2023Activity reportProject-TeamPASTA

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 and involves collaboration with researchers from Archimède (Universities of Strasbourg and Haute-Alsace), IRIMAS and CRESAT (Université de Haute-Alsace) and University of Pavia.

6.1 Fragmentation equation

Participants: Madalina Deaconu, Antoine Lejay, Victor Hoffmann.

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 new stochastic process that represents the typical evolution of the mass of a rock or of a snow aggregate subject to successive random breakages.

In a survey article 16, we present various probabilistic representations of the fragmentation equation, and show how they are connected. We focus on the stochastic process which represents the evolution of the mass of a typical particle subject to a fragmentation process. These probabilistic representations range from Markov chains to Stochastic Differential Equations with jumps. In particular, we show how these representations lead to easy numerical simulations.

Further, with Gaetano Agazzotti (former intern in the team), we have studied from an analytic viewpoint the evolution of the moments of a fragmentation equation, and obtain its asymptotic behavior, by expanding his Master thesis  37.

We have also explored, using machine learning techniques, the median size of rocks in blasting operations. In particular, we have used ensemble techniques to combine various approaches to predict the median size 36.

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 (University of Burgundy) 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 15. 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.

We develop also 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 14. 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 pursued our work on the exact simulation of the hitting times of multi-dimensional diffusions. A one-week workshop meeting was held in Torino in November.

6.3 Self exciting threshold model and singular diffusion

Participants: Antoine Lejay, Sara Mazzonetto.

In collaboration with Benoit Nieto (École Centrale Lyon) we consider several-regimes CKLS (Chan– Karolyi–Longstaff–Sanders) dynamics (including Cox-Ingersoll-Ross model) and we study parameter estimation from high-frequency observations, extending the published work 20 about threshold Vasicek model. The model fits well the behavior in financial markets related to crisis periods.

In a collaboration with Paolo Pigato (University Tor Vergata, Roma), we study new estimators from low frequency observations for the parameters of several regimes threshold models which show mean-reversions features.

Together with Alexis Anagnostakis (LJK Grenoble), we are extending our respective results on high-frequency approximation of the local time of sticky-oscillating-skew diffusion processes. The purpose is to estimate the parameters of stickiness and/or skewness and to model some critical behaviors in financial markets related to crisis periods. Inspired by a previous work  43, we extend the results obtained during the PhD thesis of Alexis Anagnostakis  38 in the context of sticky Brownian motion to more general estimators of local time and to oscillating-skew-sticky Brownian motion. This is a work on its final stage of editing. Our main goal is now to reach rates of convergence for sticky diffusions and so extend the results in 43.

We are continuing our work on an expansion of the maximum likelihood estimator using formal series expansions  41. The aim of this work is to understand the lack of Gaussianity in the non-asymptotic regime.

In 19, 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 $1/2$.

6.4 Diffusion equations with singular coefficients

Participants: Antoine Lejay, Sara Mazzonetto.

With Géraldine Pichot (Serena, 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 26, 27.

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 Hug model: parameter estimation via the ABC Shadow algorithm

Participants: Madalina Deaconu, Christophe Reype, Radu Stoica.

Studying geological fluids mixing systems allows us to understand the interaction between water sources. The Hug model is an interaction point process model that can be used to estimate the number and the chemical composition of the water sources involved in a geological fluids mixing system from the chemical composition of samples  44. In 24, we construct priors for the parameters of the Hug model using the ABC shadows algorithm  45. The long term perspective of this work is to integrate geological expertise within fully unsupervised models.

This work is a collaboration with Didier Gemmerlé (IECL, Université de Lorraine) and Antonin Richard (GeoRessources, Université de Lorraine).

6.6 Point processes in cosmology

Participants: Nathan Gillot, Radu Stoica.

Marked point processes and Bayesian inference are powerful tools for analysing spatial data. With respect to a previous work  40, we proposed a new inhomogeneous point process with superposed interaction. The results indicate a correct fit of the model and allow the study of the significance of the parameter at the corresponding prefixed interaction ranges 23. This work is a collaboration with Didier Gemmerlé (IECL, Université de Lorraine).

With Jenny Sorce (Cristal, Lille) and Elmo Tempel (Tartu Observatory, University of Tartu), we develop in 21, 25 a new algorithm based on an object point process model that can reduce biases and uncertainties in the measurement of peculiar velocities of galaxies. The algorithm uses simulated annealing to maximize the probability density of the point process model, resulting in bias-minimized catalogs. We conducted tests on synthetic catalogs mimicking the second and third distance modulus catalogs of the Cosmicflows project from which peculiar velocity catalogs are derived. By reducing the local peculiar velocity variance in catalogs by an order of magnitude, the algorithm permits the recovery of the expected one, while preserving the small-scale velocity correlation. The expected clustering was also retrieved. The resulting bias-minimized catalogs allowed for the recovery of expected statistical properties and the reconstruction of large-scale structures that matched well with existing redshift surveys of local galaxies. These bias-minimized catalogs can be used for various cosmological studies and simulations of the local Universe.

6.7 Spatial-point processes in geophysics

Participants: Radu Stoica.

Faults are crucial subsurface features that significantly influence the mechanical behavior and hydraulic properties of rock masses. Interpreting them from seismic data may lead to various scenarios due to uncertainties arising from limited seismic bandwidth and possible imaging errors. Only a few methods addressing fault uncertainties can produce curved and sub-seismic faults at once while quantitatively honoring seismic images and avoiding anchoring in a reference interpretation.

With Fabrice Taty-Moukati, François Bonneau, Guillaume Caumon (GeoRessources, Université de Lorraine) and Xinming Wu (Hefei, University of Science and Technology of China), we use a mathematical framework, namely the Candy model, of marked point processes with interactions to approximate fault networks in two dimensions with a set of line segments. The novelty of this approach lies in using the input image of fault probabilities computed by a Convolutional Neural Network (CNN). The Metropolis-Hastings algorithm is used to generate various scenarios of fault network configurations, thereby exploring the model space and reflecting the uncertainty. The empty space function produces a ranking of the generated fault networks against an existing interpretation by testing and quantifying their spatial variability. The approach is applied on two-dimensional sections of seismic data, acquired in the Central North Sea.

During geological exploration, the interpretation of faults can be ambiguous and uncertain because of disparate and often sparse observations such as fault traces on 2D seismic images or outcrops. With Amandine Fratani, Guillaume Caumon and Jéremie Guirad (GeoRessources, Université de Lorraine), we propose a hypergraph formalism to generalize the higher-order interactions between fault observations in the multiple-point association problem, by dealing with the likelihood of multiple-point fault data association. A machine learning approach is proposed to enhance or replace the expert geological rules in determining the likelihood of multiple-point fault data association. This involves training a supervised classifier using fault features extracted from known 3D geological models. By formulating the problem as a classification task, the model can determine the probability that fault observations belong to the same fault objects. We also develop a specific technique to prevent overfitting. Overall, this hypergraph-based approach with machine learning integration provides a more advanced and flexible method for interpreting and associating fault observations in geological exploration.

Fractures from systems of complex mechanical discontinuities dramatically impact the physical behavior of rock masses. With François Bonneau and Guillaume Caumon (GeoRessources, Université de Lorraine), we use the mathematical framework of marked point processes to approximate fracture networks in two dimensions with a collection of straight-line segments. Whereas most fracture characterization and modeling focuses on first order statistics (density and mark distribution), and assume independent fractures, we focus on fracture interactions by providing stochastic mathematical models involving simple pairwise interactions between fractures to capture key aspects of fracture network geometry and organization. The model is calibrated using a maximum likelihood, which we apply to real data from Oman mountains.

6.8 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.

6.9 Modeling and optimization: Stochastic matching models

Participants: Loïc Jean, Thomas Masanet, Pascal Moyal.

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

• In 33, in our long standing collaboration with Ana Busic (Doxygene, Inria Paris) and Jean Mairesse (LIP6, Université Pierre et Marie Curie), we show that all stochastic matching models on general graphs present a remarkable sub-additivity property. This implies that bi-infinite perfect matchings can be constructed explicitly and uniquely. In the particular, practical, case where the buffer is finite (thereby rendering the state space finite), we deduce a perfect simulation scheme, by showing the existence of infinitely many erasing words that cancel out the content of the buffer, by combinatorics arguments.
• In a draft paper that we are completing, we have shown, by a dynamic programming approach, that a threshold-type policy minimizes the holding cost amongst admissible policies for the N graph, an approach that we aim at generalizing for a larger class of graphs, including the paw graph with four nodes.
• In 31, in collaboration with the doctors of Agence de la Biomd́ecine (ABM) we have brought to light, by an extensive set of simulations based on survival analysis for liver transplant systems, that, in the context of real-time matching models, a matching algorithm based on simulated deadlines (the so-called Earliest Simulated Deadline First) is optimal in terms of equity between indication classes (various classes of cirrhosis and liver cancer), while keeping the overall death rate equal, with respect to the existing matching policies currently implemented by the ABM.

We have focused on the study and optimization of online algorithms on large random graphs, which are known to be prevalent in practice, for various applications such as job/housing allocation, online adverts, and so on.

• In 35, in a collaboration with Matthieu Jonckheere (LAAS) and Nahuel Soprano-Loto (Univ. Buenos Aires / LAAS), we show that an online matching algorithm of the Max-Weight type is able to guarantee a perfect marriage infinitely often, under natural connectivity conditions à la Hall, for a class of stochastic block models, in the large graph limits.
• In a collaboration with Vincent Robin and Habib Dialo Aoudi (UTC), we have proved a hydrodynamic limit for local online matching algorithms depending only on the degrees of the nodes (and not on their neighborhood). By comparing their asymptotics through their hydrodynamic approximation, we have shown that a degree-greedy type algorithm is asymptotically optimal in terms of matching coverage, with respect to greedy algorithms. A paper gathering these results is about to be submitted.

6.10 Speed of convergence in functional central limit theorems

Participants: Pascal Moyal.

In 13, in collaboration with Eustache Besançon (Telecom Paris), Laurent Decreusefond (Telecom Paris) and Laure Coutin (Université Paul Sabatier, Toulouse), we have shown universal bounds for the speed of convergence in the functional Central Limit Theorems for Lipschitz continuous functionals of Poisson random measures. As a by-product, we deduce similar bounds for Continuous Time Markov Chains (CTMCs), by using various tools of stochastic analysis, among which, Malliavin calculus for point processes, and the Stein method in infinite dimension. These results allow us to characterize the accuracy (and thereby the confidence interval) in diffusion approximations of many practical processes appearing in epidemiology (Susceptible, Infectious, or Recovered [SIR] processes) biology of development (Moran process) and telecom networks (queueing processes and the Telegraph process).

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 National initiatives

Participants: Antoine Lejay, Sara 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 members: A. Lejay, S. Mazzonetto, in a collaboration with Lionel Lenôtre (Irimas, Université of Haute-Alsace), (M.-T. 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).

9.1 Promoting scientific activities

9.2 Teaching - Supervision - Juries

9.3 Popularization

• Antoine Lejay is editor in chief of the Success stories (2 pages presentation of a successful industrial collaboration, Agence Mathématiques en Entreprises et Interactions (AMIES) and Fondation Sciences Mathématiques de Paris).

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 and S.Sara Mazzonetto. Maximum likelihood estimator for skew Brownian motion: the convergence rate.Scandinavian Journal of StatisticsFebruary 2023HALDOI
• 6 miscS.Sara Mazzonetto. Rates of convergence to the local time of Oscillating and Skew Brownian Motions.October 2021HAL
• 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, 449-468HALDOI
• 8 articleY.Youssef Rahme and P.Pascal Moyal. A stochastic matching model on hypergraphs.Advances in Applied Probability5342021, 951-980HALDOI
• 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 2020HAL
• 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 2021HALDOI

10.2 Publications of the year

10.3 Cited publications

• 37 mastersthesisG.Gaetano Agazzotti. Modélisation des processus de fragmentation à l'aide des processus de branchement.MA ThesisCe mémoire a été réalisé dans le cadre d'un ''parcours recherche'' de l'École des Mines de Nancy ouverts aux étudiants de 2ème année.Ecole des Mines de NancyJune 2022HALback to text
• 38 phdthesisA.Alexis Anagnostakis. Path-wise study of singular diffusions.Université de LorraineOctober 2022HALback to text
• 39 phdthesisM. H.Mohamed Habib Aliou Diallo Aoudi. Local matching algorithms on the configuration model.Université de technologie de CompiègneJune 2023HALback to text
• 40 articleL.Lluís Hurtado-Gil, R. S.Radu S Stoica, V. J.Vicent J Martínez and P.Pablo Arnalte-Mur. Morphostatistical characterization of the spatial galaxy distribution through Gibbs point processes.Monthly Notices of the Royal Astronomical Society5072August 2021, 1710–1722URL: http://dx.doi.org/10.1093/mnras/stab2268DOIback to text
• 41 unpublishedA.Antoine Lejay and S.Sara Mazzonetto. Beyond the delta method.July 2022, working paper or preprintHALback to text
• 42 articleS.Sara Mazzonetto and P.Paolo Pigato. Drift estimation of the threshold Ornstein-Uhlenbeck process from continuous and discrete observations.Statistica Sinica3412024, 313-336HALDOIback to text
• 43 unpublishedS.Sara Mazzonetto. Rates of convergence to the local time of Oscillating and Skew Brownian Motions.October 2021, working paper or preprintHALback to textback to text
• 44 phdthesisC.Christophe Reype. Modélisation probabiliste et inférence bayésienne pour l'analyse de la dynamique des mélanges de fluides géologiques : détection des structures et estimation des paramètres.Université de LorraineDecember 2022HALback to text
• 45 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 2021HALDOIback to text