2023Activity reportProject-TeamASCII

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.

6.1 Awards

Josselin Garnier has been elected in December 2023 to the Académie des Sciences.

6.2 Scientific results

Josselin Garnier has obtained important results with Liliana Borcea on the inverse problem methods based on reduced order models. These results allow to obtain new formulations for the inversion methods for wave forms (for sismic imaging for instance). The solutions can be obtained by optimisation methods in a much more tractable way than traditional methods, since the cost function to be optimised enjoys nice convexity properties.

7.1 New platforms

8.1 Mean field games, mean field optimal stopping problems

Participants: Leila Bassou, Quentin Cormier, Nizar Touzi.

From finite population optimal stopping to mean field optimal stopping

In collaboration with Mehdi Talbi and Jianfeng Zhang, Nizar Touzi analyzed the convergence of the finite population optimal stopping problem towards the corresponding mean field limit. Building on the viscosity solution characterization of the mean field optimal stopping problem of our previous papers [Talbi, Touzi, Zhang 2021, 2022], they proved the convergence of the value functions by adapting the Barles-Souganidis [1991] monotone scheme method to this context. They next characterized the optimal stopping policies of the mean field problem by the accumulation points of the finite population optimal stopping strategies. In particular, if the limiting problem has a unique optimal stopping policy, then the finite population optimal stopping strategies do converge towards this solution. As a by-product of their analysis, they provided an extension of the standard propagation of chaos to the context of stopped McKean-Vlasov diffusions.

A Principal-Agent Framework for Optimal Incentives in Renewable Investments

René Aïd, Annika Kemper and Nizar Touzi investigated the optimal regulation of energy production reflecting the long-term goals of the Paris Climate Agreement. They analyzed the optimal regulatory incentives to foster the development of non-emissive electricity generation when the demand for power is served either by a monopoly or by two competing agents. The regulator wishes to encourage green investments to limit carbon emissions, while simultaneously reducing intermittency of the total energy production. They find that the regulation of a competitive market is more efficient than the one of the monopoly as measured with the certainty equivalent of the Principal's value function. This higher efficiency is achieved thanks to a higher degree of freedom of the incentive mechanisms which involves cross-subsidies between firms. A numerical study quantifies the impact of the designed second-best contract in both market structures compared to the business-as-usual scenario. In addition, they expanded the monopolistic and competitive setup to a more general class of tractable Principal-Multi-Agent incentives problems when both the drift and the volatility of a multi-dimensional diffusion process can be controlled by the Agents. They followed the resolution methodology of Cvitanić et al. (2018) in an extended linear quadratic setting with exponential utilities and a multi-dimensional state process of Ornstein-Uhlenbeck type. They provided closed-form expression of the second-best contracts. In particular, they show that they are in rebate form involving time-dependent prices of each state-variable.

Mean field game of mutual holding with defaultable agents, and systemic risk

Mao Fabrice Djete, Gaoyue Guo, Nizar Touzi introduced the possibility of default in the mean field game of mutual holding. This is modeled by introducing absorption at the origin of the equity process. They provided an explicit solution of this mean field game. Moreover, they provided a particle system approximation, and derived an autonomous equation for the time evolution of the default probability, or equivalently the law of the hitting time of the origin by the equity process. The systemic risk is thus described by the evolution of the default probability.

Impact of carbon market on production emissions

Arash Fahim and Nizar Touzi addressed the effect of the carbon emission allowance market on the production policy of a large polluter production firm. They investigated this effect in two cases; when the large polluter cannot affect the risk premium of the allowance market, and when it can change the risk premium by its production. In this simple model, they ignored any possible investment of the firm in pollution reducing technologies. They formulated the problem of optimal production by a stochastic optimization problem. Then, they showed that, as expected, the market reduces the optimal production policy in the first case if the firm is not given a generous initial cheap allowance package. However, when the large producer activities can change the market risk premium, the cut on the production and consequently pollution cannot be guaranteed. In fact, there are cases in this model when the optimal production is always larger than expected, and an increase in production, and thus pollution, can increase the profit of the firm. They concluded that some of the parameters of the market which contribute to this effect can be wisely controlled by the regulators in order to diminish this manipulative behavior of the firm.

Viscosity Solutions for HJB Equations on the Process Space: Application to Mean Field Control with Common Noise

Jianjun Zhou, Nizar Touzi, Jianfeng Zhang investigated a path dependent optimal control problem on the process space with both drift and volatility controls, with possibly degenerate volatility. The dynamic value function is characterized by a fully nonlinear second order path dependent HJB equation on the process space, which is by nature infinite dimensional. In particular, their model covers mean field control problems with common noise as a special case. They introduced a new notion of viscosity solutions and establish both existence and comparison principle, under merely Lipschitz continuity assumptions. The main feature of their notion is that, besides the standard smooth part, the test function consists of an extra singular component which allows us to handle the second order derivatives of the smooth test functions without invoking the Ishii's lemma. They used the doubling variable arguments, combined with the Ekeland-Borwein-Preiss Variational Principle in order to overcome the noncompactness of the state space. A smooth gauge-type function on the path space is crucial for their estimates.

Entropic optimal planning for path-dependent mean field games

In 5, in the context of mean field games, with possible control of the diffusion coefficient, Zhenjie Ren, Xiaolu Tan, Nizar Touzi and Junjian Yang considered a path-dependent version of the planning problem introduced by P.L. Lions: given a pair of marginal distributions $({\mu }_0,{\mu }_1)$, find a specification of the game problem starting from the initial distribution ${\mu }_0$, and inducing the target distribution ${\mu }_1$ at the mean field game equilibrium. Their main result reduces the path-dependent planning problem into an embedding problem, that is, constructing a McKean-Vlasov dynamics with given marginals $({\mu }_0,{\mu }_1)$. Some sufficient conditions on $({\mu }_0,{\mu }_1)$ are provided to guarantee the existence of solutions. They also characterized, up to integrability, the minimum entropy solution of the planning problem. In particular, as uniqueness does not hold anymore in their path-dependent setting, one can naturally introduce an optimal planning problem which would be reduced to an optimal transport problem along with controlled McKean-Vlasov dynamics.

Synchronization in a Kuramoto Mean Field Game

In collaboration with René Carmona and Mete Soner, Quentin Cormier has studied the classical Kuramoto model in the setting of an infinite horizon mean field game 11. The system is shown to exhibit both synchronization and phase transition. Incoherence below a critical value of the interaction parameter is demonstrated by the stability of the uniform distribution. Above this value, the game bifurcates and develops self-organizing time homogeneous Nash equilibria. As interactions become stronger, these stationary solutions become fully synchronized. Results are proved by an amalgam of techniques from nonlinear partial differential equations, viscosity solutions, stochastic optimal control and stochastic processes.

8.2 Stochastic cooperative numerical methods for complex industrial systems

Participants: Guillaume Chennetier, Josselin Garnier, Clément Cauchy, Baptiste Kerleguer, Paul Lartaud, Angèle Niclas.

Waveform inversion via reduced order modeling

Liliana Borcea, Josselin Garnier Alexander Mamonov and Jörn Zimmerling introduced a novel approach to waveform inversion based on a data-driven reduced order model (ROM) of the wave operator 9. The presentation is for the acoustic wave equation, but the approach can be extended to elastic or electromagnetic waves. The data are time resolved measurements of the pressure wave gathered by an acquisition system that probes the unknown medium with pulses and measures the generated waves. They proposed to solve the inverse problem of velocity estimation by minimizing the square misfit between the ROM computed from the recorded data and the ROM computed from the modeled data, at the current guess of the velocity. They gave a step by step computation of the ROM, which depends nonlinearly on the data and yet can be obtained from them in a noniterative fashion, using efficient methods from linear algebra. We also explain how to make the ROM robust to data inaccuracy. The ROM computation requires the full array response matrix gathered with colocated sources and receivers. However, they find that the computation can deal with an approximation of this matrix, obtained from towed-streamer data using interpolation and reciprocity on-the-fly. Although the full-waveform inversion approach of nonlinear least-squares data fitting is challenging without low-frequency information, due to multiple minima of the data fit objective function, they find that the ROM misfit objective function has better behavior, even for a poor initial guess. They also find by explicit computation of the objective functions in a simple setting that the ROM misfit objective function has convexity properties, whereas the least-squares data fit objective function displays multiple local minim

Waveform Inversion with a Data Driven Estimate of the Internal Wave

In 8 Liliana Borcea, Josselin Garnier Alexander Mamonov and Jörn Zimmerling studied an inverse problem for the wave equation, concerned with estimating the wave speed from data gathered by an array of sources and receivers that emit probing signals and measure the resulting waves. The typical approach to solving this problem is a nonlinear least squares minimization of the data misfit, over a search space. There are two main impediments to this approach, which manifest as multiple local minima of the objective function: The nonlinearity of the mapping from the wave speed to the data, which accounts for multiple scattering effects, and poor knowledge of the kinematics (smooth part of the wave speed), which causes cycle skipping. They shaw that the nonlinearity can be mitigated using a data driven estimate of the wave field at points inside the medium, also known as the “internal wave field.” This leads to improved performance of the inversion for a reasonable initial guess of the kinematics.

Paraxial Wave Propagation in Random Media with Long-Range Correlations

In 10 Liliana Borcea, Josselin Garnier and Knut Sølna studied the paraxial wave equation with a randomly perturbed index of refraction, which can model the propagation of a wave beam in a turbulent medium. The random perturbation is a stationary and isotropic process with a general form of the covariance that may be integrable or not. They focused attention mostly on the non-integrable case, which corresponds to a random perturbation with long-range correlations, that is relevant for propagation through a cloudy turbulent atmosphere. The analysis is carried out in a high-frequency regime where the forward scattering approximation holds. It reveals that the randomization of the wave field is multiscale: the travel time of the wave front is randomized at short distances of propagation and it can be described by a fractional Brownian motion. The wave field observed in the random travel time frame is affected by the random perturbations at long distances, and it is described by a Schr¨odinger-type equation driven by a standard Brownian field. They used these results to quantify how scattering leads to decorrelation of the spatial and spectral components of the wave field and to a deformation of the pulse emitted by the source. These are important questions for applications like imaging and free space communications with pulsed laser beams through a turbulent atmosphere. We also compare the results with those used in the optics literature, which are based on the Kolmogorov model of turbulence.

Well-posedness of wave scattering in perturbed elastic waveguides and plates: application to an inverse problem of shape defect detection

Eric Bonnetier Angèle Niclas and L. Seppecher worked on theoretical tools to study wave propagation in elastic waveguides and perform multi-frequency scattering inversion to reconstruct small shape defects in elastic waveguides and plates 7. Given surface multi-frequency wavefield measurements, they used a Born approximation to reconstruct localized defect in the geometry of the plate. To justify this approximation, they introduced a rigorous framework to study the propagation of elastic wavefield generated by arbitrary sources. By studying the decreasing rate of the series of inhomogeneous Lamb mode, they proved the well-posedness of the PDE that model elastic wave propagation in two- and three-dimensional planar waveguides. They also characterize the critical frequencies for which the Lamb decomposition is not valid. By using these results, they generalized the shape reconstruction method already developed for acoustic waveguide to two-dimensional elastic waveguides and provide a stable reconstruction method based on a mode-by-mode spacial Fourier inversion given by the scattered field.

Reconstruction of smooth shape defects in waveguides using locally resonant frequencies.

In 17, Angèle Niclas and Laurent Seppecher wroked on a new method to reconstruct slowly varying width defects in 2D waveguides using locally resonant frequencies. At these frequencies, locally resonant modes propagate in the waveguide under the form of Airy functions depending on a parameter called the locally resonant point. In this particular point, the local width of the waveguide is known and its location can be recovered from boundary measurements of the wavefield. Using the same process for different frequencies, they produce a good approximation of the width in all the waveguide. Given multi-frequency measurements taken at the surface of the waveguide, we provide a $L_{\infty }$-stable explicit method to reconstruct the width of the waveguide. We finally validate our method on numerical data, and we discuss its applications and limits.

Adaptive importance sampling based on fault tree analysis for piecewise deterministic Markov process

Piecewise deterministic Markov processes (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 (AIS) method based on a cross-entropy (CE) procedure. The success of this method relies on the selection of a good family of approximations of the committor function of the PDMP. In collaboration with Hassane Chraibi and Anne Dutfoy, Guillaume Chennetier and Josselin Garnier, original families are proposed in 20. They are well adapted to high-dimensional industrial systems. Their forms are based on reliability concepts related to fault tree analysis: minimal path sets and minimal cut sets. The proposed method is discussed in detail and applied to academic systems and to a realistic system from the nuclear industry.

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

Participants: Denis Talay.

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.

8.4 Stochastic cooperative communication networks; modelling and numerical issues for interacting stochastic particle systems

Participants: Quentin Cormier, Carl Graham, Denis Talay.

Communication networks and their algorithms.

An important current research topic of Carl Graham is the modeling, analysis, simulation, and performance evaluation of communication networks and their algorithms. Most of these algorithms must function in real time, in a distributed fashion, and using sparse information. In particular load balancing algorithms aim to provide better utilization of the network resources and hence better quality of service for clients by striving to avoid the starving of some servers and the build-up of queues at others by routing the clients so as to have them well spread out throughout the system. Carl Graham's recent focus of work on these networks is perfect simulation in equilibrium, and a paper on this is in its terminal phases of writing. Perfect simulation methods allow to estimate quality of service (QoS) indicators in the stationary regime by Monte Carlo methods.

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 22. In addition, he has adapted the method to study the long time behavior of a model of interacting Integrate-and-Fire neurons 21. 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.

An hypothesis test for complex stochastic simulations.

In a joint work with Héctor Olivero, 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.

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 varied. 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 the CIROQUO Research & Industry Consortium will focus on is metamodeling and related areas such as experiment planning, optimization, inversion and calibration. IRSN, STORENGY, CEA, IFPEN, BRGM are the Technological Research Partners. Mines Saint-Etienne, Centrale Lyon, CNRS, UCA, UPS, UT3 and Inria are 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. It has three objectives: - 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.

9.2 Collaboration with EdF on industrial risks

This collaboration whose Josselin Garnier is the Ascii leader, has been underway for several years. It concerns the assessment of the reliability of hydraulic and nuclear power plants built and operated by EDF (Electricite De France). The failure of a power plant is associated with major consequences (flood, dam failure, or core meltdown), for regulatory and safety reasons EDF must ensure that the probability of failure of a power plant is sufficiently low.

The failure of such systems occurs when physical variables (temperature, pressure, water level) exceed a certain critical threshold. Typically, these variables enter this critical region only 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 a Deterministic Markovian Piecewise Process (DMPP) is used. The platform called PYCATSHOO has been developed by EDF 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, which requires a lot of simulations to estimate the probabilities of rare events, is much too slow to execute in our context. It is necessary to use methods using fewer simulations to estimate the probability of system failure: variance reduction methods. Among the variance reduction methods are "importance sampling" and "splitting" methods, but these methods present difficulties when used with PDMPs.

Work has been undertaken on the subject, leading to the defense of a first CIFRE thesis (Thomas Galtier, thesis defended in 2019) and the preparation of a new CIFRE thesis (Guillaume Chennetier, from 2021). Theoretical articles have been written and submitted to journals. New theoretical works on sensitivity analysis in rare event regimes are the subject of the new thesis. The integration of the methods in the PYCATSHOO platform is progressively done.

10.1 International initiatives

10.2 International research visitors

11.1 Promoting scientific activities

11.2 Teaching - Supervision - Juries

11.3 Popularization

Josselin Garnier and Valérie Maume-Deschamps were leading the working group “Développement économique, de la compétitivité et de l’innovation” of the “Assises des mathématiques”. They have written a synthesis included in the proceedings of the Assises.

12.1 Major publications

• 1 articleL.Liliana Borcea, J.Josselin Garnier, A.Alexander Mamonov and J.Jörn Zimmerling. Waveform Inversion with a Data Driven Estimate of the Internal Wave.SIAM Journal on Imaging Sciences161March 2023, 280-312HALDOI
• 2 articleL.Liliana Borcea, J.Josselin Garnier, A.Alexander Mamonov and J.Jörn Zimmerling. Waveform inversion via reduced order modeling.Geophysics882March 2023, R175-R191HALDOI
• 3 articleL.Liliana Borcea, J.Josselin Garnier and K.Knut Sølna. Paraxial Wave Propagation in Random Media with Long-Range Correlations.SIAM Journal on Applied Mathematics831February 2023, 25-51HALDOI
• 4 articleR.Rene Carmona, Q.Quentin Cormier and H. M.H. Mete Soner. Synchronization in a Kuramoto Mean Field Game.Communications in Partial Differential Equations489September 2023, 1214-1244HALDOI
• 5 articleZ.Zhenjie Ren, X.Xiaolu Tan, N.Nizar Touzi and J.Junjian Yang. Entropic Optimal Planning for Path-Dependent Mean Field Games.SIAM Journal on Control and Optimization613June 2023, 1415-1437HALDOIback to text

12.2 Publications of the year