5.1 Footprint of research activities

Classic footprint for researchers: massive computer runs and travel for international conferences and cooperations.

5.2 Impact of research results

The research is useful for the risk managment of global interlocked economic instances and actors, of energy production in complex power plants, and of epidemics and pandemics.

This year we are particpating in a program on the numerical twin of France and its territories. We have obtained advanced results on the twins of large urban zones furnished with statistically valid synthetic populations in movement between their hourly activiities. Our main aim was and remains to develop on these twins agent-based models for epidemics, with the goal to provide tools for health deciders to evaluate the impact of varied sanitary measures on the spread of the epidemic as well as on the economy and on social structures. These twins can be useful for many other pruposes such as urban planning.

6.1 Awards

The paper 6 is on the front cover of the journal SIAM Review (Vol. 66, Issue 3).

7.1 New platforms

8.1 Stochastic cooperative numerical methods for complex industrial systems

Participants: Josselin Garnier, Guillaume Chennetier, Paul Lartaud, Charlie Sire.

Bayesian network approach to multi-fidelity surrogate modeling

In 15, Baptiste Kerleguer, Claire Cannamela and Josselin Garnier address the surrogate modeling of a computer code output in a hierarchical multi-fidelity context, i.e., when the output can be evaluated at different levels of accuracy and computational cost. Using observations of the output at low- and high-fidelity levels, they propose a method that combines Gaussian process (GP) regression and the Bayesian neural network (BNN), called the GPBNN method. The low-fidelity output is treated as a single-fidelity code using classical GP regression. The high-fidelity output is approximated by a BNN that incorporates, in addition to the high-fidelity observations, well-chosen realizations of the low-fidelity output emulator. The predictive uncertainty of the final surrogate model is then quantified by a complete characterization of the uncertainties of the different models and their interaction.

Adaptive importance sampling based on fault trees for PDMP

In 7, Guillaume Chennetier, Hassane Chraibi, Anne Dutfoy, and Josselin Garnier introduce an original adaptive importance sampling stratgey based on fault tree analysis for Piecewise Deterministic Markov Processes (PDMPs). PDMPs can be used to model complex dynamical industrial systems. The counterpart of this modeling capability is their simulation cost, which makes reliability assessment untractable with standard Monte Carlo methods. A significant variance reduction can be obtained with an adaptive importance sampling method based on a cross-entropy procedure. The success of this method relies on the selection of a good family of approximations of the committor function of the PDMP. In this paper original families are proposed. Their forms are based on reliability concepts related to fault tree analysis: minimal path sets and minimal cut sets. They are well adapted to high-dimensional industrial systems. The proposed method is discussed in detail and applied to academic systems and to a realistic system from the nuclear industry.

Reduced order modeling for full waveform inversion

In 6, 1, Liliana Borcea, Josselin Garnier, Alexander Mamonov, and Jorn Zimmerling introduce and study a new formulation of a classical inverse problem: Full Waveform Inversion. Full Waveform inversion is concerned with estimating a heterogeneous medium, modeled by variable coefficients of wave equations, using sources that emit probing signals and receivers that record the generated waves. It is an old and intensively studied inverse problem with a wide range of applications, but the existing inversion methodologies are still far from satisfactory. The typical mathematical formulation is a nonlinear least squares data fit optimization and the difficulty stems from the nonconvexity of the objective function that displays numerous local minima at which local optimization approaches stagnate. This pathological behavior has at least three unavoidable causes: (1) The mapping from the unknown coefficients to the wave field is nonlinear and complicated. (2) The sources and receivers typically lie on a single side of the medium, so only backscattered waves are measured. (3) The probing signals are band limited and with high frequency content. There is a lot of activity in the computational science and engineering communities that seeks to mitigate the difficulty of estimating the medium by data fitting. In this paper a different point of view is presented which is based on reduced order models (ROMs) of two operators that control the wave propagation. The ROMs are called data driven because they are computed directly from the measurements, without any knowledge of the wave field inside the inaccessible medium. This computation is noniterative and uses standard numerical linear algebra methods. The resulting ROMs capture features of the physics of wave propagation in a complementary way and have surprisingly good approximation properties that facilitate waveform inversion.

Mean field game model for renewable investment

In 9, Celia Escribe,Josselin Garnier, and Emmanuel Gobet study a stylized model for investment into renewable power plants under long-term uncertainty. They model risk-averse agents facing heterogeneous weather conditions and a common noise including uncertainty on demand trends, future fuel prices and the average national weather conditions. The objective of each agent is to maximize multistage profit by controlling investment in discrete time steps. They analyze this model in a noncooperative game setting with N players, where the interaction among agents occurs through the spot price mechanism. Their model extends to a mean field game with common noise when the number of agents is infinite. They prove that the N -player game admits a Nash equilibrium. Moreover, they prove that under appropriate assumptions, any sequence of Nash equilibria to the N-player game converges to the unique solution of the MFG game. Numerical experiments highlight the impact of the risk aversion parameter and the importance of correctly specifying the distribution of the heterogeneity among agents. Moreover, they demonstrate that the results obtained by their model cannot be replicated by a model based on a representative agent with a unique parameter that would represent homogenized weather conditions. This emphasizes the importance of including explicit modeling of heterogeneity in prospective models when a heterogeneous parameter is expected to have a significant influence on the outcomes.

Sensitivity analysis of interacting particles driven by colored noise

In 11, Josselin Garnier, Harry Ip, and Laurent Mertz propose an efficient sensitivity analysis method for a wide class of colored-noise-driven interacting particle systems (IPSs). Their method is based on unperturbed simulations and significantly extends the Malliavin weight sampling method proposed by Szamel in 2017 for evaluating sensitivities such as linear response functions of IPSs driven by simple Ornstein-Uhlenbeck processes. They show that the sensitivity index depends not only on two effective parameters that characterize the variance and correlation time of the noise, but also on the noise spectrum. In the case of a single particle in a harmonic potential, they obtain exact analytical formulas for two types of linear response functions. By applying their method to a system of many particles interacting via a repulsive screened Coulomb potential, they compute the mobility and effective temperature of the system. Their results show that the system dynamics depend, in a nontrivial way, on the noise spectrum.

Model error quantification for Bayesian calibration

In 29, Gilles Defaux, Cédric Durantin, Josselin Garnier, Baptiste Kerleguer, Guillaume Perrin and Charlie Sire deal with Bayesian calibration, which is used to ensure that numerical simulations accurately reflect the behavior of physical systems. An important issue is the quantification of the model error, i.e., the discrepancy between the model outputs and the observed data. Conventional methods cannot be implemented in transposition situations, such as when a model has multiple outputs but only one is experimentally observed. To account for the model error in this context, the authors augment the calibration process by introducing additional input numerical parameters through a hierarchical Bayesian model, which includes hyperparameters for the prior distribution of the calibration variables. Importance sampling estimators are used to avoid increasing computational costs. Performance metrics are introduced to assess the proposed probabilistic model and the accuracy of its predictions. The method is applied on a computer code with three outputs that models the Taylor cylinder impact test. The outputs are considered as the observed variables one at a time, to work with three different transposition situations. The proposed method is compared with other approaches that embed model errors to demonstrate the significance of the hierarchical formulation.

8.2 Modeling, analysis, and simulation of cooperative stochastic systems

Participants: Quentin Cormier, Carl Graham, Denis Talay.

Communication networks and their algorithms.

Carl Graham studies communication networks and the algorithms (or protocols) used to manage efficiently their resources in real-time in a distributed and cooperative fashion. For instance, load balancing algorithms (LBA) strive to avoid server idleness and queue build-up, and are the topic of a lively example-based litterature.

In 30, Carl Graham has rigorously defined a wide class of LBA for which he has devised perfect simulation methods. The state space is infinite, and the methods use dominated coupling from the past. The dominating process is a network with uniform routing in a coupling preserving a preorder related to the increasing convex order. The use of a preorder is novel in this context. This allows performance evaluation in equilibrium of any LBA in the class using Monte Carlo estimation.

A stochastic numerical method for the parabolic-parabolic Keller-Segel system.

The parabolic-parabolic Keller-Segel model is a set of equations that model the process of cell movement. It takes into account the evolution of different chemical components that can aid, hinder or change the direction of movement, a process called chemotaxis.

In collaboration with Radu Maftei (who is a past Inria post-doc student) Milica Tomasevic (CMAP, Ecole Polytechnique) and Denis Talay have continued to analyse the numerical performances of a stochastic particle numerical method for the parabolic-parabolic Keller-Segel model. They also propose and test various algorithmic improvements to the method in order to substantially decrease its execution time without altering its global accuracy.

An hypothesis test for complex stochastic simulations.

In a joint work with Héctor Olivero 18, D. Talay has proposed and analyzed an asymptotic hypothesis test for independent copies of a given random variable which is supposed to belong to an unknown domain of attraction of a stable law. The null hypothesis ${𝐇}_{\mathbf{0}}$ is: `$X=\sqrt{V}$ is in the domain of attraction of the Normal law' and the alternative hypothesis is ${𝐇}_{\mathbf{1}}$: `$X$ is in the domain of attraction of a stable law with index smaller than 2'.

Surprisingly, the proposed hypothesis test is based on a statistic which is inspired by methodologies to determine whether a semimartingale has jumps from the observation of one single path at discrete times. The authors have justified their test by proving asymptotic properties of discrete time functionals of Brownian bridges. They also have discussed many numerical experiments which allowed them to illustrate satisfying properties of the proposed test.

Long time behavior of particle systems and their mean-field limit, application to spiking neural networks

Quentin Cormier has studied the long time behavior of a family of McKean-Vlasov stochastic differential equations. He has given conditions ensuring the local stability of an invariant probability measure. The criterion involves the location of the roots of an explicit holomorphic function associated to the dynamics. When all the roots lie on the left-half plane, local stability holds and convergence is proven in Wasserstein norms. The optimal rate of convergence is provided. This method is then applied to study a large class of models of interacting particles on the torus, see 27. This work will appear in Annales de l'Institut Henri Poincaré. In addition, he has adapted the method to study the long time behavior of a model of interacting Integrate-and-Fire neurons 8, 3. This last result provides new insights concerning the bistability behavior of such networks. In particular, a conjecture formulated by P. Robert and J. Touboul regarding the number of stationary solutions is proved. This last work has been published in the journal Mathematical Neuroscience and Applications.

Renewal theorems in a periodic environment

Quentin Cormier has studied a renewal problem within a periodic environment, departing from the classical renewal theory by relaxing the assumption of independent and identically distributed inter-arrival times, see 28. In this work, the conditional distribution of the next arrival time, given the current one, is governed by a periodic kernel, denoted as $H$. The periodicity property of $H$ is expressed as $\mathbb{P}(T_{k+1}>t|T_k)=H(t,T_k)$, where $H(t+T,s+T)=H(t,s)$. For a fixed time $t$, we define $N_t$ as the count of events occurring up to time $t$. The focus is on two temporal aspects: $Y_t$, the time elapsed since the last event, and $X_t$, the time until the next event occurs, given by $Y_t=t-T_{N_t}$ and $X_t=T_{N_t+1}-t$. The study explores the long-term behavior of the distributions of $X_t$ and $Y_t$. As an application, we describe the long time behavior of a spiking neuron subject to a time-periodic input current.

Optimal control under unknown intensity with Bayesian learning

Nicolas Baradel and Quentin Cormier have studied an optimal control problem inspired by neuroscience, where the dynamics is driven by a Poisson process with a controlled stochastic intensity and an uncertain parameter, see 25. Given a prior distribution for the unknown parameter, we describe its evolution according to Bayes' rule. We reformulate the optimization problem using Girsanov's theorem and establish a dynamic programming principle. Finally, we characterize the value function as the unique viscosity solution to a finite-dimensional Hamilton-Jacobi-Bellman equation, which can be solved numerically.

9.1 CIROQUO Research & Industry Consortium

Several INRIA teams, including ASCII, are involved in the CIROQUO Research & Industry Consortium – Consortium Industrie Recherche pour l'Optimisation et la QUantification d'incertitude pour les données Onéreuses – (Industry Research Consortium for the Optimization and QUantification of Uncertainty for Onerous Data). Josselin Garnier is the INRIA Saclay representative on the steering committee.

The principle of the CIROQUO Research & Industry Consortium is to bring together academic and technological research partners to solve problems related to the exploitation of numerical simulators, such as code transposition (how to go from small to large scale when only small-scale simulations are possible), taking into account the uncertainties that affect the result of the simulation, validation and calibration (how to validate and calibrate a computer code from collected experimental data).

This project is the result of a simple observation: industries using computer codes are often confronted with similar problems during the exploitation of these codes, even if their fields of application are very diverse. Indeed, the increase in the availability of computing cores is counterbalanced by the growing complexity of the simulations, whose computational times are usually of the order of an hour or a day. In practice, this limits the number of simulations. This is why the development of mathematical methods to make the best use of simulators and the data they produce is a source of progress. The experience acquired over the last thirteen years in the DICE and ReDICE projects and the OQUAIDO Chair shows that the formalization of real industrial problems often gives rise to first-rate theoretical problems that can feed scientific and technical advances.

The creation of the CIROQUO Research & Industry Consortium, led by the Ecole Centrale de Lyon and co-animated with the IFPEN, follows these observations and responds to a desire for collaboration between technological research partners and academics in order to meet the challenges of exploiting large computing codes.

Scientific approach. The limitation of the number of calls to simulators implies that some information – even the most basic information such as the mean value, the influence of a variable or the minimum value of a criterion – cannot be obtained directly by the usual methods. The international scientific community, structured around computer experiments and uncertainty quantification, took up this problem more than twenty years ago, but a large number of problems remain open. On the academic level, this is a dynamic field which is notably the subject of the French CNRS Research Group MascotNum since 2006 and renewed in 2020.

Composition. The CIROQUO Research & Industry Consortium aims to bring together a limited number of participants in order to make joint progress on test cases from the industrial world and on the upstream research that their treatment requires. The overall approach that it will focus on is metamodeling and related areas such as experiment planning, optimization, inversion and calibration. IRSN, STORENGY, CEA, IFPEN, and BRGM are the Technological Research Partners and Mines Saint-Etienne, Centrale Lyon, CNRS, UCA, UPS, UT3 and Inria the Academic Partners of the consortium.

Scientific objectives. On the practical level, the expected impacts of the project are a concretization of the progress of numerical simulation by a better use of computational time, which allows the determination of better solutions and associated uncertainties. On the theoretical level, this project will allow to create an emulation around the major scientific locks of the discipline such as code transposition/calibration/validation, modeling for complex environments, or stochastic codes. In each of these scientific axes, a particular attention will be paid to the large dimension. Real problems sometimes involve several tens or hundreds of inputs. Methodological advances will be proposed to take into account this additional difficulty. The work expected from the consortium differs from the dominant research in machine learning by specificities linked to the exploration of expensive numerical simulations. However, it seems important to build bridges between the many recent developments in machine learning and the field of numerical simulation.

Philosophy. The CIROQUO Research & Industry Consortium is a scientific collaboration project aiming to mobilize means to achieve methodological advances. The project promotes cross-fertilization between partners coming from different backgrounds but confronted with problems related to a common methodology. Its main goals are:

• The development of exchanges between technological research partners and academic partners on issues, practices and solutions through periodic scientific meetings and collaborative work, particularly through the co-supervision of students,
• The contribution of common scientific skills thanks to regular training in mathematics and computer science,
• The recognition of the Consortium at the highest level thanks to publications in international journals and the diffusion of free reference software.

Example of a case study. Calibrating a numerical code means determining the best values for the input parameters by comparing its predictions with the experimental data available. In some cases, physical experiments cannot be carried out at full scale or on all the observables we are interested in. Only small scale or partial experiments can be carried out, and these are known as simplified experiments. The aim of Charlie Sire's postdoc was to develop a methodology, known as transposition, enabling a code to be calibrated in the context of a simplified experiment and then transposed to the real case. The postdoc was short (10 months) due to Charlie Sire finding a position at Ecole des Mines. In spite of this, 29 has been written in collaboration with CEA (who brought a real test case, a Taylor cylinder impact test) and submitted in October 2024.

Extension. CIROQUO was supposed to end in December 2024, but has been extended for four years under the same terms. New partners will be included, in particular, EDF (Électricité de France).

9.2 Collaboration with EdF on industrial risks

This collaboration has been going on for several years, with Josselin Garnier as the leader for ASCII. It concerns the assessment of the reliability of hydraulic and nuclear power plants built and operated by EDF (Électricité de France). The failure of a power plant may have major consequences, such as floods, dam failures, or core meltdowns. For regulatory and safety reasons EDF must ensure that the probability of failure of a power plant is suitably small.

The failure of such systems occurs when the values of certain physical variables (temperature, pressure, water level) exceed some critical thresholds. Typically, these variables enter this critical region when several components of the system are deteriorated. Therefore, in order to estimate the probability of system failure, it is necessary to model jointly the behavior of the components and of the physical variables.

For this purpose, a model based on Piecewise Deterministic Markovian Piecewise Processes (PDMP) is devised and used. The platform called PYCATSHOO has been developed by EDF in order to simulate this type of process. This platform allows to estimate the probability of failure of the system by Monte Carlo simulation as long as the probability of failure is not too low. When the probability becomes too low, the classical Monte Carlo estimation method requires a very large number of simulations in order to estimate the probabilities of such rare events, and is much too slow to be used in our context. It is then necessary to use methods using fewer simulations to estimate the probability of system failure, such as variance reduction methods. Among the variance reduction methods are “importance sampling” and “splitting” methods, but these methods present difficulties when used with PDMPs.

This had first lead to the defense of a CIFRE thesis by Thomas Galtier in 2019. It was followed by a CIFRE thesis by Guillaume Chennetier, defended on September 24, 2024, after which he moved to a postdoc position at Ecole des Ponts. In his thesis Guillaume has proposed new methods for estimating rare event probabilities for PDMPs, which offer the flexibility needed to represent complex dynamic industrial systems. The industrial challenge was to enable the tool PyCATSHOO, used by EDF for its probabilistic safety assessment studies, to efficiently estimate the failure probability of such systems with guaranteed accuracy. Guillaume has proposed a theoretical framework for implementing importance sampling of PDMPs, and has highlighted the connection between the optimal biased distribution and the so-called “committor function" of the process. Using tools from reliability analysis and the theory of random walks on graphs, new families of approximations of the committor function were introduced in the thesis. The proposed methodology is adaptive: an approximation of the committor function is constructed a priori and then refined during the simulations of a cross-entropy procedure. The simulations are then recycled to produce an importance sampling estimator of the target probability. Convergence results have been obtained, making it possible to overcome the dependence between simulations and to construct asymptotic confidence intervals. The method produced excellent results on the tested industrial systems. Theoretical articles have been written and submitted to journals.

10.1 International initiatives

10.2 International research visitors

10.3 National initiatives

Quentin Cormier visited Julien Tugaut at Université Jean Monnet (UJM, Saint-Étienne). They work together on the long time behavior of some McKean-Vlasov equations.

11.1 Promoting scientific activities

11.2 Teaching - Supervision - Juries

12.1 Major publications

• 1 articleL.Liliana Borcea, J.Josselin Garnier, A.Alexander Mamonov and J.Jörn Zimmerling. When Data Driven Reduced Order Modeling Meets Full Waveform Inversion.SIAM Review663August 2024, 501-532HALDOIback to text
• 2 inproceedingsM.Maxime Colomb, Q.Quentin Cormier, J.Josselin Garnier, C.Carl Graham, J.Julien Perret and D.Denis Talay. ICI Project: Individual-based epidemic propagation simulator based on a Digital Twin.Health GIS: Spatial Thinking in Applied research 2024Calgari, CanadaNovember 2024HAL
• 3 articleQ.Quentin Cormier. A mean-field model of Integrate-and-Fire neurons: non-linear stability of the stationary solutions.Mathematical Neuroscience and ApplicationsSeptember 2024HALDOIback to text

12.2 Publications of the year