2021 - 2022 Program

Title : Urns in Lévy processes.

Speaker: Philippe Marchal, Paris 13 University. France.

Abstract: We show how generalized Polyà urns can be used in the anaysis of some Lévy processes in the domain of attraction of a stable processes. This enables us to recover the result of Pitman and Yor on interval partitions derived from a stable subordinator and to show that the associated fragmention process is the Ruelle cascade. This also allows explicit computations of entrance times for some subordinators.

Title : Truchet Tiles and Combinatorial Arabesque. (link to slides)

Speaker: Hedi Nabli, Faculte des Sciences de Sfax. Tunisia.

Abstract: We introduce a new topic on geometric patterns, issued from Truchet tile, that we agree to call combinatorial arabesque. The originally Truchet tile, by reference to the French scientist of 17th century Sébastien Truchet, is a square split along the diagonal into two triangles of contrasting colors. We define an equivalence relation on the set of all square tiling of same size, leading naturally to investigate the equivalence classes and their cardinality. Thanks to this class notion, it will be possible to measure the beauty degree of a Truchet square tiling by means of an appropriate algebraic group. Also, we define many specific arabesques such as entirely symmetric, magic and hyper-maximal arabesques. Mathematical characterizations of such arabesques are established facilitating thereby their enumeration and their algorithmic generating. Finally the notion of irreducibility is introduced on arabesques.

Title : The inflation phenomenon.

Speaker: Claude Lobry, France.

Abstract: Work in progress with M. Benaim (Neuchatel), T. Sari (Montpellier) et E.

Strickler (Nancy).

Although long suspected by ecological theorists, it is only recently that the

phenomenon of inflation which will be reported in the conference has been the

subject of mathematical studies [11, 9]. It was brought to light some twenty years

ago by the well known theorist of ecology R. Holt [6]. It is the following property ... (see more on Pdf File).

Title: Interacting urns on a finite directed graph. (Link to the publication) (Link to slides)

Speaker: Gursharn Kaur. Biocomplexity institute, Virginia University. USA.

Abstract: We introduce a general two-colour interacting urn model on a finite directed graph, where each urn at a node reinforces all the urns in its out-neighbours according to a fixed, non-negative and balanced reinforcement matrix. We show that the fraction of balls of either colour converges almost surely to a deterministic limit if either the reinforcement is not of Polya type or if the graph is such that every vertex with non-zero in-degree can be reached from some vertex with zero in-degree. We also obtain joint central limit theorems with appropriate scaling. Further, in the remaining case when there are no vertices with zero in-degree and the reinforcement is of Polya type, we restrict our analysis to a regular graph and show that the fraction of balls of either colour converges almost surely to a finite random limit, which is the same across all the urns.

Title : Tumor growth simulation using partial differential equations and agent- based modeling. (Common work with Salma Chabbar, Abderrahmane Habbal, and El Mahdi EL Guarmah). Link to video

Speaker: Rajae Aboulaich. Mohammadia School of Engineering, LERMA, Rabat, Morocco.

Abstract: In this work, two approaches for tumor growth modeling are presented. In the first one, a macroscopic approach using a PDE model is implemented using the level set method to track the tumor moving boundary in one hand by using Darcy’s law to compute the normal velocity of the free boundary and on the other hand using the shape optimization to draw the normal velocity. In the second one, a microscopic approach which focuses on the cellular scale is presented. A hybrid model using agent-based modeling for the cell behavior and a PDE for the description of the tumor environment is presented, numerical experiments are provided. A sensitivity analysis is conducted on the hybrid model for a better understanding of its parameters impact on the tumor development.

Title : Directed Hybrid Random Networks Mixing Preferential Attachment with Uniform Attachment Mechanisms. Link To Video -

Speaker: Panpan Zhang. University of Pennsylvania, Philadelphia, USA.

Abstract: Motivated by the complexity of network data, we propose a directed hybrid random network that mixes preferential attachment (PA) rules with uniform attachment (UA) rules. When a new edge is created, with probability p in [0,1], it follows the PA rule. Otherwise, this new edge is added between two uniformly chosen nodes. Such mixture makes the in- and out-degrees of a fixed node grow at a slower rate, compared to the pure PA case, thus leading to lighter distributional tails. For estimation and inference, we develop two numerical methods which are applied to both synthetic and real network data. We see that with extra flexibility given by the parameter p, the hybrid random network provides a better fit to real-world scenarios, where lighter tails from in- and out-degrees are observed.

Title : Explicit local time-stepping methods for wave propagation. Link To Video.

Speaker:Marcus J. Grote, University of Basel, Switzerland.

Abstract: In the presence of complex geometry, adaptivity and mesh refinement are certainly key for the efficient numerical simulation of wave phenomena. Locally refined meshes, however, impose severe stability constraints on any explicit time-marching scheme, where the maximal time-step allowed by the CFL condition is dictated by the smallest elements in the mesh. When mesh refinement is restricted to a small subregion, the use of implicit methods, or a very small time-step in the entire computational domain, are very high a price to pay.

Explicit local time-stepping schemes (LTS) overcome the bottleneck due to a few small elements by using smaller time-steps precisely where the smallest elements in the mesh are located. When combined with a finite element discretization in space with an essentially diagonal mass matrix, the resulting time-marching schemes are fully explicit and thus inherently parallel.

Title : The multi-patch logistic equation. Link To slides. Link to Video.

Speaker: Tewfik Sari, INRAE, France.

Work with : Roger Arditi (Fribourg, Switzerland and Paris, France), Tounsia Benzekri (Algiers, Algeria),

Bilal Elbetch (Saïda, Algeria), Claude Lobry (Nice, France) and Daniel Massart (Montpellier, France).

Abstract: We consider a multi-patch model, each patch following a logistic law, the patches being coupled by migration terms. First, in the case of perfect mixing, i.e. when the migration rate tends to infinity, the total population follows a logistic law with a carrying capacity which in general is different from the sum of the carrying capacities, and depends on the migration terms. Second, we determine, in some particular cases, the conditions under which fragmentation and migration can lead to a total equilibrium population greater or smaller than the sum of the carrying capacities. Finally, for the three-patch model, we show numerically the existence of at least three critical values of the migration rate for which the total equilibrium population equals the sum of the carrying capacities. The results are related to the SLOSS (single large or several small) debate and extend previous studies on the two patch case.

Title : MEAN FIELD ANALYSIS OF STOCHASTIC NETWORKS MODELING VEHICLE-SHARING SYSTEMS.

Speaker: Hanen Mohamed, Paris Nanterre University.

Abstract: Vehicle-sharing systems are becoming important for urban transportation. I will first focus on bike-sharing systems where users arrive at a station, take a bike and use it for a while, then return it to another station of their choice. Each station cannot host more bikes than its finite capacity. I will present a stochastic model for an homogeneous bike-sharing system introduced by Fricker & Gast where the dimensioning system problem is investigated. The aim is to derive the fleet size (number of bikes per station) giving the best system performance measured in term of the proportions of unsatisfied users facing an empty or a saturated station, i.e. stations that, (at a given time) at long time, have no bikes available or no available spots for bikes to be returned to. A large-scale analysis is (conducted) provided using a mean field approach and leads to simple expressions (that give qualitative and quantitative results) for optimal performance. This method works even if there is no closed-form expression for the model. It allows to study several variants taking into account user's preferences or incentive policies proposed by the operator. I will present some variants (heterogeneous model, supermarket model, influence of locality). It is quite amazing that a similar modeling of car-sharing systems presents a wealth of behaviors as soon as the reservation of either the vehicle or the car, or both, is added. More refined probabilistic tools are then sometimes needed.

Title : Geospatial Data Science for Public Health Surveillance

Speaker: Paula Moraga, King Abdullah University of Science and technology, KSA.

Abstract: Geospatial health data are essential to inform public health and policy. These data can be used to quantify disease burden, understand geographic and temporal patterns, identify risk factors, and measure inequalities. In this talk, I will give an overview of my research which focuses on the development of geospatial methods and interactive visualization applications for health surveillance. I will present disease risk models where environmental, demographic and climatic data are used to predict the risk and identify targets for intervention of lymphatic filariasis in sub-Saharan Africa, and leptospirosis in a Brazilian urban slum. I will also show the R packages epiflows for risk assessment of travel-related spread of disease, and SpatialEpiApp for disease mapping and the detection of clusters. Finally, I will describe my future research and how it can inform better surveillance and improve population health globally.

Title: Asymptotic behavior of one-dimensional wave equations with set-valued boundary damping.

Speaker: Yacine Chitour, Paris Saclay University. France.

Abstract: This talk concerns one dimensional wave equations with nonlinear boundary damping . After providing a brief summary of some important previous works, we will present a new framework for addressing well-posedness and stability issues for this PDE. We shall consider wave equations in Lp functional spaces and with set-valued boundary dampings, which are a natural generalization of nonlinear dampings allowing to fully exploit some symmetry properties previously observed and for which we can provide the most general well-posedness and existence results. We will show how our techniques allow us to retrieve known results on the asymptotic behavior and provide answers to previously open questions. In particular, we provide a complete characterization of the asymptotic behavior of systems in which the boundary condition is described by the sign function and we also address input-to-state stability with respect to boundary perturbations. This talk is based on joint works with Swann Marx and Guilherme Mazanti.

Title: Stiffness Recovery of Euler-Bernouilli beams from in-situ data.

Speaker: Faouzi Ghrib. University of Windsor. Canada.

Abstract: Recent bridge failures have raised concerns about the health of aging infrastructures across the world. During its life service, a structure should meet a series of safety requirements. The degradation of the structural integrity increases with the age of the infrastructure. The aging of infrastructures has called for the development of accurate structural damage detection methods. Structural Health Monitoring (SHM) has gained a wide research interest in the last two decades with the advances of sensors technologies, data processing and information management. Damage assessment of existing infrastructure is a fundamental tool in SHM. To prevent the occurrence of catastrophic failures, it is imperative to develop damage detection techniques that are simple, robust, and capable of accurately characterizing the extent and location of existing damage in a structure. We introduce two techniques for assessing the damage in Euler-Bernoulli beams from in-situ measurements. The first technique uses the SIFT algorithm to extract the displacement along the beam. The stiffness recovery problem is then formulated as an inverse problem. In the second technique, we present a new algorithm for optimizing the placement of sensors on beams for an effective damage assessment. Numerical and experimental results are illustrated to show the efficiency of the two techniques.

Title : Vector fields and large combinatorial objects

Speaker: Philippe Chassaing, University of Lorraine and Institut Elie Cartan. France

Abstract: We will show that the asymptotic enumeration of accessible deterministic complete automata, which are the input of Moore's algorithm, can be carried out through the analysis of the trajectories of a vector field closely related to the Stirling numbers of the second species, and we will examine from this point of view some other combinatorial triangles.

(In collaboration with Jules Flin and Alexis Zevio)

Title : Certified randomness generation using quantum theory,

Speaker: Omar Fawzi, Haute Alsace University, France.

Abstract: Quantum information aims to exploit quantum theory to achieve improved information processing. In this talk, I will focus on the task of generating certified randomness using quantum devices. I will start by introducing Bell's theorem and how it can be exploited to generate certified randomness. Then, I will discuss a mathematical tool that we called the entropy accumulation theorem which allows one to precisely quantify the randomness generated by such a protocol. The entropy accumulation theorem can be seen as an extension of the asymptotic equipartition property in information theory where the random variables are not required to be independent and identically distributed.

The talk will be mostly based on the papers

https://arxiv.org/abs/1607.01796

https://www.nature.com/articles/s41467-017-02307-4 and follow up works.

Title : Fast Signal Recovery from Quadratic Measurements.

Speaker: Chrysoula Tsogka. University Of California, Merced, USA.

Abstract: We present a novel approach for recovering a sparse signal from quadratic measurements corresponding to a rank-one tensorization of the data vector. Such quadratic measurements, referred to as interferometric or cross-correlated data, naturally arise in many fields such as remote sensing, spectroscopy, holography and seismology. Compared to the sparse signal recovery problem that uses linear measurements, the unknown in this case is a matrix, $X=\rho \rho^*$, formed by the cross correlations of $\rho \in C^K$. This creates a bottleneck for the inversion since the number of unknowns grows quadratically in $K$. The main idea of the proposed approach is to reduce the dimensionality of the problem by recovering only the diagonal elements of the unknown matrix, $| \rho_i|^2$, $i=1,\ldots,K$. The contribution of the off-diagonal terms $\rho_i \rho_j^*$ for $i \neq j$ to the data is treated as noise and is absorbed using the Noise Collector approach introduced in [Moscoso et al, The noise collector for sparse recovery in high dimensions, PNAS 117 (2020)]. With this strategy, we recover the unknown by solving a convex linear problem whose cost is similar to the one that uses linear measurements. The proposed approach provides exact support recovery when the data is not too noisy, and there are no false positives for any level of noise.

Joint work with A. Novikov, M. Moscoso and G. Papanicolaou

Title : Non-asymptotic High-probability Bounds for Polyak-Ruppert Averaged Iterates of Linear Stochastic Approximation,

Speaker: ُEric Moulines. Ecole Polytechnique, Palaiseau, France.

Abstract: