces modèles. Cet exposé a pour but de présenter certains de ces modèles, ainsi que de résultats mathématiques (très) récents sur le sujet... Nous discuterons aussi de développement numériques récents ayant permis d'énoncer des conjectures encore ouvertes. Cet exposé sera basé sur un travail en collaboration avec J. A. Carrillo, J. Hu et Z. Ma.

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Dans ce travail nous proposons un schéma numérique, basé sur des éléments finis P1 avec condensation de masse, pour une équation parabolique non linéaire. Ce schéma permet de prendre en compte à la fois la dégénérescence et l'anisotropie sur maillages généraux. De plus, il préserve au niveau discret certaines des propriétés essentielles du système continu, telle que par exemple la dissipation de l'énergie physique. Nous nous intéressons également à l'analyse d'erreur a posteriori de cette méthode. Nous présentons plusieurs résultats numériques montrant l’efficacité de la méthode.
Ce travail a été réalisé en collaboration avec Clément Cancès et Martin Vohralik.

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Proposée récemment comme une alternative à la DFT PPLB dans l'optique d'étudier des systèmes à nombre fractionnaire d'électrons, nous détaillerons la construction mathématique de la DFT N-centrée afin de pouvoir y étendre méthodes Kohn-Sham et Kohn-Sham généralisée utilisées en DFT N-électronique et aborder des problématiques annexes (limite semi-classique, dimère de Hubbard).

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Nous nous intéresserons dans cet exposé à l'équation de Schrödinger logarithmique (abrégé en logNLS), équation non-linéaire introduite en 1976 par Białynicki-Birula et Mycielski dans un modèle de mécanique des ondes linéaires en physique. Longtemps oublié par les mathématiciens, cette équation présente une dynamique originale, parfois surprenante comparée à celle des équations de Schrödinger non-linéaires usuellement étudiées, dont les non-linéarités sont régulières voire lisses (typiquement du type puissance). J'exposerai quelques propriétés de logNLS qui attestent de cette originalité, en me focalisant sur les résultats de comportement en temps long. En particulier, sera présenté plus en profondeur le cas du régime dispersif, dont la compréhension du comportement en temps grand est très avancée : la vitesse de dispersion est plus rapide d'un facteur logarithmique et le carré du module de la solution renormalisée converge faiblement dans L^1 vers une gaussienne universelle, ne dépendant pas des conditions initiales. Je montrerai que cette description peut être améliorée par une vitesse de convergence explicite et optimale en distance de Wasserstein-1 (aussi appelé métrique de Kantorovich-Rubinstein), indépendante de la constante semi-classique, et que cette convergence est également valable à la limite semi-classique.

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À l’échelle microscopique, la locomotion à travers un fluide suit des lois différentes de celles de notre échelle (régime de Stokes). La recherche sur la natation de micro-organismes et la conception de micro-robots nageurs est notamment motivée par des applications biomédicales : par exemple, chirurgie non-invasive ou livraison d’une molécule à un endroit précis du corps. Cela nécessite bien sûr d’être capable de propulser et contrôler efficacement et précisément le micro-robot. La théorie du contrôle offre un bon cadre théorique pour répondre à ces problématiques. Ainsi, dans cet exposé, je présenterai quelques résultats de contrôlabilité et de contrôle de robots micro-nageurs. Dans un premier temps, j’étudierai la propulsion par champ magnétique externe pour un robot élastique, et montrerai qu’il n’est en général pas contrôlable au voisinage de sa position d’équilibre. Ensuite, je m’intéresserai au contrôle d’une particule à l’aide de la variation du flux environnant générée par un autre nageur ; je discuterai de la contrôlabilité du système associé et présenterai une méthode de planification de trajectoires.

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