4.1 Geophysics

Geophysics is a domain which requires the application of a broad range of mathematical tools related to probability and statistics while more and more data are collected. There are several directions in which we develop our methodology in relation with practitioners in the field.

• Avalanches (snow or rock) present intricate dynamical properties, with a wide variety of behaviors that largely depend on their environments. To model such phenomena, we apply tools from fragmentation theory, stochastic calculus, partial differential equations and branching processes. Our approach is new and paves the way to considering and constructing rigorous mathematical models and simulation procedures able to reproduce and control the real phenomenon by introducing more and more issues in the models.
• Understanding the behavior of subsurface and surface fluids is a major challenge in geophysics. We deal with two main axes: (1) using tools for spatial Bayesian statistics which consists in detecting the sources of the various components of fluids from their hydrogeochemical data, and (2) developing the suitable methodological and numerical tools to simulate diffusion processes (pollutant, water...) moving in heterogenous media in the presence of interfaces.
• Earthquake forecasting is notoriously difficult. To grasp the statistical distribution of seismic hazards, we consider setting up tools to detect seismic faults using marked point processes. Such a project presents challenging aspects concerning both the inference and the simulation of the processes.

On such topics, we hold long standing interdisciplinary collaborations with INRAE Grenoble, the RING Team (GeoRessources, University of Lorraine), IMAR (Institute of Mathematics of the Romanian Academy) in Bucharest. We also have the support of the interdisciplinary LUE Deepsurf project (University of Lorraine).

4.2 Astronomy and cosmology

We have longstanding and continuous cooperation with astronomers and cosmologists in France, Spain and Estonia. In particular, we are interested in using spatial statistics tools to detect galaxies and other star patterns such as filaments detection. Such developments require to design specific point processes giving appropriate morpho-statistical distributions, as well as specific inference algorithms which are based on Monte Carlo simulations and able to handle the large volume of data.

4.3 Complex systems for healthcare, insurance, social networks and telecomnunication networks

Graphs are essential to model complex systems such as the relations between agents, the spatial distribution of points that are connected such as stars, the connections in telecommunication networks, and so on. We develop various directions of the study of random graphs that are motivated by a large class of applications:

• The success of organ transplant operations depends on their capacity to comply in real time, with sharp compatibility constraints. Here, vertices represents at any given time receivers and donors, while edges represent compatilities. To improve the quality of such life-saving medical acts, we work on the optimization and control of organ transplant systems by stochastic matching models, namely, queueing models in which elements are matched in real time, following prescribed compatibility constraints.
• The modeling of epidemics, viruses on computer networks and message percolation on large social networks can be addressed using “large graph asymptotic” of random graphs. In particular, we work on Markov exploration algorithms on large Configuration Model graphs, to propose weak, but tractable approximations of such propagation phenomena on large networks.
• We have longstanding collaborations in the domain of performance analysis of telecommunication networks. In particular, we have pursued an intensive research activity on the modeling and analysis of queuing systems with reneging with applications to real-time networking; on the performance analysis of parallel service systems, which are a natural model for server farms and call centers, and the large-network analysis of CDMA-type (Code Division Multiple Access) communication protocols, using random graph modeling (representing the spatial interactions between agents). Telecomnunication and peer-to-peer networks are now completed by the rise of small connected devices and the need to provide appropriate and reliable communication protocols. We also recently moved toward ad-hoc networking and the Internet of Things (IoT). Using graph and game theory techniques, we aim at a proper definition, and dynamical analysis, of the notion of trust between agents of these networks.
• Using random field models on graphs, we have considered the simulation and inference of the relations between bibliographical data related to scientific literature. This provides us with an application of our techniques in the field of dynamical evolution of networks.
• We study the spatial distribution of random $T$-tessellation with the aim of providing models for agricultural parcels. Again, such a problem presents challenging aspects both for simulation and inference.
• Finally, we consider personalized recommendation systems for insurance which are based on life events, using self-excited processes.

We have longstanding collaborations on these topics with Agence de Biomédecine (ABM), Le Foyer (insurance company, Luxembourg), INRAE (Avignon), Dyogene (Inria Paris), Lip 6, UTC, LORIA (computer science laboratory, Nancy), University of Buenos Aires, Northwestern University and LAAS (CNRS, Toulouse).

5.1 Modeling, simulation and inference: Fragmentation equation

Participants: Gaetano Agazzotti, Jean Armel Bra, Madalina Deaconu, Antoine Lejay.

We have a strong interest in the fragmentation equation for understanding snow or rock avalanches. Our point of view is to explore the probabilistic representations of transport equations in this framework as well as the possibilities they offer. We developed a stochastic process that represents the typical evolution of the mass of a rock or of a snow aggregate subject to successive random breakages.

In particular, we have studied from analytic viewpoint the size of a population with a fixed rate of fragmentation, as well as models with a diffusion in space and a fragmentation in size 35.

We have also set-up a preliminary program toward the use of neural networks to recover the fragmentation rate and the fragmentation kernel from data using deep recurrent neural networks.

Furthermore, we obtained new results with Oana Lupaşcu-Stamate (Institute of Mathematical Statistics and Applied Mathematics, Bucharest) for a binary coagulation-fragmentation equation which describes the avalanches phenomena. We use tools from self-organized critical systems and construct an adapted stochastic process for this phenomenon. We obtain in particular the asymptotic behaviour of the stochastic process and develop a numerical method in order to approximate the solution 28.

5.2 Modeling, simulation and inference: Hawkes process

Participants: Madalina Deaconu, Antoine Lejay, Laurent Lesage.

Hawkes processes represent a common class of self-excited stochastic processes.

We have studied the use of Hawkes processes in the context of insurance. In particular, we built a recommendation system for insurance products based on individual probabilities of life events. This system was successfully tested on the database from the insurance company Le Foyer (Luxembourg) 24 and it is now used by the company. Other applications of Hawkes processes in insurance have also been studied by introducing and developing the study of multi-variate Hawkes processes 17.

5.3 Modeling and simulation: Hitting times for stochastic differential equations

Participants: Madalina Deaconu.

The numerical approximation of stochastic differential equations (SDEs) and in particular new methodologies to approximate hitting times of SDEs is a challenging problem which is important for a large class of practical issues such as: geophysics, finance, insurance, biology, etc.

With Samuel Herrmann (University of Burgundy) we made important progress on this topic by developing new methods. Recently we introduced methods for the strong convergence and pathwise approximation of one-dimensional SDEs.

In particular, we develop new techniques for the path approximation of Bessel processes of arbitrary dimension, as such a process represents the norm of a multi-dimensional Brownian motion 13. These are part of the family of the so-called ε-strong approximations. Our approach constructs jointly the sequences of exit times and corresponding exit positions of some well-chosen domains, the construction of these domains being an important step. We construct the algorithm for any dimension and treat separately the integer dimension case and the non integer framework, each situation requiring appropriate techniques. We prove the convergence of the scheme and provide the control of the efficiency with respect to the parameter ε. We expand the theoretical part by a series of numerical developments.

Together with Samuel Herrmann (University of Burgundy) and Cristina Zucca (University of Torino) we pursue our work on the exact simulation of the hitting times of multi-dimensional diffusions.

5.4 Inference: Sticky Brownian motion

Participants: Alexis Anagnostakis, Antoine Lejay.

With Denis Villemonais (IECL, University of Lorraine), we constructed an estimator of the stickyness parameter of the sticky Brownian motion and other general diffusion processes from high frequency observations. This work is based on the construction of suitable estimators for the local time and the occupation time. Besides, this work provides a construction of sticky stochastic differential equations 22, 26, 11.

5.5 Modeling: Rough invariant embedding

Participants: Antoine Lejay.

With Renaud Marty (IECL, University of Lorraine), we are studying an invariant embedding problem, which consists in solving a differential equation whose initial and terminal conditions are linked by a linear relation. Using tools from rough paths an global analysis theories, we consider Rough Differential Equations which extend ordinary differential equations driven by rough signals. In particular, we use our development in the context of equations driven by a Brownian motion while avoiding all the difficulties related to the use of anticipative stochastic calculus 29.

5.6 Inference: expansion of the maximum likelihood estimator

Participants: Antoine Lejay, Sara Mazzonetto.

We are studying an expansion of the maximum likelihood estimator using formal series expansions. The aim of this work is to understand the lack of Gaussianity in the non-asymptotic regime 30. We apply this expansion to the estimator of the skewness parameter of a skew Brownian motion, whose asymptotic mixed normality is also proved with a rate of convergence of order $1/4$ unlike the usual cases where it is of order $1/2$.

5.7 Inference: self exciting threshold model

Participants: Sara Mazzonetto.

After considering high frequency observations, we study new estimators for low frequency observations and the presence of several regimes for different threshold models which show mean-reversions features. This work is in progress.

5.8 Modeling and simulation: stochastic matching models and service systems

Participants: Jocelyn Begeot, Thomas Masanet, Pascal Moyal.

We have made the following advances regarding stochastic matching models:

In 31 we have constructed an original perfect sampling algorithm for CTMC's under a given control condition. This condition is reminiscent of the so-called “Dominated coupling from the past” (DCFTP) of Kendall et al. With this algorithm we are able to perfectly simulate the stationary state of stochastic matching models with reneging, allowing to estimate, by Monte Carlo techniques, the loss probability at equilibrium under various matching policies, a result that have crucial applications in practice, for instance for organ transplants models, which are subject to heavy time constraints.

In 18, with Jean Mairesse (CNRS/Lip6), we build a research program on the general topic of stochastic matching models with reneging, by proposing a new approach of this class of models, generating additional, fictional, classes in the compatibility graph, to account for the balking of items.

In the thesis of Jocelyn Begeot 23, we build on a previous article 36 to obtain an explicit characterization of the stability regions of a wide class of stochastic matching models and skill-based queueing systems.

In 19, with Ohad Perry (Northwestern University) we draw a research program on the essential aspect of correlations of patience times and service times in service systems. To model this, we propose a measure-valued Markov modeling to address services systems with reneging, to account for the correlation between the patience times of the tasks and their service times. This is a theoretically tedious, but essential, task for practical purposes, as most models proposed in the literature address independent patience and service times.

5.9 Modeling and simulation: Trust modeling and management in the Internet of Things

Participants: Pascal Moyal.

In the article 21 we propose a first modeling of the Trustworthiness in the IOT by game-theoretical approaches, in the presence of intelligent and malicious intrusions. Our construction is based in a dynamic phase-based modeling of the behavior of malicious nodes.

Following this line of thought, in the submitted papers 34, 19, we propose a framework for extensive simulations of concrete networks, showing the efficiency of our trust allocation management algorithm, based on game theory.

5.10 Modeling and simulation: Dynamics on large random graphs

Participants: Pascal Moyal.

In 12 with Mohamed Habib Diallo Aoudi and Vincent Robin (University of Technology of Compiègne), we obtain a large graph estimate of the matching coverage of local matching algorithms (greedy and minimal residual) on large bipartite random graphs constructed by the configuration model. We proceed by deriving the hydrodynamic limit, in the large-graph asymptotic, of a measure-valued Markov process representing the exploration of the graph together with its sequential construction. We are currently completing a second paper including the proof of convergence (in spaces of measure-valued processes) to the hydrodynamic limit, for general, non-necessarily bipartite, graphs, and for a large class of matching algorithms.

In an ongoing collaboration with Matthieu Jonckheere and Nahuel Soprano-Loto (CNRs/LAAS), we address the question of maximal marriages on the stochastic block model (SBM), in the large graph asymptotic. Here again, we proceed by a sequential construction of the graph, coupled with its exploration, and propose, under certain explicit feasibility condition on the connectivity matrix of the SBM, a matching algorithm that is able to achieve, infinitely often with probability 1, a perfect marriage. A draft is in preparation and will be submitted shortly for publication.

5.11 An interaction point process for Bayesian detection of multiple sources in groundwaters from hydrochemical data

Participants: Madalina Deaconu, Christophe Reype, Radu Stoica.

Detecting the number and composition of multiple sources in groundwaters from hydrochemical data has remained highly challenging. In this work we develop a new interaction point process that integrates geological knowledge for the purpose of automatic sources detection of multiple sources in groundwaters from hydrochemical data. The key assumption of this approach is to consider the unknown sources to be the realization of a point process. The probability density describing the sources distribution is built in order to take into account the multidimensional nature of the data and specific physical rules. The method was first calibrated on synthetic data and then tested on real data from geothermal and ore-forming hydrothermal systems 33, 25.

This work is a collaboration with Antonin Richard (University of Lorraine, GeoRessources).

5.12 Minimizing peculiar velocity catalog biases with point process modeling and Bayesian inference

Participants: Radu Stoica.

Galaxy peculiar velocities are excellent cosmological probes provided that biases inherent to their measurements are contained before any study. We propose a new algorithm based on an object point process model whose probability density is built to minimize the effects of Malmquist biases and uncertainties due to lognormal errors in radial peculiar velocity catalogs. The resulting configurations are bias-minimized catalogs. Tests are conducted on synthetic catalogs mimicking the second and third distance modulus catalogs of the Cosmicflows project, and then on observational catalogs. The large scale structure reconstructed with the Wiener filter technique applied to the bias-minimized observational catalogs matches with great success the local Universe cosmic web as depicted by redshift surveys of local galaxies. These new bias-minimized versions of Cosmicflows catalogs can now be used as a starting point for several studies including the production of simulations constrained to reproduce the local Universe.

This work is a collaboration with Jenny Sorce (CNRS Cristal - Lille) and Elmo Tempel (Tartu Observatory - University of Tartu).

5.13 Spatio-temporal modelling of the spread of Armillaria Ostoyae: a fungus in coniferous forests

Participants: Katarzyna Adamczyk Chauvat, Nathan Gillot, Radu Stoica.

Economic forests represent today an important part of the wood production on a global scale. Measuring and understanding the impact of diseases that occur in this type of forest with a very low variety of species is therefore an economic matter today. Our interest is focused on a parasitic fungus impacting coniferous forests, Armillaria Ostoyae.

The approach consists in considering a Markovian model with multiple labels on a lattice of the same size as the data, a Potts-type model: each tree will have a life span and a spatial neighborhood. We use statistical inference techniques, both classical (pseudo-likelihood) and recent (ABC methods: ABC Shadow), which we apply to a 20-year dataset.

5.14 Two dimensional seismic fault network interpretation using marked point processes

Participants: Radu Stoica.

Seismic fault interpretation is an important input to subsurface models. Since in seismic images the dominant features are reflection events corresponding to horizons, fault interpretation can be achieved by computing a fault probability image. We aim at quantifying the uncertainties related to the number and connectivity of faults honoring a given probability image, as all the possible fault networks can yield different outcomes in terms of subsurface behavior (e.g., reservoir flow). Fault networks are seen as realization of marked-point processes whose density are defined by an energy function. We use a simulated annealing framework based on Metropolis-Hastings algorithms, which makes it possible to find the global maximum of the probability density, built in the form of a Gibbs density. We apply the proposed approach to a 2-D seismic cross-section extracted from the Volve seismic cube (provided by Equinor).

This work is a collaboration with Fabrice Taty-Moukati, François Bonneau, Guillaume Caumon (University of Lorraine, GeoRessources) and Xinming Wu (University of Science and Technology of China, Hefei).

5.15 Statistical analysis and stochastic simulation of fracture networks

Participants: Radu Stoica.

Fracture networks (FN) are systems of complex mechanical discontinuities in rocks. They dramatically impact fluid flow acting as a drain or a barrier. With François Bonneau (University of Lorraine, Georessources), we propose to use marked-point processes to build a stochastic mathematical model for fracture characterization and to approximate FN with a collection of marked-points standing for straight-line segment with a record of the length and the strike azimuth of the horizon. Geologists usually use a characterization workflow integrating density or mark distributions, which are first-order metrics, to describe the number and the geometry of fractures. Recently, second-order characteristics have been used to characterize FN's inner correlation and spatial variability. We then investigate stochastic models whose realizations reproduce the first and second order characteristics of the observations. This work may open the path to a thinner classification of FN and to predictive stochastic simulations.

5.16 Modeling and inference: Estimation of singular diffusions

Participants: Alexis Anagnostakis, Pierre Etoré, Sara Mazzonetto.

We are extending our respective results on high frequency approximation of the local time of oscillating-skew-sticky diffusion processes. The purpose is to estimate the parameters of stickiness and skewness separately and to model some critical behavior in financial markets related to crisis periods. We managed to extend the results obtained during the PhD thesis of Alexis Anagnostakis in the context of sticky BM to more general estimators of local time and to oscillating-skew-sticky Brownian motion. Our goal is to reach rates of convergence obtained in 37. We are also dealing with the different problem non-uniqueness of solutions in this context.

5.17 Modeling and simulation: Navier-Stokes equation - stochastic modeling

Participants: Madalina Deaconu.

With Lucian Beznea (IMAR, Bucharest) and Oana Lupaşcu-Stamate (Institute of Mathematical Statistics and Applied Mathematics, Bucharest) we are developing a stochastic approach for the two-dimensional Navier-Stokes equation in a bounded domain. More precisely we consider the vorticity equation and construct a specific non-local branching process. This approach is new and can conduct to important advances as it will give also a new numerical algorithm if successful.

In particular, we obtained several results concerning the construction of a duality - time reversal process and also in the development of a numerical algorithm with a non-local branching process involving creation and disapearance of particles that mimic the physics of the vorticity in the boundary layer.

6.1 International initiatives

6.2 International research visitors

6.3 National initiatives

Participants: Alexis Anagnostakis, Madalina Deaconu, Antoine Lejay, Sara Mazzonetto.

• Since October 2020, M. Deaconu is developing with S. Herrmann (University of Burgundy) and C. Zucca (University of Torino), a BQR project, funded by the University of Burgundy, on the Exact simulation of the exit time of a bounded domain for a random process.
• Projet Innovant et Risqué (PIR) on Parallélisation Simplifiée de Méthodes de Monte-Carlo Complexes (Simplified Parallelization on Monte Carlo methods) was obtained from the University of Haute Alsace by L. Lenôtre and S. Mazzonetto.
• Inria Exploratory Research Action Apollon: The goal is to automate the creation of a lexicon of ideas from the Politics of Aristotle. This interdisciplinary project mixes machine learning, history and philology. This project involves the PASTA project-team membres: A. Lejay, S. Mazzonetto, L. Lenôtre (external, Irimas, University of Haute-Alsace) in a collaboration with M.-T. Schettino, A. Pollini, C. Urlacher-Becht (Archimède, University of Haute-Alsace).

7.1 Promoting scientific activities

7.2 Teaching - Supervision - Juries

7.3 Popularization

8.1 Major publications

• 1 articleL.Lucian Beznea, M.Madalina Deaconu and O.Oana Lupascu. Stochastic equation of fragmentation and branching processes related to avalanches.Journal of Statistical Physics1624February 2016, 824-841
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• 2 articleM.Madalina Deaconu and S.Samuel Herrmann. Initial-boundary value problem for the heat equation - A stochastic algorithm.Annals of Applied Probability2832018, 1943-1976
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• 3 articleA.Anselm Hudde, M.Martin Hutzenthaler and S.Sara Mazzonetto. A stochastic Gronwall inequality and applications to moments, strong completeness, strong local Lipschitz continuity, and perturbations.Annales de l'Institut Henri Poincaré, Probabilités et Statistiques572May 2021
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• 4 articleA.Antoine Lejay. Constructing general rough differential equations through flow approximations.Electronic Journal of Probability272021, 1-24
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• 5 articleA.Antoine Lejay and P.Paolo Pigato. Statistical estimation of the Oscillating Brownian Motion.Bernoulli244B2018, 3568-3602
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• 6 miscS.Sara Mazzonetto. Rates of convergence to the local time of Oscillating and Skew Brownian Motions.October 2021
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• 7 articleP.Pascal Moyal, A.Ana Bušić and J.Jean Mairesse. A product form for the general stochastic matching model.Journal of Applied Probability582June 2021, 449-468
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• 8 articleY.Youssef Rahme and P.Pascal Moyal. A stochastic matching model on hypergraphs.Advances in Applied Probability5342021, 951-980
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• 9 inproceedingsC.Christophe Reype, A.Antonin Richard, M.Madalina Deaconu and R. S.Radu S. Stoica. Bayesian statistical analysis of hydrogeochemical data using point processes: a new tool for source detection in multicomponent fluid mixtures.RING Meeting 2020Nancy, FranceSeptember 2020
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• 10 articleR.Radu Stoica, M.Madalina Deaconu, A.Anne Philippe and L.Lluis Hurtado-Gil. Shadow Simulated Annealing: A new algorithm for approximate Bayesian inference of Gibbs point processes.Spatial StatisticsApril 2021
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8.2 Publications of the year

8.3 Cited publications

• 36 miscJ.Jocelyn Begeot, I.Irène Marcovici and P.Pascal Moyal. Stability regions of systems with compatibilities, and ubiquitous measures on graphs.November 2021
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• 37 unpublishedS.Sara Mazzonetto. Rates of convergence to the local time of Oscillating and Skew Brownian Motions.October 2021, working paper or preprint
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