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, Université de Lorraine), IMAR (Institute of Mathematics of the Romanian Academy) in Bucharest.

4.2 Astronomy

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 us 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 telecommunication 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 represent at any given time receivers and donors, while edges represent compatibilities. 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 the theory of large graph asymptotic on 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). Telecommunication 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).

4.4 Digital Humanities

Digital Humanities represents an interdisciplinary field of research. We are interested in developing suitable, automatic tools to help experts to study the ideas contained in antique texts. Together with historians of antiquity, we consider one of the founding texts of political sciences, the Politics of Aristotle. To fulfill our purposes, we consider techniques both from the history of antiquity, machine learning, and statistics. This also presents some technological challenges to develop suitable tools to load and manipulate the data.

This research is supported by the Inria Exploratory Reasearch Action Apollon (2022-2024) and involves collaboration with researchers from Archimède (Universities of Strasbourg and Haute-Alsace), IRIMAS and CRESAT (Université de Haute-Alsace) and University of Pavia.

5.1 New platforms

Participants: Antoine Lejay, Sara Mazzonetto, Radu Stoica, Saïd Toubra.

DRLIB is a C++ library built for performing modelling, simulation and statistical inference based on marked point processes with interaction. This library is the result of a joint project of Radu Stoica with Didier Gemmerlé (CNRS research engineer at IECL).

Palamède aims at being a collaborative plateform to vizualize, annotate and analyze texts from the point of view of experts in history, philology or more generally in social sciences. The short term goals are to incorporate tools from Artifical Intelligence to ease the study of texts. This plateform is supported by the ADT Apollon.

6.1 Fragmentation equation

Participants: Madalina Deaconu, Antoine Lejay, Anton Conrad.

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.

With Gaetano Agazzotti (former intern in the team), we have studied in 29 the evolution of the moments of a self-similar fragmentation equation from an analytic viewpoint. In particular, we have shown existence for an initial condition which is a measure. We have proved rigorously its asymptotic behavior.

During the internship of Anton Conrad , we have developed the technique of embedding in the shape of polytopes. The main idea is to handle the analysis of a population of geometric shapes, such as one obtained through successive fragmentations, through their embedding. This way, classical techniques of clustering, or other, may be applied to such objects. With Didier Gemmerlé (IECL), we are developing a code on this topic. We are in particular interested in studying the convergence toward some universal shapes.

6.2 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 (Université de Bourgogne) we made important progress on this topic by developing new methods. One main result concerns a new technique for the path approximation of one-dimensional stochastic processes 16. Our method applies to the Brownian motion and to some families of stochastic differential equations whose distributions could be represented as a function of a time-changed Brownian motion (usually known as $L$ and $G$-classes). We are interested in the $\epsilon$-strong approximation. We propose an explicit procedure that jointly constructs the sequences of exit times and corresponding exit positions of some well-chosen domains. We prove the convergence of our scheme and how to control the number of steps, which depends on the covering of a fixed time interval by intervals of random sizes. The underlying idea of our analysis is to combine results on Brownian exit times from time-depending domains (one-dimensional heat balls) and classical renewal theory. Numerical examples and issues are also developed in order to complete the theoretical results.

Together with Samuel Herrmann (Université de Bourgogne) and Cristina Zucca (University of Torino) we pursued our work on the exact simulation of the hitting times of multi-dimensional diffusions. Recenlty, with Samuel Herrmann we started to construct random walks on truncated spheroids with the objective of improving existing results.

6.3 Statistics for self exciting threshold model and singular diffusions

Participants: Antoine Lejay, Sara Mazzonetto.

In a collaboration with Benoit Nieto (former member of École Centrale Lyon, now member of École Polytechnique), we consider several-regimes CKLS (Chan– Karolyi–Longstaff–Sanders) dynamics (including Cox-Ingersoll-Ross model) and we study parameter estimation from high-frequency observations 36. In an ongoing collaboration we are considering a theoretical result on existence and uniqueness of solutions to stochastic differential equations admitting several regimes. These questions are important because lack of uniqueness may affect approximation or inference results.

With Paolo Pigato (University Tor Vergata, Roma), we studied new estimators from low frequency observations for the parameters of several regimes threshold models which show mean-reversions features 37.

Together with Alexis Anagnostakis (former member LJK Grenoble, now member of IECL Metz), we extended our respective results on high-frequency approximation of the local time of sticky-oscillating-skew diffusion processes. We estimate the parameters of stickiness and/or skewness 35. Our main goal is now to reach rates of convergence for sticky diffusions and so extend the results in 38. We obtained a partial interesting result for sticky Brownian motion in 34.

We continued our work on an expansion of the maximum likelihood estimator using formal series expansions 32 (preprint submitted to a journal). The aim of this work is to understand the lack of Gaussianity in the non-asymptotic regime.

6.4 Diffusion equations with singular coefficients

Participants: Antoine Lejay, Sara Mazzonetto.

With Géraldine Pichot (Serena project-team, Inria Paris), Giovanni Michele Porta and Elisa Baioni (Politecnico di Milano), we have provided an extension of a Monte Carlo method that allows for the simulation of a diffusion process in a one-dimensional discontinuous media. Using the method of images, the extension consists in finding an approximation of the fundamental solution associated with the process which is suitable for a fast simulation. Our method may be applied to situations in which both the solution and its gradient are discontinuous at some point. In particular, we may consider the case of the Fourier equation with discontinuous coefficients 13, 14.

Together with Alexis Anagnostakis and Pierre Etoré (LJK Grenoble) we are dealing with different questions about the non-uniqueness of solutions for processes solution to stochastic differential equations with a diffusion coefficient admitting jumps and becoming negative. We tackle a conjecture open since the 80's. We have obtained a partial answer and we are seeking for the link with sticky-skew diffusions.

6.5 Spatial point process modelling and Bayesian inference for large data sets

Participants: Nathan Gillot, Radu Stoica.

Modelling the galaxy distribution in our Universe is with no doubt a very important statistical challenge since the Universe contains around 200 billion galaxies. Among the typical available characteristics for the galaxies one must consider their position, mass, luminosity, and shape. Due to this, marked point processes appear as a natural modelling tool. There exists statistical methodology able to extract relevant information from marked point configurations. We take the first step in 26 and propose to use non-parametric exploratory analysis and Bayesian posterior based inference in order to explore the first characteristic, namely the positions of more than 30,000 galaxies. A new parametric multi-interaction point process model is introduced and fitted to the selected galaxy patterns. The quality of the estimation procedure and the significance of the estimated parameters is also assessed. Analysing several patterns allows us to have more insight into the stationary character of the entire observed data set and to depict perspectives with respect to the possible strategies for the general model fitting challenge.

This work is a collaboration with Didier Gemmerlé (IECL, Université de Lorraine) and Aila Särkkä (Chalmers University, Sweden).

6.6 Inhomogeneous interacting marked point processes for studying morphostructures in paleobiological data

Participants: Diego Astaburuaga, Radu Stoica.

The work 25 develops inhomogeneous marked point processes with interactions that are applied to the analysis of morphostructures exhibited by a paleo-biological dataset presented in Kolesnikov (2018). Specifically, due to the nature of the dataset, we model the probability density function describing the models by considering three effects: the distance to the nearest edge, the distance to the lower right corner, and the distance to a reference circle. Furthermore, interactions between the points through the observed marks are introduced. This is done using the Strauss and Area-Interaction processes. The C++ library DRLib is the main programming tool used to perform model simulations, while the R package spatstat is used for the exploratory analysis, the ABC Shadow algorithm, the model verification analysis by global envelope tests and the graphical presentation of the results.

This work is a collaboration with Francisco Cuevas (Universidad Tecnica Federico Santa Maria, Chile) and Didier Gemmerlé (IECL, Université de Lorraine).

6.7 Statistical inference for random T-tessellations models: application to agricultural landscape modeling

Participants: Radu Stoica.

The Gibbsian T-tessellation models allow the representation of a wide range of spatial patterns. We propose in 12 an integrated approach for statistical inference. Model parameters are estimated via Monte Carlo maximum likelihood. The simulations needed for likelihood computation are produced using an adapted Metropolis-Hastings-Green dynamics initialized using pseudolikelihood estimates. A real data application is performed on three French agricultural landscapes. The Gibbs T-tessellation models simultaneously provide a morphological and statistical characterization of these data.

This work is a collaboration with Katarzyna Adamczyk-Chauvat (INRAe Jouy-en-Josas, Université Paris Saclay).

6.8 From fault likelihood to fault networks: stochastic seismic interpretation through a marked point process with interactions

Participants: Radu Stoica.

Faults are critical subsurface features influencing rock mass mechanical and hydraulic properties. Interpreting them from seismic data involves uncertainties from limited bandwidth and imaging errors. In 23, we use a marked point process framework to approximate fault networks in two dimensions, introducing the Candy Model that captures fault segment interactions. The approach innovatively conditions the stochastic model using fault probability images from a Convolutional Neural Network. The Metropolis-Hastings algorithm generates fault network scenarios, exploring model space and uncertainty. Probability level sets and empty space function provide insights into fault network realizations and parameters, with the method applied to seismic data from the Central North Sea.

This work is a collaboration with Fabrice Taty-Moukati, François Bonneau, and Guillaume Caumon (GeoRessources, Université de Lorraine).

6.9 Multiple point fault observation association using random forest from analog structural models

Participants: Radu Stoica.

During geological modelling, 3D fault interpretation can be ambiguous from incomplete observations like fault traces in 2D seismic images. The problem of associating partial fault observations has been formalized using a graph where nodes represent observations and edges show potential associations. We propose extending this approach with a multiple-point likelihood computation and using machine learning to infer association probabilities. By training on fault features from known 3D geological models and splitting the domain into training and testing sectors, we aim to create a probabilistic representation of fault associations that improves upon existing pairwise methods. To mitigate the problem of expert rules defined on highly dimensional problems, we propose to augment or replace them by inference from analog or partly observed data.

This work is a collaboration with Amandine Fratani, Guillaume Caumon (GeoRessources, Université de Lorraine) and Jeremie Giraud (Centre for Exploration Targeting (School of Earth Sciences), University of Western Australia).

6.10 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 also results in 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 the creation and disappearance of particles that mimic the physics of the vorticity in the boundary layer.

6.11 Modeling and optimization: Stochastic matching models

Participants: Pascal Moyal.

We have made various advances in the analysis and optimization of stochastic matching models:

• In collaboration with Ana Busic (Dyogene project-team, Inria Paris) and Jean Mairesse (LIP6, Université Pierre et Marie Curie), where we show a remarkable sub-additivity property for general stochastic matching models on general graphs 21.
• In 19 we analyze the performance of a similar perfect simulation scheme for general stochastic matching models with reneging.

6.12 Markovian algorithms on large random graphs

Participants: Pascal Moyal.

We have pursued an intensive research activity on the Markovian analysis of various Markovian exploration algorithms on random graphs, analyzed and/or approximated to the large graph limits, using scaling limits of stochastic processes. These various techniques apply to three distinct classes of random graphs: Stochastic block models, Configuration models and Preferential attachment models.

• With Vincent Robin and Mohamed Habib Dialo Aoudi (UTC), we prove a hydrodynamic large-graph limit for various local online matching algorithms depending only on the degrees of the nodes (and not on their neighborhood), among which, greedy and degree-greedy algorithms, on Configuration model (CM) random graphs 31. For doing so, we follow the so-called constructing while exploring approach of the CM. By comparing their asymptotic behaviors through the hydrodynamic limits of a suitable sequence of measure-valued processes (rather than local limits), we have shown that the degree-greedy type algorithm is asymptotically optimal in terms of matching coverage, with respect to greedy algorithms.
• In an ongoing collaboration with Mariana Olvera-Cravioto (University of North Carolina), we investigate the connected out-component of a typical vertex in a large, oriented, preferential attachment random graph (Barabasí- Albert model). By a Markov in-depth exploration and coupling methods, we show that the local limits of the construction of the out-component is a suitable Galton-Watson process. This result has crucial implications for the asymptotics analysis of the main properties of preferential attachment models, which are prevalent in many applications, such as epidemiological models and social media. An article gathering these advances is under preparation.

6.13 Reinforcement learning, and applications to queueing

Participants: Pascal Moyal.

With Céline Comte (LAAS-CNRS), we work on an optimization scheme for the access control of various queueing systems by using reinforcement learning techniques. We have shown that a wide class of systems (encountered in telecom networking, supply chains or call centers) exhibit the same type of quasi-reversibility property, leading to a “universal” product-form type stationary distributions. Then, if the access control enjoys a general balance property, we show this product-form structure remains. We then resort to a model-free (or partially observable model) approach, and apply a class of reinforcement learning algorithms called Policy-gradient, which are able to optimize the access control of the considered models. We are currently finishing the redaction of a paper gathering these results, which we will soon submit for publication.

6.14 Speed of convergence in functional central limit theorems

Participants: Pascal Moyal.

In collaboration with Eustache Besançon (Telecom Paris), Laurent Decreusefond (Telecom Paris) and Laure Coutin (Université Paul Sabatier, Toulouse), in which we have shown universal bounds for the speed of convergence in the functional Central Limit Theorems for Lipschitz continuous functionals of Poisson random measures 15. These results allow us to characterize the accuracy of diffusion approximations of many practical processes appearing in epidemiology, biology of development and telecom networks.

6.15 Game theoretical modeling for trust management in IoT networks

Participants: Pascal Moyal.

• In 27, we propose a novel game-theoretical approach for the modeling and analysis of the so-called crowdsourcing IoT (Internet of Things), by an evolutionary game in which the agents (in particular, the service provider and the service requestor) are possibly irrational, non-homogeneous and change strategy over time. We conduct a theoretical analysis and numerical results, to analyze the influence of the strategy changes.
• In 22, we introduce another game-theoretical role-based attack-resilient trust management (TM) model for community-driven IoT, which takes in account in particular the trust between different communities in terms of cooperativeness. We thoroughly analyze and simulate this TM under various scenario, to show the effectiveness in evaluating both intra and inter-community trustworthiness, and to validate the model in practice.

7.1 Scientific expertise

Participants: Pascal Moyal.

• Pascal Moyal has collaborated, as a scientific expert in Stochastic modeling and Machine learning, with the Start-Up mAIedge.

8.1 International initiatives

8.2 International research visitors

8.3 European initiatives

8.4 National initiatives

Participants: Antoine Lejay, Sara Mazzonetto, Saïd Toubra.

• 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 members: Antoine Lejay , Amélie Ferstler , Sara Mazzonetto andSaïd Toubra in a collaboration with Lionel Lenotre (Irimas, Université of Haute-Alsace), Maria-Teresa Schettino (Archimède, Université de Haute-Alsace), Catherine Roth (CRESAT, Université of Haute-Alsace), Cesare Zizza (Department of Humanistic Studies, Università degli Studi di Pavia), Didier Gemmerlé (IECL, Université de Lorraine).
• Programme Blanc MITI Némésis (funding CNRS). This grant, whose PI are Antoine Lejay and Maria-Teresa Schettino (Archimède, Université de Haute-Alsace), supports the interdisciplinary research in the field of Digital Humanities, namely the development of the Palamède software within the Apollon project and the PhD thesis of Saïd Toubra .
• Insmi PEPS project on Local time approximation. PIs Alexis Anagnostakis (LJK, Grenoble) and Pasta project-team member Sara Mazzonetto .

9.1 Promoting scientific activities

9.2 Teaching - Supervision - Juries

9.3 Popularization

10.1 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, 824-841HALDOI
• 2 articleM.Madalina Deaconu and S.Samuel Herrmann. Initial-boundary value problem for the heat equation - A stochastic algorithm.Annals of Applied Probability2832018, 1943-1976HALDOI
• 3 articleM.Madalina Deaconu and A.Antoine Lejay. Probabilistic representations of fragmentation equations.Probability Surveys202023, 226-290HALDOI
• 4 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 2021HALDOI
• 5 articleA.Antoine Lejay. Constructing general rough differential equations through flow approximations.Electronic Journal of Probability272021, 1-24HALDOI
• 6 articleA.Antoine Lejay and S.Sara Mazzonetto. Maximum likelihood estimator for skew Brownian motion: the convergence rate.Scandinavian Journal of StatisticsFebruary 2023HALDOI
• 7 miscS.Sara Mazzonetto. Rates of convergence to the local time of Oscillating and Skew Brownian Motions.October 2021HAL
• 8 articleP.Pascal Moyal, A.Ana Bušić and J.Jean Mairesse. A product form for the general stochastic matching model.Journal of Applied Probability582June 2021, 449-468HALDOI
• 9 articleY.Youssef Rahme and P.Pascal Moyal. A stochastic matching model on hypergraphs.Advances in Applied Probability5342021, 951-980HALDOI
• 10 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 2020HAL
• 11 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 2021HALDOI

10.2 Publications of the year