Séminaire Equations aux dérivées partielles
organisé par l'équipe Modélisation et contrôle
-
Corentin Houpert
Physics-Informed Autoencoders for filling missing values in CO2 measurements
13 janvier 2026 - 14:00Salle de conférences IRMA
Missing values in measurements for carbon dioxide emissions on drained peatlands remains an open challenge for training forecasting techniques to achieve net zero. At the field scale, existing methods struggle to model CO_2 emissions to fill gaps, especially in nighttime measurements. We propose robust Physics-Informed Autoencoders (PIAEs), which combine the generative capabilities of Autoencoders with the reliability of physical models of Net Ecosystem Exchange (NEE) that quantify CO_2 exchanges between the atmosphere and major carbon pools. Our method integrates equations describing the physical processes and associated uncertainties to fill gaps in NEE measurements from eddy covariance (EC) flux towers. In the PIAE, various sensor measurements are encoded into the latent space, and a set of decoders is then used to approximate the ecosystem parameters and the optimal NEE forecast, directed by dynamics described by a stochastic differential equation. These decoders utilize nighttime and daytime NEE models that describe carbon transfer as a Wiener process. Finally, we use a two-phased training routine with two loss functions describing each phase: Mean Squared Error (MSE) and Maximum Mean Discrepancy (MMD) between the measurements and the reconstructed samples. PIAE outperforms the current state-of-the-art Random Forest Robust on the prediction of nighttime NEE measurements on various distribution-based and data-fitting metrics. We present significant improvement in capturing temporal trends in the NEE at daily, weekly, monthly and quarterly scales. -
Raphaël Bulle
Adaptive multi-mesh finite element method for the spectral fractional Laplacian
27 janvier 2026 - 14:00Salle 301
Fractional partial differential equations have gained interest in the last 15 years due to their ability to model non-local behavior (such as e.g. anomalous diffusion) with a relatively small number of parameters. These advantages come with drawbacks from the numerical perspective as these problems raise new challenges regarding simulations. In this talk we are interested in a particular fractional problem, based on the spectral fractional Laplacian operator. More specifically, we look at the discretization and the design of adaptive mesh refinement strategies to efficiently solve such problem numerically. To do so we consider a particular framework based on a rational scheme coupled with a finite element method. We derive an a posteriori error estimator for the finite element discretization error which is then used to steer an adaptive refinement loop. Finally, we will introduce a novel multi-mesh adaptive refinement algorithm taking advantage of the rational scheme to further optimize the discretization. -
Barbara Verfürth
Multiscale methods for wave propagation in spatio-temporal metamaterials
10 février 2026 - 14:00Salle de conférences IRMA
Time modulation of materials has become increasingly popular as a further design approach for metamaterials to induce unusual wave phenomena. In the mathematical modelling, this leads to partial differential equations with coefficients that are highly varying in space and/or time. Since direct numerical simulations are prohibitively expensive, multiscale methods are required to efficiently approximate at least the macroscopic behavior. At the example of the classical wave equation, we will discuss two recent approaches in that direction. One approach is more analytical and aims to deduce higher-order effective equations for temporal multiscale coefficients using asymptotic expansions. The other approach is more numerical, where we present the construction of non-polynomial, multiscale basis functions combined with standard time stepping schemes numerical for spatial multiscale coefficients with additional slow time variations. -
Marie Boussard
Estimation of numerical entropy loss via a projection method
3 mars 2026 - 14:00Salle de conférences IRMA
Ocean circulation models cover a spatial domain up to the planetary scale. Due to computational constraints, numerical simulations therefore rely on relatively coarse meshes, with cell areas of the order of 10 000 km^2. As a consequence, finite volume schemes used for discretization induce a significant numerical entropy loss. This phenomenon leads to unphysical behaviours in the simulation result, such as the mixing of water masses with distinct temperatures and salinities. To address this problem, referred to as "spurious mixing" by oceanographers, localizing and quantifying this numerical entropy loss in high-order codes is crucial. A method was proposed by Aguillon, Audusse, Desveaux and Salomon, but is limited to the one-dimensional case.
I will introduce a method suitable for simulations in two dimensions of space, including high order schemes and in the presence of source terms. This approach relies on projecting a consistent flux onto the set of fluxes satisfying a discrete entropy inequality. The projection is carried out by an optimization algorithm with inequality constraints. Then, I will discuss the theoretical guarantees provided by the method, in particular the establishment of a Lax-Wendroff-type theorem, and conclude with numerical results for the shallow water equations.
This is a joint work with Nina Aguillon and Julien Salomon. -
Guillaume De Romémont
A data-driven learned discretization approach for Finite Volume schemes
10 mars 2026 - 14:00Salle de conférences IRMA
The recent development of Machine Learning (ML) and Deep Learning methods, coupled with recent advances in GPU-based computing defined new promising techniques for the numerical resolution of PDEs entirely solved with ML, as well as tuning existing algorithms for learning corrections or discretizations. In this work, we combine finite volume numerical schemes and neural networks to learn the discretization of the spatial derivatives of partial differential equations (PDEs) in order to better resolve the small spatial scales. We use approximate solutions of the 1D and 2D Euler equations obtained on a finer grid for the reference database in order to learn an optimal spatial discretization on a coarse grid, even with discontinuities in the solution. We post-process the outputs of the neural network to propose an interpretable, entropy consistent subgrid-model with super-resolution capabilities within a second-order finite volume solver without violating physical constraints. -
Corentin Gentil
A linear model of ocean western boundary currents with bathymetry and topography
17 mars 2026 - 14:00Salle de conférences IRMA
The trajectory of ocean western boundary currents is crucial in climate simulations, but the contribution of each physical effect to its path, like wind forcing, stratification, rotation, inertia, geometry of the coast, etc. remains an open question. In this presentation, I will introduce a simplified model for oceanic motion close to the Boussinesq equations, which takes into account two effects that are essential for predicting the trajectory of ocean western boundary currents: stratification and topography. We will see how to construct an approximate solution to this system as a superposition of interior terms on the one hand, and boundary layer terms of different natures on the other hand, due to small parameters. We will study how topography and stratification affect the solution and discuss different asymptotics created by small and large parameters. -
Mabrouk Ben Jaba
À l'optimum, peut-on entendre la ventilation du poumon humain ? Une approche « contrôle optimal ».
24 mars 2026 - 14:00Salle 301
Le poumon constitue une interface d’échange essentielle entre l’air ambiant et le sang, jouant un rôle crucial dans l’oxygénation de ce dernier et l’élimination du dioxyde de carbone. Différentes modélisations mathématiques dans la littérature permettent d’étudier son fonctionnement, certaines faisant intervenir des équations aux dérivées partielles complexes. Une approche alternative consiste à considérer des modèles intégrant l’arbre bronchique dans son ensemble, ce qui constitue le point de vue adopté ici. Notre démarche repose sur l’hypothèse selon laquelle les échanges gazeux sont optimisés pour maximiser l’efficacité du poumon, en accord avec des principes tels que la théorie de l’évolution. Nous cherchons ainsi à retrouver les caractéristiques du cycle respiratoire en formulant un problème d’optimisation (modélisation inverse). Afin d’explorer cette hypothèse et d’évaluer ce principe d'optimalité, nous proposons un modèle basé sur des équations différentielles ordinaires décrivant l’évolution de la concentration de dioxygène dans le poumon et son transport. Dans ce cadre, nous introduisons, analysons et étudions un problème de contrôle optimal visant à caractériser les dynamiques du cycle respiratoire. ——— Travail en commun avec Zakaria BELHACHMI (Univ. Haute-Alsace), Benjamin MAUROY (Univ. Côte d’Azur), Yannick PRIVAT (Univ. Lorraine) et Jean-François SCHEID (Univ. Lorraine). -
Takéo Takahashi
Comportement en temps long pour un système solide rigide / fluide visqueux compressible
31 mars 2026 - 14:00Salle de conférences IRMA
Lors de cet exposé, je présenterai des résultats récents obtenus en collaboration avec Debayan Maity concernant l'interaction entre un solide rigide et un fluide visqueux compressible dans un domaine extérieur. Notre analyse porte sur l'existence de solutions fortes et leur comportement asymptotique en temps long. En combinant un changement de variables approprié avec des estimations de décroissance $L_p-L_q$, nous établissons le caractère bien posé global du système couplé pour des données initiales petites. Dans cette démonstration, nous obtenons un taux de décroissance pour les vitesses du fluide et du solide lorsque $t \to \infty$, qui nous permet en particulier de montrer que le solide rigide tend vers une position finale. -
Tiphaine Delaunay
Observers for wave equations and associated discretizations
7 avril 2026 - 14:00Salle de conférences IRMA
The context of this talk is the study of inverse problems for wave propagation phenomena. The goal is to formalize, analyze, and discretize sequential data assimilation strategies, in which measurements are incorporated as they become available. The resulting system, called an observer, stabilizes on the observed trajectory and hence reconstructs the state and possibly some unknown parameters of the system. In the first part of the talk, I focus on source reconstruction, an estimation problem intermediate in complexity between state estimation and the more general problem of parameter identification. In this setting, we define, in a deterministic infinite-dimensional framework, a so-called Kalman estimator that sequentially estimates the source term to be identified. Using tools from dynamic programming, we show that this sequential estimator is equivalent to the minimization of a functional. This equivalence allows us to establish convergence results under observability conditions. We demonstrate these observability inequalities for different source types by combining functional analysis, multiplier methods, and Carleman estimates. The second part concerns discretization issues and their analysis. We propose a discretize-then-optimize approach, where the observability inequalities need to be extended to discretized systems. In particular, in this context, we extend uniform exponential stabilization results to the discretization for high-order finite element discretizations of a 1D wave equation. Joint work with S. Imperiale and P. Moireau. -
Chloé Mimeau
Simulation of incompressible flows with a Remeshed Vortex Method - a hybrid Lagrangian/Eulerian approach
28 avril 2026 - 14:00Salle de conférences IRMA
In particle methods, also known as Lagrangian methods, the discretization of the simulation domain is realized on a set of numerical particles that follow the dynamics of the system and are displaced with respect to the flow velocity. Vortex methods belong to particle methods and allow to discretize in a Lagrangian way the vorticity-velocity formulation of the Navier-Stokes equations, for the simulation of incompressible flows. This work is focused on a semi-Lagrangian variant of the Vortex Methods, the so-called « Remeshed Vortex Methods » (RVM). In this approach, the Lagrangian particles are regularly re-located on a Cartesian grid (through a « particle-remeshing » procedure) in order to avoid one of the major drawback of purely mesh-free Lagrangian schemes, namely the distorsion of particle distribution, leading to convergence loss. These methods constitute an extension of the « Particle-In-Cell » methods and can be seen as a hybrid Lagrangian-Eulerian approach; they take advantage from both the low numerical dissipation brought by the Lagrangian treatment of the flow advection and the efficiency of Eulerian schemes to handle the Poisson equation of the problem, the viscous term or the external forces. The present talk will present the numerical characteristics of the RVM, through a comparison of Direct Numerical Simulations (DNS) obtained with other numerical methods (like a Lattice Boltzmann method (LBM), i.e. an Eulerian method based on a mesoscopic flow description). We will then highlight the interest of RVM to afford Large Eddy Simulations (LES) and present the sub-grid scale models suited to RVM for the LES of homogeneous isototropic turbulents flows and of turbulent flows over solid walls. -
Aline Lefebvre-Lepot
Interactions proches dans les écoulements granulaires immergés
5 mai 2026 - 14:00Salle de conférences IRMA
Cette présentation porte sur la simulation des milieux granulaires denses composés de particules macroscopiques en suspension dans un fluide visqueux, comme les boues. Ces systèmes présentent des comportements macroscopiques extrêmement complexes, incluant des phénomènes tels que les instabilités de concentration et les blocages d’écoulement. Au cœur de ces comportements se trouvent les interactions physiques microscopiques entre particules voisines, en particulier les effets de lubrification induits par le fluide interstitiel, ainsi que les contacts solides. Ces interactions sont essentielles non seulement pour comprendre les écoulements granulaires immergés, mais aussi dans le cadre plus large des suspensions de particules. Malgré leur importance, l’influence de ces mécanismes microscopiques sur la dynamique globale du système reste mal comprise. Une difficulté majeure provient du caractère singulier de ces effets lorsque la distance entre particules tend vers zéro, introduisant des singularités temporelles. Nous présenterons dans cet exposé de nouveaux modèles de type dynamique des contacts, prenant en compte la lubrification ainsi que le contact solide, avec ou sans friction. Nous montrerons que l’on peut construire, à partir de ces modèles, des schémas numériques stables et robustes. Ces schémas mènent, à chaque itération en temps, à la résolution d’un problème d’optimisation sous contrainte. Nous illustrerons l’exposé par des résultats numériques. -
Annamaria Massimini
Structure-preserving hybrid finite volume scheme for an anisotropic cross-diffusion system
12 mai 2026 - 14:00Salle de séminaires IRMA
In this presentation, I will introduce a hybrid finite volume method on general polygonal and polyhedral meshes for the modeling of an anisotropic cross-diffusion system arising from a mesoscopic stochastic process describing diffusion in solids, under a size-exclusion constraint.
This system possesses an entropy structure, which is exploited to define the numerical scheme in terms of (discrete) entropy variables, and is thus preserved at the discrete level.
This structure makes it possible to prove the existence of nonnegative discrete solutions satisfying the size-exclusion constraint, as well as mass conservation, and to establish the convergence of the scheme under mesh refinement.
To the best of our knowledge, this is the first work proposing and analyzing a structure-preserving hybrid finite volume scheme for anisotropic cross-diffusion systems on general polygonal and polyhedral meshes.
The preprint associated with this presentation is:
V. Ehrlacher, A. Massimini, J. Moatti. Structure-preserving hybrid finite volume scheme for an anisotropic cross-diffusion system, 2026. Preprint, HAL : hal-05589824 -
Lukas Renelt
Optimal contraction of the energy difference in adaptive FEM for strongly monotone nonlinear problems
19 mai 2026 - 14:00Salle de conférences IRMA
In this talk we discuss the convergence of adaptive FEM for strongly monotone nonlinear partial differential equations arising from the minimization of a convex energy functional. Rather than focusing on specific algorithmic implementations, we establish a generalized theoretical framework that identifies sufficient conditions for an adaptive scheme to contract the energy difference at an optimal rate relative to the number of degrees of freedom. The presented abstract point of view is subsequently shown to include many existing error estimation approaches (local residuals, flux equilibration, linear residual liftings, energy descent), thereby providing a unified proof for their optimal convergence. A key contribution of our recent work is the introduction of computable local equivalence factors yielding a computable a posteriori bound on the contraction rate which can then be used for an improved local refinement procedure. Finally, we will discuss technical challenges posed by higher-order discretizations and show how the arising oscillations can be managed through local higher-order flux-equilibration. -
Teresa Malheiro Et Gaspar Machado
R-Block structural schemes of high order accuracy
26 mai 2026 - 14:00Salle de conférences IRMA
We present a compact scheme whose core concept involves decomposing it into two subsystems of equations. The Physical Equations utilise the function and its K derivatives at a node by implementing physical relations. These equations operate locally, with no exchange of information with other nodes,as the physics involved are governed by local operators. The Structural Equations depend on linear relationships between the function and its derivatives across a stencil of R points, which we call a R-block, establishing complete connections between a node and its neighbours. These relationships are independent of the physics involved since they are established regardless of the specific problem. In this presentation we address in particular the accuracy and stability of these methods. Based on joint work with S. Clain, G. J. Machado and Ricardo Costa -
Julius Ehigie
TBA
15 septembre 2026 - 14:00Salle de conférences IRMA
-
Thomas Perrin
Dynamics of the damped focusing cubic Klein-Gordon equation
22 septembre 2026 - 14:00Salle de conférences IRMA
In this talk, I will present several results on the damped focusing cubic Klein-Gordon equation on a bounded domain. First, I will present an explicit dichotomy between blow-up and stabilisation, under the geometric control condition, for solutions below the ground state energy. Second, I will present a trichotomy result for solutions slightly above the ground state energy, and a complete study of space-independent solutions. Finally, I will discuss some perspectives concerning solutions of arbitrarily large energy and generic domains. -
Mattia Manucci
Contour Integral Methods for Efficient Time Integration and Model Reduction
29 septembre 2026 - 14:00Salle de conférences IRMA
We discuss a unified framework based on contour integral methods (CIMs) for the efficient numerical treatment of linear evolution problems. The central idea is to apply the Laplace transform to the underlying evolution equation and recover the time-domain solution by numerically approximating the inverse Laplace transform. In contrast to classical time-stepping approaches, this perspective allows the solution to be computed directly at prescribed time instants or over entire time intervals, offering increased flexibility and, in suitable settings, significant computational advantages.
We first discuss the key features that make a problem particularly well suited to CIMs and then explore how the Laplace-transform framework naturally connects with projection-based model order reduction (MOR). In particular, we show how CIMs can be combined with projection-based techniques to efficiently treat parametric evolution PDEs and control systems. Since the reduction can be constructed from selected solution snapshots rather than from the full set of intermediate states generated by a time-stepping method, the computational cost of the reduction phase can be substantially reduced. For parametric problems, we further discuss a greedy construction of the projection spaces based on suitable error estimates.
Finally, we extend the CIM framework beyond ordinary differential equations to linear differential-algebraic equations (DAEs), namely, linear dynamical systems subject to algebraic constraints that must be satisfied throughout the evolution. The resulting approach provides an efficient alternative to classical Runge–Kutta methods, which can suffer from order reduction when applied to DAEs.
Overall, these results illustrate how contour integral methods provide a flexible framework for the numerical solution and model reduction of linear evolution problems, ranging from PDEs and parametric systems to constrained dynamical systems. The seminar will highlight the underlying numerical analysis as well as the computational advantages and limitations of the proposed approaches. -
Roxana Sublet
Modelling collective cell dynamics
6 octobre 2026 - 14:00Salle de conférences IRMA
This work focuses on the mathematical modeling of cellular tissue dynamics. We first propose an individual-based model that provides the dynamics of the positions, velocities, and polarities of the cells, idealized as hard spheres. Cells interact with each other through contact forces, smooth attraction, and polarity alignment. The present work is an extension of the model proposed in [3] and validated by experiments on cellular rings. We will study the well-posedness of a regularized version. We are particularly interested in the congested regime and the impact of apoptosis on the jammed state. Indeed, cell apoptosis corresponds to programmed cell death: when cells leave the tissue, they induce local contractions but also enable cellular rearrangements. To this end, we add to the previous model a microscopic description of apoptotic and proliferation events. Numerical simulations are performed to show the impact of apoptosis on collective cell dynamics. Next, we derive a macroscopic description, following the methodology proposed in [1] and [2]. We start from a mean-field dynamics of the kinetic distribution function in phase space (position, polarity, radius), where contact forces have been replaced with repulsion forces. We then introduce a specific time and space rescaling and identify the equilibrium distribution functions, which are parameterized by two macroscopic quantities: the density and the mean polarity. Based on the Generalized Collision Invariant (GCI) method [2], we are then able to identify their dynamics: the resulting description can be seen as a modified Self-Organized Hydrodynamics (SOH) model. We finally discuss the obtained model and highlight the effect of the apoptotic events on the dynamics. This is a joint work with Laurent Navoret (Université de Strasbourg) and Marcela Szopos (Université Paris Cité). It has also been carried out in collaboration with Romain Levayer (Institut Pasteur) and Daniel Riveline (IGBMC, Université de Strasbourg) in the context of the ANR project MAPEFLU. References [1] Degond, P., Dimarco, G., Mac, T. B. N., and Wang, N. Macroscopic models of collective motion with repulsion. Communications in Mathematical Sciences, 13(6), 1615–1638 (2015). [2] Degond, P., and Motsch, S. Continuum limit of self-driven particles with orientation interaction. Mathematical Models and Methods in Applied Sciences, 18(supp01), 1193–1215 (2008). [3] Vecchio, S. L., Pertz, O., Szopos, M., Navoret, L., and Riveline, D. Spontaneous rotations in epithelia as an interplay between cell polarity and boundaries. Nature Physics (2024).