I will present a work in collaboration with M. Bessemoulin-Chatard and T. Rey, in which we consider a non-linear kinetic model describing a two-species generation-recombination reaction that can be considered as a simplified version of the models describing the generation and recombination of electron-hole pairs in semiconductors. I will introduce a finite volume discretization of this model for which we can prove an exponential decay towards the steady state using discrete hypocoercivity methods. After presenting the ideas of the proof in the continuous framework, I will highlight the main difficulties induced by the discretization process. The properties of the method will then be illustrated by several numerical examples.
On considère des processus autorégressifs pour lesquels on crée un pont entre un comportement stable et un comportement instable à l'aide d'une matrice compagne $A{n}$ dépendant du temps et dont le rayon spectral $\rho(A{n}) < 1$ est tel que $\rho(A_{n}) \rightarrow 1$. Ce cadre de travail est particulièrement pertinent pour comprendre les problématiques de racines unitaires en se focalisant sur la frontière intérieure du cercle unité. On étudie le comportement asymptotique de l'estimation en termes de consistance et de normalité. On propose de plus une procédure de test statistique pour décider de la proximité du rayon spectral avec le cercle unité, afin de savoir "à quel point un processus quasi-instable est proche de l'instabilité". Quelques résultats numériques illustrent ces résultats.
In healthy hearts, the propagation of electrical waves follows a predictable pattern, whereas in people suffering from arrhythmia, the electrical waves can become chaotic and affect the heart's pumping function. The main treatment is catheter ablation, during which small areas of heart tissue are destroyed to isolate the cause of the irregular heartbeats. Most catheter ablations are performed thermally through the application of a radiofrequency electromagnetic field (RFA), but in this work, we focus on the study of a non-thermal ablation technique: pulsed electric field ablation (PFA), which utilizes irreversible electroporation, a complex cell death phenomenon that occurs when biological tissue is exposed to very intense electrical pulses. This technique has been used in oncology for more than a decade, but in cardiology it is still in its infancy due to the technical complexity of this novel approach. Mathematical models and numerical strategies could improve the understanding of PFA on the cardiac signal. In this talk, I will introduce the various mathematical challenges that we would like to address. Second, I will present a bidomain model incorporating a nonlinear transport term derived thanks to a two-scale approach that can account for the different time and length scales between cardiac electrophysiology and electroporation. Numerical simulations with industrial catheter geometries will also be presented, showing first interesting results for the determination of the electroporated area. Third, I will present a cardiac electrophysiological model of a cardiac domain containing a region ablated by PFA. Considering modeling assumptions and performing a rigorous asymptotic analysis, we determine the transmission conditions at the interface between the two regions. Numerical simulations performed thanks to well-designed Schwarz algorithms will be presented in the context of atrial fibrillation. A numerical comparison between PFA and RFA will allow to propose a numerical explanation for the higher rate of fibrillation recurrence after RFA compared to PFA
The objective of topological data analysis is to extract information of topological nature (connected components, holes, voids...) from the data and use them as features to perform various machine-learning tasks. We start by building a nested sequence of simplicial complexes on top of the data and track the evolution of its topology along the sequence. Computing the Euler characteristic of each complex in the sequence yields a descriptor called the Euler characteristic curve that we use as a feature vector. We will demonstrate that this descriptor has a very good performance in terms of accuracy, strong explainability in terms of topology, and stability with respect to the input data while having a drastically reduced computational cost. We will also study integral transforms of this descriptor and show how this "topological signal processing" enables better performance, especially in an unsupervised setting. Joint work with Vadim Lebovici (Oxford)
Références utiles :
Euler characteristic tools for topological data analysis, Hacquard and Lebovici (2023)
An introduction to topological data analysis: fundamental and practical aspects for data scientists, Chazal and Michel (2021)
Euler characteristic curves and profiles: a stable shape invariant for big data problems, Dlotko and Gurnari (2023)
Classical inference methods fail when applied to data-driven test hypotheses. Selective inference is particularly relevant post-clustering, typically when testing a difference in the mean between two clusters. Thus, dedicated methodologies are required to obtain statistical quarantees for these selective inference problems. In this work, we address convex clustering with l1 penalization, by leveraging related selective inference tools for regression, based on Gaussian vectors conditioned to polyhedral sets.
In the one-dimensional case, we prove a polyhedral characterization of obtaining given clusters, then enables us to suggest a test procedure with statistical guarantees. This characterization also allows us to provide a computationally efficient regularization path algorithm. Then, we extend the above test procedure and guarantees to some multi-dimensional clusterings. Our methods are implemented in the R package poclin.
On présente deux travaux reliés concernant des modèles cinétiques de type saut au point-milieu. Le premier (avec Pierre Degond, Gaël Raoul) est un modèle d’alignement de particules autopropulsées (la version cinétique du modèle de Bertin-Droz-Grégoire). La version homogène en espace et non-bruitée correspond à un modèle de saut au point-milieu sur la sphère unité. Le deuxième (avec Cécile Taing) est le modèle de Fisher infinitésimal en dynamique des populations sexuées, dans le cas où la variabilité est nulle. Ceci correspond également à un modèle de saut au point-milieu mais dans lequel le taux de mortalité variable. Dans ces deux modèles, une des difficultés principales et l’absence de conservation du centre de masse. Dans les deux cas on arrive à démontrer la stabilité asymptotique de masses de Dirac (qui sont des états stationnaires) en distance de Wasserstein. Dans le cas du modèle de population sexuée, on obtient également convergence de la solution vers un profil autosimilaire à queue lourde.
Le développement de système de suivi GPS léger et très autonome permet de suivre les déplacements des individus de manière précise précise et peu invasive. Ces données de trajectoires sont comprises par les écologues comme le reflet des besoins internes de l'animal, envisagé comme un problème de segmentation, et de sa réponse à l'environnement. Dans cet exposé, je présenterai plusieurs développement statistiques récents, reposant sur des approches à variables latentes, qui visent à extraire ce type d'information des données déplacement. Ces méthodes posent des questions intéressantes en terme de modélisation, d'inférence et de choix de modèles qui seront détaillées dans l'exposé.
Reférences :
Gloaguen, Pierre, Marie-Pierre Etienne, and Sylvain Le Corff. "Stochastic differential equation based on a multimodal potential to model movement data in ecology." Journal of the Royal Statistical Society Series C: Applied Statistics 67.3 (2018): 599-619.
Patin, Rémi, et al. "Identifying stationary phases in multivariate time series for highlighting behavioural modes and home range settlements." Journal of Animal Ecology 89.1 (2020): 44-56.
In this work, we construct a structure-preserving reduced-order model for the resolution of parametric cross-diffusion systems.
Cross-diffusion systems model the evolution of the concentrations or volumic
fractions of mixtures composed of different species and often read as nonlinear degenerated parabolic partial differential equations whose numerical resolutions are highly expensive from a computational point
of view.
We are interested here in cross-diffusion systems which exhibit a so-called entropic structure,
in the sense that they can be formally written as gradient flows of a certain entropy functional which is actually a Lyapunov functional of the system.
In this work, we propose a new reduced-order modelling
method, based on a reduced basis paradigm, in order to accelerate the resolution of parameter-dependent
cross-diffusion systems, which preserves, at the level of the reduced-order model, the main mathematical
properties of the continuous solution, namely mass conservation, non-negativeness, preservation of the
volume-filling property and entropy-entropy dissipation relationship.
The theoretical advantages of our approach are confirmed by several numerical experiments.
Présentation d'une version préliminaire de sa soutenance de thèse :
Les technologies de single-cell génèrent des données qui présentent de nombreux défis, il y a beaucoup d'observations, elles sont en grande dimension et souvent parcimonieuses. De nombreuses expériences de biologie consistent à comparer des conditions. L'objet de la thèse est de développer un ensemble d'outils qui permet de comparer des échantillons de données issues des technologies single-cell afin de détecter et éventuellement décrire les différences qui existent. Pour cela, j'applique les tests de comparaison de deux échantillons basés sur les méthodes à noyaux existants et propose un nouveau test qui généralise ces méthodes pour un design expérimental quelconque. Je discuterai aussi l'implémentation et l'utilisation des tests. Afin que ces outils soient accessibles et utilisables par des non-spécialistes, je propose un ensemble d'outils pour interpréter les résultats et identifier les observations ou les groupes d'observations influentes.
