- 26 janvier 2022
- Pierre Le Bris (LJLL)
*Uniform in time propagation of chaos for the 2D vortex model and other singular stochastic systems*The talk will be in two parts.

The first one will consists in a general introduction on propagation of chaos in systems of particles in mean field interaction, detailing the goal and motivations, and mentioning some approaches to the problem.

In the second part, we focus on a specific proof and show how we may obtain uniform in time propagation of chaos for a class of singular interaction kernels, extending the results of Fournier-Hauray-Mischler (JEMS) and Jabin-Wang (Inv. Math.). In particular, our models contain the Biot-Savart kernel on the torus and thus the 2D vortex model. The strategy is to combine the relative entropy approach of Jabin-Wang with functional inequalities as well as uniform bounds on all the derivatives of the solution of the non linear limit equation, in order to control both the entropy dissipation and the constants appearing in the large deviation estimates.

This is joint work with Arnaud Guillin (LMBP, Université Clermont-Auvergne) and Pierre Monmarché (LJLL, Sorbonne Université).

- 19 janvier 2022
- Louis Reboul (CMAP)
*Asymptotic-preserving schemes for kinetic and fluid equations*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 inexploitable. Asymptotic preserving schemes are designed to both lift the restrictive stability conditions and remain accurate in the asymptotic regime. In this presentation, we introduce some fluid and kinetic equations of interest and their corresponding asymptotic regimes and we present a new asymptotic-preserving strategy for a wide range of applications. Our aim includes plasma discharges with sheaths, where we have two small parameters related to Debye length and mass ratio. Numerical simulations assess and illustrate the potential of the method we have introduced. - 13 janvier 2022
- Jesús Bellver Arnau (LJLL)
*Dengue outbreak mitigation via instant releases*In the fight against arboviruses, the endosymbiotic bacterium Wolbachia has become in recent years a promising tool as it has been shown to prevent the transmission of some of these viruses between mosquitoes and humans. This method offers an alternative strategy to the more traditional sterile insect technique, which aims at reducing or suppressing entirely the population instead of replacing it.

In this presentation I will present an epidemiological model including mosquitoes and humans. I will discuss optimal ways to mitigate a Dengue outbreak using instant releases, comparing the use of mosquitoes carrying Wolbachia and that of sterile mosquitoes.

This is a joint work with Luis Almeida (Laboratoire Jacques-Louis Lions), Yannick Privat (Université de Strasbourg) and Carlota Rebelo (Universidade de Lisboa). - 5 janvier 2022
- Willy Haik (LJLL et LMT)
*A real-time variational data assimilation method with model bias identification and correction*Real-time monitoring on a physical system by means of a model-based digital twin may be difficult if occurring phenomena are multiphysics and multiscale. A main difficulty comes from the numerical complexity which is associated to an expensive computation hardly compatible with real-time. To overcome this issue, the high-fidelity parameterized physical model may be simplified which adds a model bias. Moreover, the parameter values can be inaccurate, and one part of the physics may be unknown. All those errors affect the effectiveness of numerical diagnostics and predictions and need to be corrected with assimilation techniques on observation data. Therefore, the monitoring of a process occurs in two stages: (1) the state estimation at the acquisition time which may be associated with an identification of the set of unknown parameters of the parametrized model and an update state which enriches the model; (2) a state prediction for future time steps with the updated model.

The present study is mainly denoted to perform the state estimation using an extension, for time-dependent problems, of the Parameterized Background Data-Weak (PBDW) method introduced in [1]. This method is a non-intrusive, reduced basis and in-situ data assimilation framework for physical systems modeled by parametrized Partial Differential Equations initially designed for steady-state problems. The key idea of the formulation is to seek an approximation to the true field employing projection-by-data, with a first contribution from a background estimate computed from a reduced-order method (ROM) enhanced on-the-fly, and a second contribution from an update state informed by the experimental observations (correction of model bias). Further research works [2,3] developed an extension to deal with noisy data and a nonlinear framework. Moreover, a priori error analysis was conducted by providing a bound on the state error and identifying individual contributions. In the present work, the state prediction for future time steps is also performed from an evaluation of the updated model and an extrapolation of the time function from the tensor-based decomposition (SVD) on prior updates.

Numerical experiments are conducted on a thermal conduction problem in the context of heating on a Printed Circuit Board (PCB) with different cases of model bias: a bias on heat source, a biased boundary condition and an error on the constitutive equation. These numerical experiments show that the method significantly reduces the online computational time while providing relevant state evaluations and predictions. We thus illustrate the considerable improvement in prediction provided by the hybrid integration of a best-knowledge model and experimental observations.

[1] Maday, Y., Patera, A. T., Penn, J. D., and Yano, M. (2015). A parameterized background dataweak approach to variational data assimilation: formulation, analysis, and application to acoustics. International Journal for Numerical Methods in Engineering, 102(5), 933-965.

[2] Yvon Maday and Tommaso Taddei (2017) Adaptive pbdw approach to state estimation: noisy observations; user-defined update spaces. arXiv preprint arXiv:1712.09594.

[3] Gong, H., Maday, Y., Mula, O., and Taddei, T. (2019). PBDW method for state estimation: error analysis for noisy data and nonlinear formulation. arXiv preprint arXiv:1906.00810.

- 15 décembre 2021
- Nicolás Torres (LJLL)
*On a competitive system with ideal free dispersal and global bifurcation results*In this work we study the dynamics of a diffusion-advection-competition model for two species living in a bounded region. For this model, there exists an optimal dispersal strategy called “ideal free”. An important result states that under symmetric competition, the ideal free strategy is optimal in the sense it always ensures the survival for the species which adopts it. We extend the study of this system for the non symmetric case, by proving some results about stability, multiplicity and bifurcation of equilibriums. - 8 décembre 2021
- Elisabetta Brocchieri (Paris Saclay - UEVE)
*About entropy methods to a dietary diversity cross-diffusion system*Cross-diffusion systems are non-linear parabolic systems, modelling the evolution of densities or concentrations of multicomponent populations in interaction. They may be derived by random walk on lattices, in a microscopic scaling, or as limit of linear parabolic diffusion systems, at a mesoscopic level. In this talk, we propose the rigorous passage from a weak competitive reaction-diffusion system towards a reaction cross diffusion system, in the fast reaction limit. The resulting limit system shows a starvation driven cross-diffusion term. The main ingredients used to prove the existence of global solutions are an energy functional and a compactness argument. However, the analysis of an appropriate family of energy functionals allows to improve the regularity of the solution. Furthermore, we also investigate the linear stability of homogeneous steady states of those systems and rule out the possibility of Turing instability. Then, no pattern formations occur. To conclude, numerical simulations are included, proving the compatibility with the theoretical results.

Reference:

B., E., Corrias, L., Dietert, H. and Kim, Y-J. Evolution of dietary diversity and a starvation driven cross-diffusion system as its singular limit. J. Math. Biol. 83, 58 (2021). https://doi.org/10.1007/s00285-021-01679-y - 1 décembre 2021
- Maria Cabrera Calvo (LJLL)
*Highly oscillatory integrators at low regularity for the Klein-Gordon equation*We propose a novel class of uniformly accurate integrators for the Klein–Gordon equation which capture classical $c = 1$ as well as highly-oscillatory non-relativistic regimes $c \gg 1$ and, at the same time, allow for low regularity approximations. In particular, our first- and second-order schemes require no step size restrictions and lower regularity assumptions than classical schemes, such as splitting or exponential integrator methods. The new schemes in addition preserve the nonlinear Schrodinger (NLS) limit on the discrete level. More precisely, we will design our schemes in such a way that in the limit $c \to \infty$ they converge to a recently introduced class of low regularity integrators for NLS.

This is joint work with Katharina Schratz (Sorbonne University). - 23 novembre 2021
- Thomas Borsoni (LJLL)
*A general framework for the kinetic modeling of polyatomic gases*In the framework of kinetic models of gases, we propose a Boltzmann model where we describe the internal structure of molecules in a general way, with a measured space corresponding to the internal states and a real function on this space corresponding to the energy associated to each state. We prove the H theorem in this case, which gives in particular the equilibrium distribution: the product of a Maxwellian and a Gibbs distribution. We show that this framework contains the monoatomic case, and the two existing polyatomic models, which are the models with continuous internal energy and discrete energy levels. We build new models in this framework directly from physical considerations, which allows in particular to describe also non-polytropic gases. Finally, we explain the model reduction, which allows us to get back to the model with continuous internal energy, which gives sense and a formula to calculate from the description of the molecule the integration weight present in this latter model. We briefly explain the extension of this framework to a mixture of gases with chemical reactions. - 17 novembre 2021
- Emma Leschiera (LJLL)
*Mathematical modelling of tumour-immune interaction: discrete and continuum approaches*The recent successes of immunotherapy for the treatment of tumours has highlighted the importance of the interactions between tumour cells and immune cells. However, these interactions are based on complex mechanisms, making it difficult to design an effective treatment aimed at strengthening the immune response. Therefore, mathematical models are needed to faithfully reproduce and predict the spatio-temporal dynamics of tumour growth, describing the interaction of the tumour with the immune cells.

In this talk, we start by presenting a stochastic individual-based model capturing the interactions between tumour cells and immune cells. Considering different initial compositions of the tumour, we investigate how intra-tumour heterogeneity (ITH) affects the anti-tumour immune response. In this model, ITH can vary with the number of tumour antigens (i.e. the number of sub-populations of tumour cells) and with the level of antigen presentation (i.e. the immunogenicity of tumour cells). Computational simulations show that both components play a role in the anti-tumour immune response. In the second part of the talk, we re-formulate the individual-based model, and we introduce a continuum model formally obtained as the deterministic continuum limit of such individual-based model. We report on computational results of the individual-based model, and show that there is a good agreement between them and numerical results of the continuum model.

- 10 novembre 2021
- Various speakers (LJLL)
*GTT 1A*Schedule:

10h--10h15 Presentation of the day, of the welcome booklet and of the lab

10h15--10h35 Elena Ambrogi, Degenerate parabolic models of neural networks

10h35--10h55 Robin Roussel, Notions of magnetic confinement in stellarators using Hamiltonian dynamics

10h55--11h05 GTT, masters/PhD students meeting

11h05--11h20 Coffee break

11h20--11h40 Fabrice Serret, Variational Quantum Algorithms

11h40-12h Marguerite Champion, Fluid-structure interaction with contact

12h--12h10 GT EDP, Infomaths

12h10--14h Lunch break

14h--14h10 PhD representatives, heads of offices

14h10--14h30 Edouard Timsit, Randomization for solving linear systems and compressing tensors arising in molecular simulations

14h30--14h50 Guillaume Garnier, Effects of a mutation on the selective value of a bacterium: Theoretical estimation and numerical implementation

14h50--15h Support committee, Parity committee

15h--15h20 Yvonne Alama Bronsard, Low regularity error estimates and tree series analysis for dispersive and wave type systems

15h20--15h40 Nicolai Gouraud, Quantum algorithms and non-reversible MCMC methods

15h40--15h45 Lab tea

15h45--16h15 Coffee break

16h15--16h35 Charles Elbar, Cahn Hilliard equation and Hele-Shaw models

16h35--16h55 Clément Lasuen, An asymptotic preserving scheme for a moment model on conical meshes

16h55--17h15 Lucas Perrin, Time parallelisation, observers, and data assimilation Admin