Publications

2023

• Mathematical models to study the interaction between recombination suppression and deleterious mutations near a mating-type l
  
  • Tezenas Du Montcel Emilie
  , 2023. This PhD manuscript presents the development of several stochastic models that contribute to our understanding of recombination suppression evolution on sex and mating-type chromosomes. Recombination is a mechanism that exchanges parts of chromosomes, which creates novel allelic combinations. However, there have been reports in a wide range of organisms of large regions with suppressed recombination that encompass genes involved in mating compatibility (i.e., genes determining sex or mating type). The nature of the mechanisms that induce the extension of the non-recombining zone beyond the genes involved in sex compatibility remains debated. In this PhD thesis, we use various mathematical approaches to study the dynamics of deleterious mutations and recombination suppressors. The first chapter shows, with the analysis of a simple deterministic model and the simulation of a more complex stochastic one, that deleterious mutations impact the evolution of a recombination suppressor at several stages. The second chapter focuses on deleterious mutations dynamics near a permanently heterozygous locus in selfing or outcrossing populations. We model the initial evolution of deleterious mutations with a multitype branching process and study its criticality and its extinction time distribution. In the third chapter, we compare the effect of deleterious mutation accumulation in selfing populations of recombining or non-recombining individuals. We use a measure-valued branching process on a three-dimensional trait space to study the evolution of the mutational load for populations carrying an always-heterozygous mating-type locus.
• Volume computation for Meissner polyhedra and applications
  
  • Bogosel Beniamin
  Discrete and Computational Geometry, Springer Verlag, 2023. The volume of a Meissner polyhedron is computed in terms of the lengths of its dual edges. This allows to reformulate the Meissner conjecture regarding constant width bodies with minimal volume as a series of explicit finite dimensional problems. A direct consequence is the minimality of the volume of Meissner tetrahedras among Meissner pyramids. (10.1007/s00454-024-00688-0)
  DOI : 10.1007/s00454-024-00688-0
• Computing the diffusivity of a particle subject to dry friction with colored noise
  
  • Garnier Josselin
  • Mertz Laurent
  Physical Review E, American Physical Society (APS), 2023, 108 (4), pp.045309. This paper considers the motion of an object subjected to a dry friction and an external random force. The objective is to characterize the role of the correlation time of the external random force. We develop efficient stochastic simulation methods for computing the diffusivity (the linear growth rate of the variance of the displacement) and other related quantities of interest when the external random force is white or colored. These methods are based on original representation formulas for the quantities of interest, which make it possible to build unbiased and consistent estimators. The numerical results obtained with these original methods are in perfect agreement with known closed-form formulas valid in the white-noise regime. In the colored-noise regime, the numerical results show that the predictions obtained from the white-noise approximation are reasonable for quantities such as the histograms of the stationary velocity but can be wrong for the diffusivity unless the correlation time is extremely small. (10.1103/PhysRevE.108.045309)
  DOI : 10.1103/PhysRevE.108.045309
• A tribute to Elisabeta Vergu: Multi-level contact models of epidemics
  
  • Kubasch Madeleine
  • Bansaye Vincent
  • Deslandes François
  • Vergu Elisabeta
  , 2023.
• Stackelberg games, optimal pricing and application to electricity markets
  
  • Jacquet Quentin
  , 2023. In this PhD dissertation, we combine tools from bilevel optimization, mean-field games and ergodic control to tackle challenging issues in pricing management, especially in the optimal design of contracts for retail electricity markets.First of all, we formulate the problem as a leader-follower interaction (Stackelberg game), in which the customers decision is of probabilistic nature (bounded rationality). Through the tropical geometry viewpoint, we analyze the customers choice by interpreting the latter as a polyhedral complex (cell arrangement). We develop a new algorithm that exploits this underlying geometry, and provide results on realistic instances from the pricing problem faced in retail electricity markets. We then extend this model in two ways: firstly, we incorporate inertia in the customers decision, modeled as a Markov Decision Process in which transitions correspond to bilevel problems. We prove that the associated ergodic control problem admits a solution, that can be obtained through the solving of an eigenvalue problem. Secondly, we extend the model by optimizing not only the price coefficients but also the structure of the tariff menu : the question of the optimal number of contracts is viewed as the optimal quantization of an infinite-size menu of offers, the latter being described as a convex program. We develop to this purpose new pruning procedures, inherited from max-plus based methods used in optimal control.Besides, we study incentive mechanisms through monetary rewards by defining a Principal-Agent interaction between a retailer and a field of agents, where each agent competes with similar ones to be the most energy-compliant customer (ranking games). We make explicit both the Nash equilibrium achieved by the agents and the optimal reward function to offer to a heterogeneous population with uniform price elasticity. We present numerical results on the general case, and show the potential of this ranking game as a sobriety lever. Finally, we study two frameworks that appear in pricing management, that is chance-constrained programming and sparse optimization. For the former, we analyze the tractability of convex conservative approximations based on concentration inequalities. For sparsity concerns, we introduce a family of entropic bounds -- proved to control the cardinality requirement -- that we embed into sparse optimization problems to derive nonlinear approximations to the latter.
• Étude et modélisation stochastique du comportement humain face au changement environnemental
  
  • Ecotière Claire
  , 2023. Par leur comportement, les individus peuvent contribuer à la dégradation de leur environnement. Dans le cadre du changement climatique, une meilleure compréhension de l'évolution des comportements humains est nécessaire.Dans le premier chapitre, nous développons un modèle stochastique individus centré couplant des changements de comportement dans une population de taille fixe avec une dynamique environnementale dépendant des comportements dans la population.Les individus peuvent passer d'un comportement à un autre par le biais de deux types de transfert différents: les interactions sociales ou l'évaluation individuelle de la dégradation environnementale.Le comportement actif contribue moins à la dégradation de l'environnement, mais est plus coûteux à adopter que le comportement de base.Nous étudions la dynamique déterministe associée au système et les différents types d'équilibre admissible en temps longs. Cette étude met en avant l'existence de jeux de paramètres pour lesquels le système admet trois points d'équilibre, dont deux stables, dans le système: un des équilibre admet une majorité d'individus actifs et l'autre équilibre stable favorise une majorité d'individus basiques.Il est nécessaire de comprendre le comportement en temps long du système stochastique lorsque le système déterministe qui lui est associé admet trois points d'équilibre.Le second chapitre cherche à répondre à cette question pour un système plus simple ayant un équilibre instable encadré de deux équilibres stables. Nous étudions asymptotiquement de ce système différentiel stochastique à deux composantes dont le coefficient de diffusion est dégénéré. Nous étendons les résultats de Freidlin et Wentzell et nous montrons sous quelles conditions cette mesure se concentre autour d'un seul des équilibres stables du système.Le chapitre trois permet de faire le lien entre les deux premiers chapitres.Nous reparamétrisons le système biologique introduit dans le premier chapitre et nous entendons les résultats du deuxième chapitre à ce nouveau système. Cette étude des limites asymptotiques nous permet de caractériser le comportement du système lorsqu'il admet trois points d'équilibre. Nous vérifierons les conjectures faites sur les paramètres grâce à des simulations numériques de la mesure d'occupation.
• Quantification d'incertitudes au sein des réseaux de neurones : Application à la mesure automatisée de la taille de particules de TiO2
  
  • Monchot Paul
  , 2023. L'utilisation croissante de solutions technologiques fondées sur des algorithmes d'apprentissage profond a connu une explosion ces dernières années en raison de leurs performances sur des tâches de détection d'objets, de segmentation d'images et de vidéos ou encore de classification, et ce dans de nombreux domaines tels que la médecine, la finance, la conduite autonome ... Dans ce contexte, la recherche en apprentissage profond se concentre de plus en plus sur l'amélioration des performances et une meilleure compréhension des algorithmes utilisés en essayant de quantifier l'incertitude associée à leurs prédictions. Fournir cette incertitude est clé pour une dissémination massive de ces nouveaux outils dans l'industrie et lever les freins actuels pour des systèmes critiques notamment. En effet, fournir l'information de l'incertitude peut revêtir une importance réglementaire dans certains secteurs d'activité.Ce manuscrit expose nos travaux menés sur la quantification de l'incertitude au sein des réseaux de neurones. Pour commencer, nous proposons un état des lieux approfondi en explicitant les concepts clés impliqués dans un cadre métrologique. Ensuite, nous avons fait le choix de nous concentrer sur la propagation de l'incertitude des entrées à travers un réseau de neurones d'ores-et-déjà entraîné afin de répondre à un besoin industriel pressant. La méthode de propagation de l'incertitude des entrées proposée, nommée WGMprop, modélise les sorties du réseau comme des mixtures de gaussiennes dont la propagation de l'incertitude est assurée par un algorithme Split&Merge muni d'une mesure de divergence choisie comme la distance de Wasserstein. Nous nous sommes ensuite focalisés sur la quantification de l'incertitude inhérente aux paramètres du réseau. Dans ce cadre, une étude comparative des méthodes à l'état de l'art a été réalisée. Cette étude nous a notamment conduit à proposer une méthode de caractérisation locale des ensembles profonds, méthode faisant office de référence à l'heure actuelle. Notre méthodologie, nommée WEUQ, permet une exploration des bassins d'attraction du paysage des paramètres des réseaux de neurones en prenant en compte la diversité des prédicteurs. Enfin, nous présentons notre cas d'application, consistant en la mesure automatisée de la distribution des tailles de nanoparticules de dioxyde de titane à partir d'images acquises par microscopie électronique à balayage (MEB). Nous décrivons à cette occasion le développement de la brique technologique utilisée ainsi que les choix méthodologiques de quantification d'incertitudes découlant de nos recherches.
• Nouveaux outils d’apprentissage statistiques pour l’imputation et le pronostic en conservation
  
  • ALAO AFOLABI Shadé
  • Hélou-de la Grandière Pauline
  • Coustet Chloë
  • Thoury Mathieu
  • Perruchini Elsa
  • Le Pennec Erwan
  • Cohen Serge X.
  , 2023.
• Large population limit for a multilayer SIR model including households and workplaces
  
  • Kubasch Madeleine
  , 2023. We study a multilayer SIR model with two levels of mixing, namely a global level which is uniformly mixing, and a local level with two layers distinguishing household and workplace contacts, respectively. We establish the large population convergence of the corresponding stochastic process. For this purpose, we use an individual-based model whose state space explicitly takes into account the duration of infectious periods. This allows to deal with the natural correlation of the epidemic states of individuals whose household and workplace share a common infected. In a general setting where a non-exponential distribution of infectious periods may be considered, convergence to the unique deterministic solution of a measurevalued equation is obtained. In the particular case of exponentially distributed infectious periods, we show that it is possible to further reduce the obtained deterministic limit, leading to a closed, finite dimensional dynamical system capturing the epidemic dynamics. This model reduction subsequently is studied from a numerical point of view. We illustrate that the dynamical system derived from the large population approximation is a pertinent model reduction when compared to simulations of the stochastic process or to an alternative edgebased compartmental model, both in terms of accuracy and computational cost.
• Numerical simulation of the impact of a gas jet on a free water surface
  
  • Haegeman Ward
  • Le Touze Clément
  • Dupays Joël
  • Massot Marc
  Multiphase Science and Technology, Begell House, 2023, 36 (1), pp.27-48. In this work, we are interested in the numerical simulation of a high-speed hot jet impinging on a free liquid surface at rest by means of diffuse interface models. We first consider the case of a low-temperature subsonic jet; a 4-equation model is used on a 2D axi-symmetric setup. Turbulence is accounted for by solving the Reynolds averaged equations and using a k-ω turbulence model. Numerical results are evaluated by comparing the depth of the cavity formed in the liquid surface to the predicted values using theoretical models from the literature. We then consider the case of a high-temperature jet. We start by showing equilibrium assumption between the liquid and gas phases which is no longer valid. A 5-equation model that does not rely on this assumption is presented. Both models are compared numerically on a simplified set-up. (10.1615/MultScienTechn.2023047916)
  DOI : 10.1615/MultScienTechn.2023047916
• Uniform attachment with freezing: Scaling limits
  
  • Bellin Etienne
  • Blanc-Renaudie Arthur
  • Kammerer Emmanuel
  • Kortchemski Igor
  , 2023.
• Linear algebra over T-pairs
  
  • Akian Marianne
  • Gaubert Stéphane
  • Rowen Louis
  , 2023. This paper treats linear algebra over a semiring pair, in a wide range of applications to tropical algebra and related areas such as hyperrings and fuzzy rings. First we present a more general category of pairs with their morphisms, called ``weak morphisms,'' paying special attention to supertropical pairs, hyperpairs, and the doubling functor. Then we turn to matrices and the question of whether the row rank, column rank, and submatrix rank of a matrix are equal. Often the submatrix rank is less than or equal to the row rank and the column rank, but there is a counterexample to equality, discovered some time ago by the second author, which we provide in a more general setting (``pairs of the second kind'') that includes the hyperfield of signs. Additional positive results include a version of Cramer's rule, and we find situations when equality holds, encompassing results by Akian, Gaubert, Guterman, Izhakian, Knebusch, and Rowen. We pay special attention to the question of whether $n+1$ vectors of length $n$ need be dependent. At the end, we introduce a category with stronger morphisms, that preserve a surpassing relation.
• Inverse scattering problems without phase information
  
  • Sivkin Vladimir
  , 2023. This thesis is devoted to different approaches to phaseless inverse scattering problems. Our studies are motivated by problems of tomographies which use elementary particles (for example, electrons, X-ray photons) as probing tool. In these tomographies only the absolute values of scattering data are measurable.In the framework of quantum mechanics, this limitation is related to the Born principle that complex values of the wave function don’t have direct physical interpretation, whereas its absolute values squared admit probabilistic interpretation and can be directly measured. In the framework of optics (including X-ray scattering) this limitation is related to very high wave frequencies, which don’t allow to measure wave phase directly by modern technical devices.We contribute to phaseless inverse scattering by developing the method of background scatterers and the multipoint method.The method of background scatterers uses scattering in presence of a priori known objects. By our results, in this connection, we also contribute to the phase retrieval problem for the classical Fourier transform. The multipoint method consists in finding important leading terms (not accessible for direct measurements) in asymptotic expansion of a function from several values of this function (accessible for direct measurements).For both methods, we give, in particular, new explicit formulas for various phaseless and phased inverse scattering problems. In addition, we implement numerically many of our theoretical results. In these implementations we use, in particular, old and new regularisation technics.Our algorithms can be applied, for example, to X-ray imaging and to electron tomography.
• Distributed Monte Carlo simulation with large-scale Machine Learning : Bayesian Inference and Conformal Prediction
  
  • Plassier Vincent
  , 2023. Centralizing data is impractical or undesirable in many scenarios, especially when sensitive information is involved. In such cases, the need for alternative methods becomes evident. As large datasets are known to facilitate the learning of efficient models, distributed methods have emerged as a powerful tool to overcome the challenges posed by centralized data. Consequently, this thesis introduces innovative approaches to tackle large-scale Bayesian inference and uncertainty quantification, aiming to provide effective solutions in the context of distributed data environments. The federated Monte Carlo (MC) approaches allow multiple agents/nodes to conduct computations locally and securely, with a central server combining the results to obtain samples from the global posterior distribution. Bayesian posterior sampling techniques benefit from the incorporation of prior knowledge, leading to improved results. Additionally, the uncertainty associated with the parameters and the predictions are naturally quantified, which is crucial for decision-making. Especially with limited or noisy data, the ability to quantify uncertainty becomes even more essential.The first part of this manuscript focuses on MC via Markov chains (MCMC) methods. In particular, we introduce two procedures, named DG-LMC and FALD, designed to target a global posterior distribution while ensuring scalability. Local agents are associated with a central server that aggregates information from each agent to generate samples from the posterior distribution. This approach minimizes the need to transmit large amounts of data across participating agents, making it especially advantageous in federated environments with limited bandwidth or low computational power. Considering the distributed nature of today's datasets, concerns about trust and confidence arise when transferring information to a central server. The proposed methods not only address practical applications but also extend existing learning algorithms to Bayesian inference problems. The proposed approach contributes to the development of more robust and efficient machine learning algorithms, and holds potential applications in various domains, including epidemiology and finance, where large-scale inference and data privacy are significant concerns. To demonstrate the effectiveness of the approach, real-world datasets are employed, and the results show the performance of federated MCMC simulation.The second part of the thesis focuses on uncertainty management. Initially, we present the Bayesian approach, which involves defining a prior and a likelihood. To address bandwidth bottlenecks while efficiently generating samples, our proposed approach leverages compression operators. In the final part of this thesis, we introduce a novel frequentist FL method based on conformal predictions. Unlike other methods, our model-agnostic approach does not rely on specific model assumptions and can be applied to any underlying prediction model. Referred to as DP-FedCP, this method leverages quantile regression techniques to generate personalized prediction sets while maintaining robustness to outliers. The label shift between agents is addressed by determining quantiles based on importance weights. One crucial aspect of our approach is the preservation of differential privacy, it allows users to assess the confidence level of predictions and make informed decisions based on the associated level of uncertainty. By incorporating this privacy measure, we ensure safeguarding the user' sensitive information.
• Numerical analysis and methods for mean-field-type optimization problems
  
  • Liu Kang
  , 2023. This thesis deals with the numerical analysis and methods for optimization problems and potential games involving a large number of agents. We consider asymptotic models obtained through a mean-field approximation; they exhibit convexity properties of great interest. We focus on large-scale aggregative optimization problems, for which the objective function depends on an aggregate term, which is the sum of the contributions of the agents to some common good. We also focus on potential Mean Field Game (MFG) models, which are limit models for differential games. The thesis consists of four contributions.1) We propose a mean-field relaxation for aggregative optimization problems, obtained by randomization. The relaxation gap is estimated to be of order O(1/N), where N represents the number of individuals. We develop and prove the convergence of a stochastic variant of the Frank-Wolfe algorithm, called SFW algorithm, to address the original aggregative problem.2) We formulate a general class of optimization problems involving a set of probability distributions with a prescribed marginal m. We call them Mean Field Optimization (MFO) problems. Our framework contains the relaxed aggregative problems as well as some Lagrangian potential MFGs. We demonstrate a stability result with respect to perturbations of m. It enables us to derive an error estimate for a numerical method relying on a discretization of m and the SFW algorithm.3) We introduce a novel finite-difference scheme, called theta-scheme, for solving monotone second-order MFGs. We give a precise convergence result for the theta-scheme, of order O(h^r), where h is the step length of the space variable and 0<r<1 is related to the Hölder continuity of the solution of the continuous problem and some of its derivatives.4) We consider the resolution of potential second-order MFGs with the generalized Frank-Wolfe algorithm, combined with the theta-scheme. We prove a sublinear and a linear rate of convergence for this algorithm. More importantly, these rates possess the mesh-independence property, i.e., the convergence constants are independent of the discretization parameters.
• Strong error bounds for the convergence to its mean field limit for systems of interacting neurons in a diffusive scaling
  
  • Erny Xavier
  • Löcherbach Eva
  • Loukianova Dasha
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2023, 33 (5). We consider the stochastic system of interacting neurons introduced in De Masi et al. (2015) and in Fournier and L\"ocherbach (2016) and then further studied in Erny, L\"ocherbach and Loukianova (2021) in a diffusive scaling. The system consists of $N$ neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the potential of the spiking neuron is reset to $0$ and all other neurons receive an additional amount of potential which is a centred random variable of order $ 1 / \sqrt{N}.$ In between successive spikes, each neuron's potential follows a deterministic flow. In our previous article Erny, L\"ocherbach and Loukianova (2021) we proved the convergence of the system, as $N \to \infty$, to a limit nonlinear jumping stochastic differential equation. In the present article we complete this study by establishing a strong convergence result, stated with respect to an appropriate distance, with an explicit rate of convergence. The main technical ingredient of our proof is the coupling introduced in Koml\'os, Major and Tusn\'ady (1976) of the point process representing the small jumps of the particle system with the limit Brownian motion. (10.1214/22-AAP1900)
  DOI : 10.1214/22-AAP1900
• Large deviation Local Limit Theorems and limits of biconditioned Trees and Maps
  
  • Kortchemski Igor
  • Marzouk Cyril
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2023, 33 (5). We first establish new local limit estimates for the probability that a nondecreasing integer-valued random walk lies at time $n$ at an arbitrary value, encompassing in particular large deviation regimes. This enables us to derive scaling limits of such random walks conditioned by their terminal value at time $n$ in various regimes. We believe both to be of independent interest. We then apply these results to obtain invariance principles for the Lukasiewicz path of Bienaym\'e-Galton-Watson trees conditioned on having a fixed number of leaves and of vertices at the same time, which constitutes a first step towards understanding their large scale geometry. We finally deduce from this scaling limit theorems for random bipartite planar maps under a new conditioning by fixing their number of vertices, edges, and faces at the same time. In the particular case of the uniform distribution, our results confirm a prediction of Fusy & Guitter on the growth of the typical distances and show furthermore that in all regimes, the scaling limit is the celebrated Brownian map. (10.1214/22-AAP1906)
  DOI : 10.1214/22-AAP1906
• Approximations déterministes de modèles stochastiques multi-niveaux en épidémiologie
  
  • Kubasch Madeleine
  • Bansaye Vincent
  • Deslandes François
  • Vergu Elisabeta
  , 2023.
• The Physisorbate-Layer Problem Arising in Kinetic Theory of Gas-Surface Interaction
  
  • Aoki Kazuo
  • Giovangigli Vincent
  • Golse François
  • Kosuge Shingo
  , 2023. A half-space problem of a linear kinetic equation for gas molecules physisorbed close to a solid surface, relevant to a kinetic model of gas-surface interactions and derived by Aoki et al. (K. Aoki et al., in: Phys. Rev. E 106:035306, 2022), is considered. The equation contains a confinement potential in the vicinity of the solid surface and an interaction term between gas molecules and phonons. It is proved that a unique solution exists when the incoming molecular flux is specified at infinity. This validates the natural observation that the half-space problem serves as the boundary condition for the Boltzmann equation. It is also proved that the sequence of approximate solutions used for the existence proof converges exponentially fast. In addition, numerical results showing the details of the solution to the half-space problem are presented.
• Equilibrium Configurations For Nonhomogeneous Linearly Elastic Materials With Surface Discontinuities
  
  • Chambolle Antonin
  • Crismale Vito
  Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Scuola Normale Superiore, 2023, XXIV (3), pp.1575-1610. We prove a compactness and semicontinuity result that applies to minimisation problems in nonhomogeneous linear elasticity under Dirichlet boundary conditions. This generalises a previous compactness theorem that we proved and employed to show existence of minimisers for the Dirichlet problem for the (homogeneous) Griffith energy. (10.2422/2036-2145.202006_002)
  DOI : 10.2422/2036-2145.202006_002
• Dynamic mixture models and longitudinal monitoring for mixed-type and spatio-temporal data inference : application in Public Health
  
  • Pruilh Solange
  , 2023. In this thesis, we focus on statistical learning methods for spatio-temporal and mixed-type data. With the rapid growth of public health information systems, a wide range of real-time data is now available for many diseases. The aim is to develop methods for using this data to build operational decision support systems.We first propose a spatio-temporal pipeline for estimating the distribution of a population and highlighting temporal differences. This pipeline is a first step towards a decision support and alert system for the spatio-temporal analysis of population trends. This pipeline is designed so that different distributions and therefore different algorithms can be considered. For an initial application, this pipeline is combined with robust EM algorithms for estimating Gaussian mixture models. It is evaluated using data from Paris hospitals corresponding to people who tested positive for SARS-CoV-2 infection over eleven weeks in 2020.In the second part, we propose a set of algorithms for estimating statistical models on mixed data. We consider that mixed-type data are distributed according to mixtures of laws. We first describe mixture models for various continuous and discrete laws, assuming local independence between discrete and continuous variables. We then propose dynamic algorithms of the EM type, allowing the estimation of all the parameters of the mixture as well as the estimation of the number of classes. We show that our different dynamic algorithms allow us to reach the real number of classes and to correctly estimate the parameters of the discrete and continuous laws. We also highlight the benefits of introducing regularizations to improve performance in situations where the sample size is insufficient for the complexity of the model. These dynamic algorithms are then validated on real data from the literature.
• Scattered wavefield in the stochastic homogenization regime
  
  • Garnier Josselin
  • Giovangigli Laure
  • Goepfert Quentin
  • Millien Pierre
  , 2023. In the context of providing a mathematical framework for the propagation of ultrasound waves in a random multiscale medium, we consider the scattering of classical waves (modeled by a divergence form scalar Helmholtz equation) by a bounded object with a random composite micro-structure embedded in an unbounded homogeneous background medium. Using quantitative stochastic homogenization techniques, we provide asymptotic expansions of the scattered field in the background medium with respect to a scaling parameter describing the spatial random oscillations of the micro-structure. Introducing a boundary layer corrector to compensate the breakdown of stationarity assumptions at the boundary of the scattering medium, we prove quantitative $L^2$- and $H^1$- error estimates for the asymptotic first-order expansion. The theoretical results are supported by numerical experiments.
• Symmetrization and local existence of strong solutions for diffuse interface fluid models
  
  • Giovangigli Vincent
  • Calvez Yoann Le
  • Nabet Flore
  Journal of Mathematical Fluid Mechanics, Springer Verlag, 2023, 25 (4), pp.82. We investigate compressible nonisothermal diffuse interface fluid models also termed capillary fluids. Such fluid models involve van der Waals' gradient energy, Korteweg's tensor, Dunn and Serrin's heat flux as well as diffusive fluxes. The density gradient is added as an extra variable and the convective and capillary fluxes of the augmented system are identified by using the Legendre transform of entropy. The augmented system of equations is recast into a normal form with symmetric hyperbolic first order terms, symmetric dissipative second order terms and antisymmetric capillary second order terms. New a priori estimates are obtained for such augmented system of equations in normal form. The time derivatives of the parabolic components are less regular than for standard hyperbolic-parabolic systems and the strongly coupling antisymmetric fluxes yields new majorizing terms. Using the augmented system in normal form and the a priori estimates, local existence of strong solutions is established in an Hilbertian framework. (10.1007/s00021-023-00825-4)
  DOI : 10.1007/s00021-023-00825-4
• Introduction
  
  • Bansaye Vincent
  • Kuhn Estelle
  • Moireau Philippe
  MathematicS In Action, Société de Mathématiques Appliquées et Industrielles (SMAI), 2023, 12, pp.1-1. At first glance, mathematics and life sciences may appear to be two distant fields. However, history has shown the importance of mathematical modeling in understanding and analyzing biological phenomena. Research has evolved and developed considerably recently, with numerous collaborations between mathematicians and biologists, stimulating both communities. The interface is very active and diversified today, from highly theoretical questions to the most applied aspects. This interface is stimulated by the need for and complexity of models and technological advances and the influx of data. In this special issue of the recent SMAI Math’s in Action open journal, we aim to show the vitality and scientific richness of the work being done today at this interface, where original mathematical results have helped to answer questions raised by biological, ecological or medical applications. Indeed, the reader will find a wide range of applications in this issue, from epidemiology to medical imaging, from subcellular biology to organ physiology. But also a wide range of mathematical tools from probability to statistics, from mathematical analysis of partial differential equations to theoretical computer science using formal methods, from theory to numerics. We would like to dedicate this special issue to Elisabeta Vergu, whose research and scientific leadership have contributed significantly to the development of the interface between mathematics and epidemiology. (10.5802/msia.28)
  DOI : 10.5802/msia.28
• Resolving a Clearing Member’s Default, A Radner Equilibrium Approach
  
  • Bastide Dorinel
  • Crépey Stéphane
  • Drapeau Samuel
  • Tadese Mekonnen
  , 2023. For vanilla derivatives that constitute the bulk of investment banks’ hedging portfolios, central clearing through central counterparties (CCPs) has become hegemonic. A key mandate of a CCP is to provide an efficient and proper clearing member default resolution procedure. When a clearing member defaults, the CCP can hedge and auction or liquidate its positions. The counterparty credit risk cost of auctioning has been analyzed in terms of XVA metrics in Bastide, Crépey, Drapeau, and Tadese (2023). In this work we assess the costs of hedging or liquidating. This is done by comparing pre- and post-default market equilibria, using a Radner equilibrium approach for portfolio allocation and price discovery in each case. We show that the Radner equilibria uniquely exist and we provide both analytical and numerical solutions for the latter in elliptically distributed markets. Using such tools, a CCP could decide rationally on which market to hedge and auction or liquidate defaulted portfolios.