Séminaires

This presentation delves into the spectral instability of periodic waves for the electronic Euler-Poisson system, a hydrodynamical model for understanding electron dynamics in plasmas.
It is found that even small-amplitude periodic traveling waves in this system exhibit spectral instability, which is neither modulation nor co-periodic, necessitating an usual spectral analysis approach.
The lecture will provide an explanation of fundamental tools employed to investigate the stability of periodic waves, including Bloch-Floquet theory, Kato’s perturbation theory, while outlining the primary proof strategy.

Oratrice : Laetitia Colombani (CMAP, École Polytechnique) FitzHugh-Nagumo equations have been suggested in 1961 to model neurons. Stochastic versions of these equations have since been developed. The specificity of these SDE is a cubic term in the drift, which needs us to pay attention. With Pierre Le Bris, we have studied the behavior of a network on N neurons, interacting with each other, when N tends to infinity. We prove an uniform in time propagation of chaos in a mean-field framework, with a coupling method suggested by Eberle (2016). During this talk, I will present this model and the idea of the method.

Le système de la chromatographie est un système d’équations aux dérivées partielles, Après quelques définitions, propriétés sur les lois de conservations et la construction de solutions, nous nous concentrerons sur la reconstruction rétrograde, l’optimisation et le contrôle aux bords du système de la chromatographie. Ce système est composé d’une loi de conservation scalaire couplée avec une (ou plusieurs) équation de continuité. La reconstruction rétrograde de lois de conservations se résume au problème suivant : étant donné un instant T > 0, il faut d’abord déterminer si un état UT est atteignable, et si oui, quelles données initiales U0 peuvent nous amener au profil sélectionné, et comment reconstruire les solutions numériquement. L’optimisation sur un problème rétrograde consiste à trouver, partant d’un profil non atteignable, le profil atteignable le plus proche de celui ci

Orateur : Ahmed Zaoui (LmB, Besançon) This presentation focuses on the estimation of the variance function in regression and its applications in regression with reject option and prediction intervals. First, we are interested in estimating the variance function through two methods: model selection (MS) and convex aggregation (C). The goal of the MS procedure is to select the best estimator from a set of predictors, while the C procedure aims to choose the best convex combination among the predictors. The selected predictors are then referred to as MS-estimator and C-estimator, respectively. The construction of both the MS-estimator and C-estimator is based on a two-step procedure. In the first step, using the first sample, we construct estimators of the variance function through a residual-based method. In the second step, we aggregate these estimators using a second sample. We establish the consistency of both the MS-estimator and C-estimator with respect to the L2-risk. Next, we shift our focus to the regression problem, where one is allowed to abstain from predicting. We focus on the case where the rejection rate is fixed and derive the optimal rule which relies on thresholding the conditional variance function. We provide a semi-supervised estimation procedure for this optimal rule. The resulting predictor with reject option is shown to be almost as good as the optimal predictor with reject option both in terms of risk and rejection rate. We additionally apply our methodology with kNN algorithm and establish rates of convergence for the resulting kNN predictor under mild conditions. Finally, a numerical study is performed to illustrate the benefit of using the proposed procedure. Finally, we tackle the problem of building a prediction interval in heteroscedastic Gaussian regression. We focus on prediction intervals with constrained expected length in order to guarantee interpretability of the output. In this framework, we derive a closed form expression of the optimal prediction interval that allows for the development a data-driven prediction interval based on plug-in. The construction of the proposed algorithm is based on two samples, one labeled and another unlabeled. Under mild conditions, we show that our procedure is asymptotically as good as the optimal prediction interval both in terms of expected length and error rate. We conduct a numerical analysis that exhibits the good performance of our method.

Oratrice : Sophie Dabo (Lille) Multivariate principal component analysis and a multivariate functional binary choice model is explored in a case-control or choice-based sample design context. In other words a model is considered in which the response is binary, the explanatory variables are functional, and the sample is stratified with respect to the values of the response variable. A dimension reduction of the space of the explanatory random functions based on a Karhunen–Loève expansion is used to define a conditional maximum likelihood estimate of the model. Based on this formulation, several asymptotic properties are given. A simulation study and an application to real data are used to compare the proposed method with the ordinary maximum likelihood method, which ignores the nature of the sampling. The proposed model yields encouraging results. The potential of the functional choice-based sampling model for integrating special non-random features of the sample, which would have been difficult to see otherwise, is also outlined.

I would like to discuss the continuum limit of discrete Dirac operators on the square lattice in [katex]R^2[/katex] as the mesh size tends to zero. To this end, I propose the most natural and simplest embedding of [katex]\ell^2( Z^d_h )[/katex] into [katex]L^2( R^d)[/katex].
This approach enables one to define difference operators in a subspace of step functions in [katex]L^2( R^d)[/katex] and to compare the discrete Dirac operators with the continuum Dirac operators in the same Hilbert space [katex]L^2(R^2)^2[/katex]. In particular, I can prove that discrete Dirac operators converge to the continuum Dirac operators in the strong resolvent sense. Potentials are assumed to be bounded and uniformly continuous functions on [katex]R^2[/katex] and allowed to be complex matrix-valued. I also prove that the discrete Dirac operators do not converge to the continuum Dirac operators in the norm resolvent sense.
This finding is closely related to the observation that the Liouville theorem does not hold in discrete complex analysis. If time allows, I would also like to discuss the continuum limit of discrete Dirac operators on the 1D lattice.
(This is joint work with Karl Michael Schmidt.)