Keywords
 A6. Modeling, simulation and control
 A6.1. Methods in mathematical modeling
 A6.1.1. Continuous Modeling (PDE, ODE)
 A6.1.4. Multiscale modeling
 A6.1.5. Multiphysics modeling
 A6.2. Scientific computing, Numerical Analysis & Optimization
 A6.2.1. Numerical analysis of PDE and ODE
 A6.2.4. Statistical methods
 A6.2.6. Optimization
 A6.3. Computationdata interaction
 A6.3.1. Inverse problems
 A6.3.2. Data assimilation
 A6.3.5. Uncertainty Quantification
 A9. Artificial intelligence
 A9.2. Machine learning
 B1.1.8. Mathematical biology
 B1.1.11. Plant Biology
 B2.2.1. Cardiovascular and respiratory diseases
 B5.2.1. Road vehicles
 B5.2.3. Aviation
 B5.3. Nanotechnology
 B7.1.1. Pedestrian traffic and crowds
 B7.1.2. Road traffic
 B8.1.1. Energy for smart buildings
1 Team members, visitors, external collaborators
Research Scientists
 Paola Goatin [Team leader, Inria, Senior Researcher, HDR]
 Mickael Binois [Inria, Researcher]
 JeanAntoine Desideri [Inria, Emeritus, HDR]
 Regis Duvigneau [Inria, Researcher, HDR]
Faculty Member
 Abderrahmane Habbal [Univ Côte d'Azur, Associate Professor, HDR]
PostDoctoral Fellow
 Khadija Musayeva [Univ de Nice  Sophia Antipolis, from Oct 2020]
PhD Students
 Salma Chabbar [Ecole Mohammedia d'Ingénieurs, Université Mohamed V, Rabat, 01/01/2020  31/12/2022, coadvise EIFFEL Scholarship ]
 Marwa Ouni [Ecole Nationale d'ingénieurs de Tunis, Université Tunis al Manar, Tunis, 01/10/2017  31/03/2021, PHC Utique coadvise]
 Stefano Pezzano [Inria]
Interns and Apprentices
 Alexandra Wuerth [Inria, from Apr 2020 until Sep 2020]
Administrative Assistant
 Montserrat Argente [Inria]
Visiting Scientist
 Brahim El Alaoui [Université Mohamed V, , until Aug 2020, Erasmus++ doctoral ]
2 Overall objectives
ACUMES aims at developing a rigorous framework for numerical simulations and optimal control for transportation and buildings, with focus on multiscale, heterogeneous, unsteady phenomena subject to uncertainty. Starting from established macroscopic Partial Differential Equation (PDE) models, we pursue a set of innovative approaches to include smallscale phenomena, which impact the whole system. Targeting applications contributing to sustainability of urban environments, we couple the resulting models with robust control and optimization techniques.
Modern engineering sciences make an important use of mathematical models and numerical simulations at the conception stage. Effective models and efficient numerical tools allow for optimization before production and to avoid the construction of expensive prototypes or costly postprocess adjustments. Most uptodate modeling techniques aim at helping engineers to increase performances and safety and reduce costs and pollutant emissions of their products. For example, mathematical traffic flow models are used by civil engineers to test new management strategies in order to reduce congestion on the existing road networks and improve crowd evacuation from buildings or other confined spaces without constructing new infrastructures. Similar models are also used in mechanical engineering, in conjunction with concurrent optimization methods, to reduce energy consumption, noise and pollutant emissions of cars, or to increase thermal and structural efficiency of buildings while, in both cases, reducing ecological costs.
Nevertheless, current models and numerical methods exhibit some limitations:
 Most simulationbased design procedures used in engineering still rely on steady (timeaveraged) state models. Significant improvements have already been obtained with such a modeling level, for instance by optimizing car shapes, but finer models taking into account unsteady phenomena are required in the design phase for further improvements.
 The classical purely macroscopic approach, while offering a framework with a sound analytical basis, performing numerical techniques and good modeling features to some extent, is not able to reproduce some particular phenomena related to specific interactions occurring at lower (possibly micro) level. We refer for example to selforganizing phenomena observed in pedestrian flows, or to the dynamics of turbulent flows for which large scale / small scale vortical structures interfere. These flow characteristics need to be taken into account to obtain more precise models and improved optimal solutions.
 Uncertainty related to operational conditions (e.g. inflow velocity in aerodynamics), or models (e.g. individual behavior in crowds) is still rarely considered in engineering analysis and design, yielding solutions of poor robustness.
This project focuses on the analysis and optimal control of classical and nonclassical evolutionary systems of Partial Differential Equations (PDEs) arising in the modeling and optimization of engineering problems related to safety and sustainability of urban environments, mostly involving fluiddynamics and structural mechanics. The complexity of the involved dynamical systems is expressed by multiscale, timedependent phenomena, possibly subject to uncertainty, which can hardly be tackled using classical approaches, and require the development of unconventional techniques.
3 Research program
3.1 Research directions
The project develops along the following two axes:
 modeling complex systems through novel (unconventional) PDE systems, accounting for multiscale phenomena and uncertainty;
 optimization and optimal control algorithms for systems governed by the above PDE systems.
These themes are motivated by the specific problems treated in the applications, and represent important and uptodate issues in engineering sciences. For example, improving the design of transportation means and civil buildings, and the control of traffic flows, would result not only in better performances of the object of the optimization strategy (vehicles, buildings or road networks level of service), but also in enhanced safety and lower energy consumption, contributing to reduce costs and pollutant emissions.
3.2 PDE models accounting for multiscale phenomena and uncertainties
Dynamical models consisting of evolutionary PDEs, mainly of hyperbolic type, appear classically in the applications studied by the previous ProjectTeam Opale (compressible flows, traffic, celldynamics, medicine, etc). Yet, the classical purely macroscopic approach is not able to account for some particular phenomena related to specific interactions occurring at smaller scales. These phenomena can be of greater importance when dealing with particular applications, where the "first order" approximation given by the purely macroscopic approach reveals to be inadequate. We refer for example to selforganizing phenomena observed in pedestrian flows 110, or to the dynamics of turbulent flows for which large scale / small scale vortical structures interfere 138.
Nevertheless, macroscopic models offer well known advantages, namely a sound analytical framework, fast numerical schemes, the presence of a low number of parameters to be calibrated, and efficient optimization procedures. Therefore, we are convinced of the interest of keeping this point of view as dominant, while completing the models with information on the dynamics at the small scale / microscopic level. This can be achieved through several techniques, like hybrid models, homogenization, mean field games. In this project, we will focus on the aspects detailed below.
The development of adapted and efficient numerical schemes is a mandatory completion, and sometimes ingredient, of all the approaches listed below. The numerical schemes developed by the team are based on finite volumes or finite elements techniques, and constitute an important tool in the study of the considered models, providing a necessary step towards the design and implementation of the corresponding optimization algorithms, see Section 3.3.
3.2.1 Micromacro couplings
Modeling of complex problems with a dominant macroscopic point of view often requires couplings with small scale descriptions. Accounting for systems heterogeneity or different degrees of accuracy usually leads to coupled PDEODE systems.
In the case of heterogeneous problems the coupling is "intrinsic", i.e. the two models evolve together and mutually affect eachother. For example, accounting for the impact of a large and slow vehicle (like a bus or a truck) on traffic flow leads to a strongly coupled system consisting of a (system of) conservation law(s) coupled with an ODE describing the bus trajectory, which acts as a moving bottleneck. The coupling is realized through a local unilateral moving constraint on the flow at the bus location, see 80 for an existence result and 64, 79 for numerical schemes.
If the coupling is intended to offer higher degree of accuracy at some locations, a macroscopic and a microscopic model are connected through an artificial boundary, and exchange information across it through suitable boundary conditions. See 70, 98 for some applications in traffic flow modelling, and 90, 95, 97 for applications to cell dynamics.
The corresponding numerical schemes are usually based on classical finite volume or finite element methods for the PDE, and Euler or RungeKutta schemes for the ODE, coupled in order to take into account the interaction fronts. In particular, the dynamics of the coupling boundaries require an accurate handling capturing the possible presence of nonclassical shocks and preventing diffusion, which could produce wrong solutions, see for example 64, 79.
We plan to pursue our activity in this framework, also extending the above mentioned approaches to problems in two or higher space dimensions, to cover applications to crowd dynamics or fluidstructure interaction.
3.2.2 Micromacro limits
Rigorous derivation of macroscopic models from microscopic ones offers a sound basis for the proposed modeling approach, and can provide alternative numerical schemes, see for example 71, 85 for the derivation of LighthillWhithamRichards 122, 137 traffic flow model from FollowtheLeader and 91 for results on crowd motion models (see also 113). To tackle this aspect, we will rely mainly on two (interconnected) concepts: measurevalued solutions and meanfield limits.
The notion of measurevalued solutions for conservation laws was first introduced by DiPerna 86, and extensively used since then to prove convergence of approximate solutions and deduce existence results, see for example 92 and references therein. Measurevalued functions have been recently advocated as the appropriate notion of solution to tackle problems for which analytical results (such as existence and uniqueness of weak solutions in distributional sense) and numerical convergence are missing 52, 94. We refer, for example, to the notion of solution for nonhyperbolic systems 100, for which no general theoretical result is available at present, and to the convergence of finite volume schemes for systems of hyperbolic conservation laws in several space dimensions, see 94.
In this framework, we plan to investigate and make use of measurebased PDE models for vehicular and pedestrian traffic flows. Indeed, a modeling approach based on (multiscale) timeevolving measures (expressing the agents probability distribution in space) has been recently introduced (see the monograph 75), and proved to be successful for studying emerging selforganised flow patterns 74. The theoretical measure framework proves to be also relevant in addressing micromacro limiting procedures of mean field type 101, where one lets the number of agents going to infinity, while keeping the total mass constant. In this case, one must prove that the empirical measure, corresponding to the sum of Dirac measures concentrated at the agents positions, converges to a measurevalued solution of the corresponding macroscopic evolution equation. We recall that a key ingredient in this approach is the use of the Wasserstein distances 147, 146. Indeed, as observed in 130, the usual ${L}^{1}$ spaces are not natural in this context, since they don't guarantee uniqueness of solutions.
This procedure can potentially be extended to more complex configurations, like for example road networks or different classes of interacting agents, or to other application domains, like celldynamics.
Another powerful tool we shall consider to deal with micromacro limits is the socalled Mean Field Games (MFG) technique (see the seminal paper 121). This approach has been recently applied to some of the systems studied by the team, such as traffic flow and cell dynamics. In the context of crowd dynamics, including the case of several populations with different targets, the mean field game approach has been adopted in 60, 61, 87, 120, under the assumption that the individual behavior evolves according to a stochastic process, which gives rise to parabolic equations greatly simplifying the analysis of the system. Besides, a deterministic context is studied in 133, which considers a nonlocal velocity field. For cell dynamics, in order to take into account the fast processes that occur in the migrationrelated machinery, a framework such the one developed in 78 to handle games "where agents evolve their strategies according to the bestreply scheme on a much faster time scale than their social configuration variables" may turn out to be suitable. An alternative framework to MFG is also considered. This framework is based on the formulation of Nash games constrained by the FokkerPlanck (FP, 50) partial differential equations that govern the time evolution of the probability density functions PDF of stochastic systems and on objectives that may require to follow a given PDF trajectory or to minimize an expectation functional.
3.2.3 Nonlocal flows
Nonlocal interactions can be described through macroscopic models based on integrodifferential equations. Systems of the type
where $u=u(t,\mathbf{x})\in {R}^{N}$, $N\ge 1$ is the vector of conserved quantities and the variable $W=W(t,\mathbf{x},u)$ depends on an integral evaluation of $u$, arise in a variety of physical applications. Spaceintegral terms are considered for example in models for granular flows 47, sedimentation 54, supply chains 105, conveyor belts 103, biological applications like structured populations dynamics 129, or more general problems like gradient constrained equations 49. Also, nonlocal in time terms arise in conservation laws with memory, starting from 77. In particular, equations with nonlocal flux have been recently introduced in traffic flow modeling to account for the reaction of drivers or pedestrians to the surrounding density of other individuals, see 56, 63, 67, 102, 141. While pedestrians are likely to react to the presence of people all around them, drivers will mainly adapt their velocity to the downstream traffic, assigning a greater importance to closer vehicles. In particular, and in contrast to classical (without integral terms) macroscopic equations, these models are able to display finite acceleration of vehicles through Lipschitz bounds on the mean velocity 56, 102 and lane formation in crossing pedestrian flows.
General analytical results on nonlocal conservation laws, proving existence and eventually uniqueness of solutions of the Cauchy problem for 1, can be found in 48 for scalar equations in one space dimension ($N=d=1$), in 68 for scalar equations in several space dimensions ($N=1$, $d\ge 1$) and in 43, 69, 73 for multidimensional systems of conservation laws. Besides, specific finite volume numerical methods have been developed recently in 43, 102 and 119.
Relying on these encouraging results, we aim to push a step further the analytical and numerical study of nonlocal models of type 1, in particular concerning wellposedness of initial  regularity of solutions, boundary value problems and highorder numerical schemes.
3.2.4 Uncertainty in parameters and initialboundary data
Different sources of uncertainty can be identified in PDE models, related to the fact that the problem of interest is not perfectly known. At first, initial and boundary condition values can be uncertain. For instance, in traffic flows, the timedependent value of inlet and outlet fluxes, as well as the initial distribution of vehicles density, are not perfectly determined 62. In aerodynamics, inflow conditions like velocity modulus and direction, are subject to fluctuations 109, 128. For some engineering problems, the geometry of the boundary can also be uncertain, due to structural deformation, mechanical wear or disregard of some details 89. Another source of uncertainty is related to the value of some parameters in the PDE models. This is typically the case of parameters in turbulence models in fluid mechanics, which have been calibrated according to some reference flows but are not universal 139, 145, or in traffic flow models, which may depend on the type of road, weather conditions, or even the country of interest (due to differences in driving rules and conductors behaviour). This leads to equations with flux functions depending on random parameters 140, 143, for which the mean and the variance of the solutions can be computed using different techniques. Indeed, uncertainty quantification for systems governed by PDEs has become a very active research topic in the last years. Most approaches are embedded in a probabilistic framework and aim at quantifying statistical moments of the PDE solutions, under the assumption that the characteristics of uncertain parameters are known. Note that classical MonteCarlo approaches exhibit low convergence rate and consequently accurate simulations require huge computational times. In this respect, some enhanced algorithms have been proposed, for example in the balance law framework 126. Different approaches propose to modify the PDE solvers to account for this probabilistic context, for instance by defining the nondeterministic part of the solution on an orthogonal basis (Polynomial Chaos decomposition) and using a Galerkin projection 109, 118, 123, 150 or an entropy closure method 84, or by discretizing the probability space and extending the numerical schemes to the stochastic components 42. Alternatively, some other approaches maintain a fully deterministic PDE resolution, but approximate the solution in the vicinity of the reference parameter values by Taylor series expansions based on first or secondorder sensitivities 134, 145, 148.
Our objective regarding this topic is twofold. In a pure modeling perspective, we aim at including uncertainty quantification in models calibration and validation for predictive use. In this case, the choice of the techniques will depend on the specific problem considered 53. Besides, we plan to extend previous works on sensitivity analysis 89, 124 to more complex and more demanding problems. In particular, highorder Taylor expansions of the solution (greater than two) will be considered in the framework of the Sensitivity Equation Method 57 (SEM) for unsteady aerodynamic applications, to improve the accuracy of mean and variance estimations. A second targeted topic in this context is the study of the uncertainty related to turbulence closure parameters, in the sequel of 145. We aim at exploring the capability of the SEM approach to detect a change of flow topology, in case of detached flows. Our ambition is to contribute to the emergence of a new generation of simulation tools, which will provide solution densities rather than values, to tackle reallife uncertain problems. This task will also include a reflection about numerical schemes used to solve PDE systems, in the perspective of constructing a unified numerical framework able to account for exact geometries (isogeometric methods), uncertainty propagation and sensitivity analysis w.r.t. control parameters.
3.3 Optimization and control algorithms for systems governed by PDEs
The nonclassical models described above are developed in the perspective of design improvement for reallife applications. Therefore, control and optimization algorithms are also developed in conjunction with these models. The focus here is on the methodological development and analysis of optimization algorithms for PDE systems in general, keeping in mind the application domains in the way the problems are mathematically formulated.
3.3.1 Sensitivity vs. adjoint equation
Adjoint methods (achieved at continuous or discrete level) are now commonly used in industry for steady PDE problems. Our recent developments 136 have shown that the (discrete) adjoint method can be efficiently applied to cost gradient computations for timeevolving traffic flow on networks, thanks to the special structure of the associated linear systems and the underlying one dimensionality of the problem. However, this strategy is questionable for more complex (e.g. 2D/3D) unsteady problems, because it requires sophisticated and timeconsuming checkpointing and/or recomputing strategies 51, 104 for the backward time integration of the adjoint variables. The sensitivity equation method (SEM) offers a promising alternative 88, 114, if the number of design parameters is moderate. Moreover, this approach can be employed for other goals, like fast evaluation of neighboring solutions or uncertainty propagation 89.
Regarding this topic, we intend to apply the continuous sensitivity equation method to challenging problems. In particular, in aerodynamics, multiscale turbulence models like LargeEddy Simulation (LES) 138 , DetachedEddy Simulation (DES) 142 or OrganizedEddy Simulation (OES) 58, are more and more employed to analyse the unsteady dynamics of the flows around bluffbodies, because they have the ability to compute the interactions of vortices at different scales, contrary to classical ReynoldsAveraged NavierStokes models. However, their use in design optimization is tedious, due to the long time integration required. In collaboration with turbulence specialists (M. Braza, CNRS  IMFT), we aim at developing numerical methods for effective sensitivity analysis in this context, and apply them to realistic problems, like the optimization of active flow control devices. Note that the use of SEM allows computing cost functional gradients at any time, which permits to construct new gradientbased optimization strategies like instantaneousfeedback method 116 or multiobjective optimization algorithm (see section below).
3.3.2 Multiobjective descent algorithms for multidisciplinary, multipoint, unsteady optimization or robustdesign
In differentiable optimization, multidisciplinary, multipoint, unsteady optimization or robustdesign can all be formulated as multiobjective optimization problems. In this area, we have proposed the MultipleGradient Descent Algorithm (MGDA) to handle all criteria concurrently 8281. Originally, we have stated a principle according which, given a family of local gradients, a descent direction common to all considered objectivefunctions simultaneously is identified, assuming the Paretostationarity condition is not satisfied. When the family is linearlyindependent, we dispose of a direct algorithm. Inversely, when the family is linearlydependent, a quadraticprogramming problem should be solved. Hence, the technical difficulty is mostly conditioned by the number $m$ of objective functions relative to the search space dimension $n$. In this respect, the basic algorithm has recently been revised 83 to handle the case where $m>n$, and even $m\gg n$, and is currently being tested on a testcase of robust design subject to a periodic timedependent NavierStokes flow.
The multipoint situation is very similar and, being of great importance for engineering applications, will be treated at large.
Moreover, we intend to develop and test a new methodology for robust design that will include uncertainty effects. More precisely, we propose to employ MGDA to achieve an effective improvement of all criteria simultaneously, which can be of statistical nature or discrete functional values evaluated in confidence intervals of parameters. Some recent results obtained at ONERA 131 by a stochastic variant of our methodology confirm the viability of the approach. A PhD thesis has also been launched at ONERA/DADS.
Lastly, we note that in situations where gradients are difficult to evaluate, the method can be assisted by a metamodel 152.
3.3.3 Bayesian Optimization algorithms for efficient computation of general equilibria
Bayesian Optimization (BO) relies on Gaussian processes, which are used as emulators (or surrogates) of the blackbox model outputs based on a small set of model evaluations. Posterior distributions provided by the Gaussian process are used to design acquisition functions that guide sequential search strategies that balance between exploration and exploitation. Such approaches have been transposed to frameworks other than optimization, such as uncertainty quantification. Our aim is to investigate how the BO apparatus can be applied to the search of general game equilibria, and in particular the classical Nash equilibrium (NE). To this end, we propose two complementary acquisition functions, one based on a greedy search approach and one based on the Stepwise Uncertainty Reduction paradigm 96. Our proposal is designed to tackle derivativefree, expensive models, hence requiring very few model evaluations to converge to the solution.
3.3.4 Decentralized strategies for inverse problems
Most if not all the mathematical formulations of inverse problems (a.k.a. reconstruction, identification, data recovery, non destructive engineering,...) are known to be ill posed in the Hadamard sense. Indeed, in general, inverse problems try to fulfill (minimize) two or more very antagonistic criteria. One classical example is the Tikhonov regularization, trying to find artificially smoothed solutions close to naturally nonsmooth data.
We consider here the theoretical general framework of parameter identification coupled to (missing) data recovery. Our aim is to design, study and implement algorithms derived within a game theoretic framework, which are able to find, with computational efficiency, equilibria between the "identification related players" and the "data recovery players". These two parts are known to pose many challenges, from a theoretical point of view, like the identifiability issue, and from a numerical one, like convergence, stability and robustness problems. These questions are tricky 44 and still completely open for systems like e.g. coupled heat and thermoelastic joint data and material detection.
4 Application domains
4.1 Active flow control for vehicles
The reduction of CO2 emissions represents a great challenge for the automotive and aeronautic industries, which committed respectively a decrease of 20% for 2020 and 75% for 2050. This goal will not be reachable, unless a significant improvement of the aerodynamic performance of cars and aircrafts is achieved (e.g. aerodynamic resistance represents 70% of energy losses for cars above 90 km/h). Since vehicle design cannot be significantly modified, due to marketing or structural reasons, active flow control technologies are one of the most promising approaches to improve aerodynamic performance. This consists in introducing microdevices, like pulsating jets or vibrating membranes, that can modify vortices generated by vehicles. Thanks to flow nonlinearities, a small energy expense for actuation can significantly reduce energy losses. The efficiency of this approach has been demonstrated, experimentally as well as numerically, for simple configurations 151.
However, the lack of efficient and flexible numerical tools, that allow to simulate and optimize a large number of such devices on realistic configurations, is still a bottleneck for the emergence of this technology in industry. The main issue is the necessity of using highorder schemes and complex models to simulate actuated flows, accounting for phenomena occurring at different scales. In this context, we intend to contribute to the following research axes:
 Sensitivity analysis for actuated flows. Adjointbased (reverse) approaches, classically employed in design optimization procedure to compute functional gradients, are not well suited to this context. Therefore, we propose to explore the alternative (direct) formulation, which is not so much used, in the perspective of a better characterization of actuated flows and optimization of control devices.
 Isogeometric simulation of control devices. To simulate flows perturbed by smallscale actuators, we investigate the use of isogeometric analysis methods, which allow to account exactly for CADbased geometries in a highorder hierarchical representation framework. In particular, we try to exploit the features of the method to simulate more accurately complex flows including moving devices and multiscale phenomena.
4.2 Vehicular and pedestrian traffic flows
Intelligent Transportation Systems (ITS) is nowadays a booming sector, where the contribution of mathematical modeling and optimization is widely recognized. In this perspective, traffic flow models are a commonly cited example of "complex systems", in which individual behavior and selforganization phenomena must be taken into account to obtain a realistic description of the observed macroscopic dynamics 111. Further improvements require more advanced models, keeping into better account interactions at the microscopic scale, and adapted control techniques, see 59 and references therein.
In particular, we will focus on the following aspects:
 Junction models. We are interested in designing a general junction model both satisfying basic analytical properties guaranteeing wellposedness and being realistic for traffic applications. In particular, the model should be able to overcome severe drawbacks of existing models, such as restrictions on the number of involved roads and prescribed split ratios 72, 99, which limit their applicability to real world situations. HamiltonJacobi equations could be also an interesting direction of research, following the recent results obtained in 115.
 Data assimilation. In traffic flow modeling, the capability of correctly estimating and predicting the state of the system depends on the availability of rich and accurate data on the network. Up to now, the most classical sensors are fixed ones. They are composed of inductive loops (electrical wires) that are installed at different spatial positions of the network and that can measure the traffic flow, the occupancy rate (i.e. the proportion of time during which a vehicle is detected to be over the loop) and the speed (in case of a system of two distant loops). These data are useful / essential to calibrate the phenomenological relationship between flow and density which is known in the traffic literature as the Fundamental Diagram. Nowadays, thanks to the wide development of mobile internet and geolocalization techniques and its increasing adoption by the road users, smartphones have turned into perfect mobile sensors in many domains, including in traffic flow management. They can provide the research community with a large database of individual trajectory sets that are known as Floating Car Data (FCD), see 112 for a real field experiment. Classical macroscopic models, say (hyperbolic systems of) conservation laws, are not designed to take into account this new kind of microscopic data. Other formulations, like HamiltonJacobi partial differential equations, are most suited and have been intensively studied in the past five years (see 66, 65), with a stress on the (fixed) Eulerian framework. Up to our knowledge, there exist a few studies in the timeLagrangian as well as spaceLagrangian frameworks, where data coming from mobile sensors could be easily assimilated, due to the fact that the Lagrangian coordinate (say the label of a vehicle) is fixed.
 Control of autonomous vehicles. Traffic flow is usually controlled via traffic lights or variable speed limits, which have fixed space locations. The deployment of autonomous vehicles opens new perspectives in traffic management, as the use of a small fraction of cars to optimize the overall traffic. In this perspective, the possibility to track vehicles trajectories either by coupled micromacro models 80, 98 or via the HamiltonJacobi approach 66, 65 could allow to optimize the flow by controlling some specific vehicles corresponding to internal conditions.
4.3 Virtual Fractional Flow Reserve in coronary stenting
Atherosclerosis is a chronic inflammatory disease that affects the entire arterial network and especially the coronary arteries. It is an accumulation of lipids over the arterial surface due to a dysfunction of this latter. The objective of clinical intervention, in this case, is to establish a revascularization using different angioplasty techniques, among which the implantation of stents is the most widespread. This intervention involves introducing a stent into the damaged portion in order to allow the blood to circulate in a normal way over all the vessels. Revascularization is based on the principle of remedying ischemia, which is a decrease or an interruption of the supply of oxygen to the various organs. This anomaly is attenuated by the presence of several lesions (multivessel disease patients), which can lead to several complications. The key of a good medical intervention is the fact of establishing a good diagnosis, in order to decide which lesion requires to be treated. In the diagnosis phase, the clinician uses several techniques, among which angiography is the most popular. Angiography is an Xray technique to show the inside (the lumen) of blood vessels, in order to identify vessel narrowing: stenosis. Despite its widespread use, angiography is often imperfect in determining the physiological significance of coronary stenosis. If the problem remains simple for non significant lesions ($\le 40\%$) or very severe ( $\ge 70\%$), a very important category of intermediate lesions must benefit from a functional evaluation which will determine the strategy of treatment 76.
The technique of the Fractional Flow Reserve (FFR) has derived from the initial coronary physical approaches decades ago. Since then, many studies have demonstrated its effectiveness in improving the patients prognosis, by applying the appropriate approach. Its contribution in the reduction of mortality was statistically proved by the FAME (Fractional Flow Reserve Versus Angiography for Multivessel Evaluation) study 153. It is established that the FFR can be easily measured during coronary angiography by calculating the ratio of distal coronary pressure ${P}_{d}$ to aortic pressure ${P}_{a}$. These pressures are measured simultaneously with a special guidewire. FFR in a normal coronary artery equals to 1.0. FFR value of 0.80 or less identifies ischemiacausing coronary lesions with an accuracy of more than 90% 153.
Obviously, from an interventional point of view, the FFR is binding since it is invasive. It should also be noted that this technique induces additional costs, which are not covered by insurances in several countries. For these reasons, it is used only in less than 10% of the cases.
In this perspective, a new virtual version of the FFR, entitled VFFR, has emerged as an attractive and noninvasive alternative to standard FFR, see 144, 127. VFFR is based on computational modeling, mainly fluid and fluidstructural dynamics. However, there are key scientific, logistic and commercial challenges that need to be overcome before VFFR can be translated into routine clinical practice.
While most of the studies related to VFFR use NavierStokes models, we focus on the nonnewtonian case, starting with a generalized fluid flow approach. These models are more relevant for the coronary arteries, and we expect that the computation of the FFR should then be more accurate. We are also leading numerical studies to assess the impact (on the FFR) of the interaction of the physical devices (catheter, optical captors, spheroids) with the blood flow.
4.4 Other application fields
Besides the above mentioned axes, which constitute the project's identity, the methodological tools described in Section have a wider range of application. We currently carry on also the following research actions, in collaboration with external partners.

Modeling cell dynamics. Migration and proliferation of epithelial cell sheets are the two keystone aspects of the collective cell dynamics in most biological processes such as morphogenesis, embryogenesis, cancer and wound healing. It is then of utmost importance to understand their underlying mechanisms.
Semilinear reactiondiffusion equations are widely used to give a phenomenological description of the temporal and spatial changes occurring within cell populations that undergo scattering (moving), spreading (expanding cell surface) and proliferation. We have followed the same methodology and contributed to assess the validity of such approaches in different settings (cell sheets 106, dorsal closure 46, actin organization 45). However, epithelial cellsheet movement is complex enough to undermine most of the mathematical approaches based on locality, that is mainly traveling wavefrontlike partial differential equations. In 93 it is shown that MadinDarby Canine Kidney (MDCK) cells extend cryptic lamellipodia to drive the migration, several rows behind the wound edge. In 132 MDCK monolayers are shown to exhibit similar nonlocal behavior (long range velocity fields, very active borderlocalized leader cells).
Our aim is to start from a mesoscopic description of cell interaction: considering cells as independent anonymous agents, we plan to investigate the use of mathematical techniques adapted from the meanfield game theory. Otherwise, looking at them as interacting particles, we will use a multiagent approach (at least for the actin dynamics). We intend also to consider approaches stemming from compartmentbased simulation in the spirit of those developed in 90, 95, 97.

Game strategies for thermoelastography. Thermoelastography is an innovative noninvasive control technology, which has numerous advantages over other techniques, notably in medical imaging 125. Indeed, it is well known that most pathological changes are associated with changes in tissue stiffness, while remaining isoechoic, and hence difficult to detect by ultrasound techniques. Based on elastic waves and heat flux reconstruction, thermoelastography shows no destructive or aggressive medical sequel, unlike Xray and comparables techniques, making it a potentially prominent choice for patients.
Physical principles of thermoelastography originally rely on dynamical structural responses of tissues, but as a first approach, we only consider static responses of linear elastic structures.
The mathematical formulation of the thermoelasticity reconstruction is based on data completion and material identification, making it a harsh ill posed inverse problem. In previous works 107, 117, we have demonstrated that Nash game approaches are efficient to tackle illposedness. We intend to extend the results obtained for Laplace equations in 107, and the algorithms developed in Section 3.3.4 to the following problems (of increasing difficulty):
 Simultaneous data and parameter recovery in linear elasticity, using the socalled Kohn and Vogelius functional (ongoing work, some promising results obtained).
 Data recovery in coupled heatthermoelasticity systems.
 Data recovery in linear thermoelasticity under stochastic heat flux, where the imposed flux is stochastic.
 Data recovery in coupled heatthermoelasticity systems under stochastic heat flux, formulated as an incomplete information Nash game.
 Application to robust identification of cracks.

Constraint elimination in QuasiNewton methods. In singleobjective differentiable optimization, Newton's method requires the specification of both gradient and Hessian. As a result, the convergence is quadratic, and Newton's method is often considered as the target reference. However, in applications to distributed systems, the functions to be minimized are usually “functionals”, which depend on the optimization variables by the solution of an often complex set of PDE's, through a chain of computational procedures. Hence, the exact calculation of the full Hessian becomes a complex and costly computational endeavor.
This has fostered the development of quasiNewton's methods that mimic Newton's method but use only the gradient, the Hessian being iteratively constructed by successive approximations inside the algorithm itself. Among such methods, the BroydenFletcherGoldfarbShanno (BFGS) algorithm is wellknown and commonly employed. In this method, the Hessian is corrected at each new iteration by rankone matrices defined from several evaluations of the gradient only. The BFGS method has "superlinear convergence".
For constrained problems, certain authors have developed socalled Riemannian BFGS, e.g. 135, that have the desirable convergence property in constrained problems. However, in this approach, the constraints are assumed to be known formally, by explicit expressions.
In collaboration with ONERAMeudon, we are exploring the possibility of representing constraints, in successive iterations, through local approximations of the constraint surfaces, splitting the design space locally into tangent and normal subspaces, and eliminating the normal coordinates through a linearization, or more generally a finite expansion, and applying the BFGS method through dependencies on the coordinates in the tangent subspace only. Preliminary experiments on the difficult Rosenbrock testcase, although in low dimensions, demonstrate the feasibility of this approach. Ongoing research is on theorizing this method, and testing cases of higher dimensions.

Multiobjective optimization for nanotechnologies. Our team takes part in a larger collaboration with CEA/LETI (Grenoble), initiated by the Inria ProjectTeam Nachos, and related to the Maxwell equations. Our component in this activity relates to the optimization of nanophotonic devices, in particular with respect to the control of thermal loads. We have first identified a gradation of representative testcases of increasing complexity:
 infrared microsource;
 microphotoacoustic cell;
 nanophotonic device.
These cases involve from a few geometric parameters to be optimized to a functional minimization subject to a finiteelement solution involving a large number of dof's. CEA disposes of such codes, but considering the computational cost of the objective functions in the complex cases, the first part of our study is focused on the construction and validation of metamodels, typically of RBFtype. Multiobjective optimization will be carried out subsequently by MGDA, and possibly Nash games.
5 Social and environmental responsibility
5.1 Impact of research results
The research conducted with the startup Mycophyto aims at reducing the use of chemical fertilisers and phytopharmaceutical products by developing natural biostimulants (mycorrhyzal fungi). It started with the arrival of Khadija Musayeva in October 2020.
Acumes's research activity in traffic modeling and control is intended to improve road network effciency, thus reducing energy consumption and pollutant emission.
From medical viewpoint, virtual fractional flow reserve vFFR is a promising technique to support clinicians in cardiostenting with cheap social costs compared to the analogic commercial solutions. Acumes has contributed to improve the involved computational apparatus (nonlinear fluid mechanics with ad hoc boundary conditions).
The research activities related to isogeometric analysis aim at facilitating the use of shape optimization methods in engineering, yielding a gain of efficiency, for instance in transportation industry (cars, aircrafts) or energy industry (air conditioning, turbines).
6 Highlights of the year
6.1 Awards
 M. Binois: 2020 Youden Award for best expository paper appearing in the 2019 issues of Technometrics (American Quality Association).
 M. Binois: Finalist for the 2020 Gordon Bell Special Prize for High Performance ComputingBased COVID19 Research (ACM Gordon Bell prize).
 P. Goatin: ANR3IA Côte d'Azur Senior Chair for the project “Data driven traffic management” (20202024).
7 New software and platforms
7.1 New software
7.1.1 MGDA
 Name: Multiple Gradient Descent Algorithm
 Keywords: Descent direction, Multiple gradients, Multiobjective differentiable optimization, Prioritized multiobjective optimization

Scientific Description:
The software relies upon a basic MGDA tool which permits to calculate a descent direction common to an arbitrary set of cost functions whose gradients at a computational point are provided by the user, as long as a solution exists, that is, with the exclusion of a Paretostationarity situation.
More specifically, the basic software computes a vector d whose scalar product with each of the given gradients (or directional derivative) is positive. When the gradients are linearly independent, the algorithm is direct following a GramSchmidt orthogonalization. Otherwise, a subfamily of the gradients is identified according to a hierarchical criterion as a basis of the spanned subspace associated with a cone that contains almost all the gradient directions. Then, one solves a quadratic programming problem formulated in this basis.
This basic tool admits the following extensions:  constrained multiobjective optimization  prioritized multiobjective optimization  stochastic multiobjective optimization.

Functional Description:
Chapter 1: Basic MGDA tool Software to compute a descent direction common to an arbitrary set of cost functions whose gradients are provided in situations other than Pareto stationarity.
Chapter 2: Directions for solving a constrained problem Guidelines and examples are provided according the Inria research report 9007 for solving constrained problems by a quasiRiemannian approach and the basic MGDA tool.
Chapter 3: Tool for prioritized optimization Software permitting to solve a multiobjective optimization problem in which the cost functions are defined by two subsets:  a primary subset of cost functions subject to constraints for which a Pareto optimal point is provided by the user (after using the previous tool or any other multiobjective method, possibly an evolutionary algorithm)  a secondary subset of cost functions to be reduced while maintaining quasi Pareto optimality of the first set. Procedures defining the cost and constraint functions, and a small set of numerical parameters are uploaded to the platform by an external user. The site returns an archive containing datafiles of results including graphics automatically generated.
Chapter 4: Stochastic MGDA Information and bibliographic references about SMGDA, an extension of MGDA applicable to certain stochastic formulations.
Concerning Chapter 1, the utilization of the platform can be made via two modes : – the interactive mode, through a web interface that facilitates the data exchange between the user and an Inria dedicated machine, – the iterative mode, in which the user downloads the object library to be included in a personal optimization software. Concerning Chapters 2 and 3, the utilizer specifies cost and constraint functions by providing procedures compatible with Fortran 90. Chapter 3 does not require the specification of gradients, but only the functions themselves that are approximated by the software by quadratic metamodels.

URL:
http://
mgda. inria. fr  Publications: hal01139994, hal01414741, hal01417428, hal02285197, hal02285899
 Authors: JeanAntoine Désidéri, Nicolas Niclausse, Thibaud Kloczko
 Contacts: JeanAntoine Désidéri, Thibaud Kloczko
 Participant: JeanAntoine Désidéri
7.1.2 Igloo
 Name: IsoGeometric anaLysis using discOntinuOus galerkin methods
 Keywords: Numerical simulations, Isogeometric analysis
 Scientific Description: Igloo contains numerical methods to solve partial differential equations of hyperbolic type, or convectiondominant type, using an isogeometric formulation (NURBS bases) with a discontinuous Galerkin method.
 Functional Description: Simulation software for NURBS meshes

URL:
https://
gitlab. inria. fr/ igloo/ igloo/ / wikis/ home  Author: Régis Duvigneau
 Contact: Régis Duvigneau
7.1.3 BuildingSmart
 Name: BuildingSmart interactive visualization
 Keywords: Physical simulation, 3D rendering, 3D interaction
 Scientific Description: The aim of the BuildingSmart project is to develop a software environment for the simulation and interactive visualisation for the design of buildings (structural safety, thermal confort).
 Functional Description: The main task of the project is to study and develop solutions dedicated to interactive visualisation of building performances (heat, structural) in relation to the Building Information Modeling BIM framework, using Occulus Rift immersion.
 News of the Year: Demo movies are available from Youtube (see web site)

URL:
http://
youtu. be/ MW_gIF8hUdk  Contact: Abderrahmane Habbal
 Participants: Régis Duvigneau, JeanLuc Szpyrka, David Rey, Clement Welsch, Abderrahmane Habbal
8 New results
8.1 Macroscopic traffic flow models on networks
Participants: Mickaël Binois, Paola Goatin, Alexandra Würth, Antonella Ferrara, Giulia Piacentini.
Traffic control by vehicle platooning.
In 33, a coupled PDEODE model describing the interaction between the bulk traffic flow and a platoon of connected vehicles is adopted to develop a control action aiming at reducing the fuel consumption of the overall traffic flow. The platoon is modeled as a capacity restriction acting on the surrounding traffic. The trajectory of the initial and final points of the platoon are optimized by means of a model predictive control strategy, acting on the speeds of the frontend and backend of the platoon, thus resulting in controlling both the speed and the length of the platoon. The approach is assessed in simulations.
This work is part of G. Piacentini's PhD thesis.
Traffic flow model calibration by statistical approaches.
In the framework of A. Würth's internship, we have employed a Bayesian approach including a bias term to estimate first and second order model parameters, based on two traffic data sets: a set of loop detector data located on the A50 highway between Marseille and Aubagne provided by DirMED, and publicy available data from the Minnesota Department of transportation (http://
8.2 Nonlocal conservation laws
Participants: Paola Goatin, Raimund Bürger, Daniel Inzunza, Luis Miguel Villada.
In the framework of the Associated Team NOLOCO, we proposed a revised version often the nonlocal macroscopic pedestrian flow model proposed in [R. M. Colombo, M. Garavello, and M. LécureuxMercier. A class of nonlocal models for pedestrian traffic. Math. Models Methods Appl. Sci., 22(4):1150023, 2012] to account for anisotropic interactions and the presence of walls or other obstacles in the walking domain. We proved the wellposedness of this extended model and we applied highresolution numerical schemes to illustrate the model characteristics. In particular, numerical simulations highlight the role of different model parameters in the observed pattern formation. The results are published in 18.
8.3 Isogeometric Discontinuous Galerkin method for compressible flows
Participants: Régis Duvigneau, Stefano Pezzano, Maxime Stauffert.
The coexistence of different geometrical representations in the design loop (CADbased and meshbased) is a real bottleneck for the application of design optimization procedures in industry, yielding a major waste of human time to convert geometrical data. Isogeometric analysis methods, which consists in using CAD bases like NURBS in a FiniteElement framework, were proposed a decade ago to facilitate interactions between geometry and simulation domains.
We investigate the extension of such methods to Discontinuous Galerkin (DG) formulations, which are better suited to hyperbolic or convectiondominated problems. Specifically, we develop a DG method for compressible Euler and NavierStokes equations, based on rational parametric elements, that preserves exactly the geometry of boundaries defined by NURBS, while the same rational approximation space is adopted for the solution. The following research axes are considered in this context:

CADconsistent adaptive refinement
Properties of NURBS functions are used to define an adaptive refinement strategy, which refines locally the discretization according to an error indicator, while describing exactly CAD geometries whatever the refinement level. The resulting approach exhibits an optimal convergence rate and capture efficiently local flow features, like shocks or vortices, avoiding refinement due to geometry approximation 24.

Arbitrary EulerianLagrangian formulation for highorder meshes
To enable the simulation of flows around moving bodies, an Arbitrary EulerianLagrangian (ALE) formulation is proposed in the context of the isogeometric DG method. It relies on a NURBSbased grid velocity field, integrated along time over moving NURBS elements. The gain of using exactgeometry representations is clearly quantified, in terms of accuracy and computational efficiency 31, 36.

Geometrically exact sliding interfaces
In the context of rotating machines (compressors, turbines, etc), computations are achieved using a rotating inner grid interfaced to an outer fixed grid. This coupling is combersome using classical piecewiselinear grids due to a lack of common geometrical interface. Thus, we develop a method based on a geometrically exact sliding interface using NURBS elements, ensuring a fully conservative scheme.

Isogeometric shape optimization
We develop an optimization procedure with shape sensitivity analysis, entirely based on NURBS representations 41. The mesh, the shape to optimize, as well as the flow solutions are represented by NURBS, which avoids any geometrical conversion and allows to exploit NURBS properties regarding regularity or hierarchy.
8.4 Sensitivity analysis for compressible flows
Participants: Régis Duvigneau, Maxime Stauffert, Camilla Fiorini, Christophe Chalons.
The adjoint equation method, classically employed in design optimization to compute functional gradients, is not well suited to complex unsteady problems, because of the necessity to solve it backward in time. Therefore, we investigate the use of the sensitivity equation method, which is integrated forward in time, in the context of compressible flows. More specifically, the following research axes are considered:

Sensitivity analysis in presence of shocks
While the sensitivity equation method is a common approach for parabolic systems, its use for hyperbolic ones is still tedious, because of the generation of discontinuities in the state solution, yielding Dirac distributions in the sensitivity solution. To overcome this difficulty, we investigate a modified sensitivity equation, that includes an additional source term when the state solution exhibits discontinuities, to avoid the generation of deltapeaks in the sensitivity solution. We consider as typical example the 1D compressible Euler equations. Different approaches are tested to integrate the additional source term: a Roe solver, a Godunov method and a moving cells approach. Applications to uncertainty quantification in presence of shocks are demonstrated and compared to the classical MonteCarlo method 26. This study is achieved in collaboration with C. Chalons and C. Fiorini from University of Versailles.

Shape sensitivity analysis
When shape parameters are considered, the evaluation of flow sensitivities is more difficult, because equations include an additional term, involving flow gradient, due to the fact that the parameter affects the boundary condition location. To overcome this difficulty, we propose to solve sensitivity equations using an isogeometric Discontinuous Galerkin (DG) method, which allows to estimate accurately flow gradients at boundary and consider boundary control points as shape parameters. First results obtained for 2D compressible Euler equations exhibit a suboptimal convergence rate, as expected, but a better accuracy with respect to a classical DG method 41.
8.5 Advanced Bayesian optimization
Participants: Mickaël Binois, Régis Duvigneau, Abderrahmane Habbal, Mahmoud Elsawy, Frédéric Hauville, Olivier Lemaitre, Stéphane Lanteri, Victor Picheny, Matthieu Sacher.
Multifidelity Bayesian optimization
The objective of multifidelity optimization strategies is to account for a set of models of different accuracies and costs to accelerate the optimization procedure. In the context of Bayesian optimization, we develop such a multifidelity approach based on nonnested evaluations: each time a new evaluation is required, the algorithm selects a new design point associated to a fidelity level to maximize the expected improvement on the finest modeling level. The proposed approach is applied to the fluidstructure optimization of a sailing boat, which is described by five modeling levels. A significant acceleration of the optimization procedure is reported, without loss of accuracy 35.
The KalaiSmorodinski solution for manyobjective Bayesian optimization
Extending the short paper 16 on the use of the gametheoretic KalaiSmorodinski solution in Bayesian optimization, we have refined the definition of solutions, discussed underlying assumptions, and shown empirically the improved performance of our proposed approach over naive heuristics. A realistic hyperparameter tuning problem with eight objectives as well as an expensive calibration problem with nine objectives have been considered as well.
In parallel, we have substantially improved the efficiency of the implementation, enabled specific treatment of calibration problems as well as handling noise in the GPGame package https://
Bayesian optimization of nanophotonic devices
In collaboration with Atlantis ProjectTeam, we consider the optimization of optical metasurface devices, which are able to alter light properties by operating at nanoscale. In the contexte of Maxwell equations, modified to account for nanoscale phenomena, the geometrical properties of materials are optimized to achieve a desired electromagnetic wave response, such as change of polarization, intensity or direction. This task is especially challenging due to the computational cost related to the 3D timeaccurate simulations, the difficulty to handle the different geometrical scales in optimization and the presence of uncertainties.
First studies achieved using Bayesian optimization algorithms, demonstrate the potentiality of the proposed approach 25, 37.
8.6 Gaussian process based sequential design
Participants: Mickaël Binois, Robert Gramacy, Michael Ludkovski, Xiong Lyu, Stefan Wild, Nathan Wycoff.
Besides Bayesian optimization as above, Gaussian processes are useful for a variety of other related tasks. Here we first present a tutorial on the subject of modeling with input dependent noise with an implementation in the hetGP R package. Then the estimation of levelset in for noisy simulators with complex input noise is studied, before treating sequential design for efficient dimension reduction.
Heteroskedastic Gaussian process modeling and sequential design
An increasing number of timeconsuming simulators exhibit a complex noise structure that depends on the inputs. For conducting studies with limited budgets of evaluations, new surrogate methods are required in order to simultaneously model the mean and variance fields. To this end, in 55 we present the hetGP package https://
Sequential learning of active subspace
Continuing a work started at Argonne National Laboratory, in 149 we consider the combination of Gaussian process regression modeling with the active subspace methods (ASMs), which have become a popular means of performing subspace sensitivity analysis on blackbox functions. Naively applied, however, ASMs require gradient evaluations of the target function. In the event of noisy, expensive, or stochastic simulators, evaluating gradients via finite differencing may be infeasible. In such cases, often a surrogate model is employed, on which finite differencing is performed. When the surrogate model is a Gaussian process, we show that the ASM estimator is available in closed form, rendering the finitedifference approximation unnecessary. We use our closedform solution to develop acquisition functions focused on sequential learning tailored to sensitivity analysis on top of ASMs. We also show that the traditional ASM estimator may be viewed as a method of moments estimator for a certain class of Gaussian processes. We demonstrate how uncertainty on Gaussian process hyperparameters may be propagated to uncertainty on the sensitivity analysis, allowing modelbased confidence intervals on the active subspace. Our methodological developments are illustrated on several examples.
Evaluating Gaussian Process Metamodels and Sequential Designs for Noisy Level Set Estimation
We consider the problem of learning the level set for which a noisy blackbox function exceeds a given threshold. To efficiently reconstruct the level set, we investigate Gaussian process (GP) metamodels. Our focus in 40 is on strongly stochastic samplers, in particular with heavytailed simulation noise and low signaltonoise ratio. To guard against noise misspecification, we assess the performance of three variants: (i) GPs with Studentt observations; (ii) Studentt processes (TPs); and (iii) classification GPs modeling the sign of the response. In conjunction with these metamodels, we analyze several acquisition functions for guiding the sequential experimental designs, extending existing stepwise uncertainty reduction criteria to the stochastic contourfinding context. This also motivates our development of (approximate) updating formulas to efficiently compute such acquisition functions. Our schemes are benchmarked by using a variety of synthetic experiments in 1–6 dimensions. We also consider an application of level set estimation for determining the optimal exercise policy of Bermudan options in finance.
8.7 Prioritized multiobjective optimization of a sandwich panel
Participants: JeanAntoine Désidéri, Pierre Leite, Quentin Mercier.
The potential of the prioritized multiobjective optimization approach has been demonstrated by applying it to the sizing of a sandwich panel with respect to mechanical criteria: mass, critical failure forces under bending load (1st and 2nd modes), and blast mitigation measured by the core energy absorption, deflection, or both. Several objective functions are defined by analytical models. Four numerical testcases were documented.
In each testcase, in a first phase of optimization, the Primary Pareto Front associated with two criteria only, considered preponderant, was first established. Mass was always retained in these primary objective functions. In a second phase, a starting point on the Primary Pareto Front is selected and the corresponding design is improved with respect to secondary criteria by the construction of a continuum of Nash equilibria tangent to the front in function space at the starting point, thus only marginally degrading the primary criteria.
The second phase of optimization can be viewed as a form of adaptation of the optimization process. Here it permits to account for four criteria, at a more economical and simpler strategy than it would be to evaluate and analyze a complete Pareto front in a fourdimensional function space. 39
(All the numerical experiments were realized using the software platform
http://
8.8 Inverse CauchyStokes problems solved as Nash games
Participants: Abderrahmane Habbal, Marwa Ouni, Moez Kallel.
We extend in two directions our results published in 108 to tackle ill posed CauchyStokes inverse problems as Nash games. First, we consider the problem of detecting unknown pointwise sources in a stationary viscous fluid, using partial boundary measurements. The considered fluid obeys a steady Stokes regime, the boundary measurements are a single compatible pair of Dirichlet and Neumann data, available only on a partial accessible part of the whole boundary. This inverse source identification for the CauchyStokes problem is illposed for both the sources and missing data reconstructions, and designing stable and efficient algorithms is challenging. We reformulate the problem as a threeplayer Nash game. Thanks to a source identifiability result derived for the CauchyStokes problem, it is enough to set up two Stokes BVP, then use them as state equations. The Nash game is then set between 3 players, the two first targeting the data completion while the third one targets the detection of the number, location and magnitude of the unknown sources. We provided the third player with the location and magnitude parameters as strategy, with a cost functional of KohnVogelius type. In particular, the location is obtained through the computation of the topological sensitivity of the latter function. We propose an original algorithm, which we implemented using Freefem++. We present 2D numerical experiments for many different testcases.The obtained results corroborate the efficiency of our 3player Nash game approach to solve parameter or shape identification for CauchyStokes problems.
The second direction is dedicated to the solution of the data completion problem for nonlinear flows. We consider two kinds of non linearities leading to either a non newtonian Stokes flow or to NavierStokes equations. Our recent numerical results show that it is possible to perform a oneshot approach using Nash games : players exchange their respective state information and solve linear systems. At convergence to a Nash equilibrium, the states converge to the solution of the non linear systems. To the best of our knowledge, this is the first time such an approach is applied to solve Inverse problems for nonlinear systems.
8.9 Virtual FFR quantified with a generalized flow model using Windkessel boundary conditions ; Application to a patientspecific coronary tree
Participants: Abderrahmane Habbal, Keltoum Chahour, Rajae Aboulaich, Nejib Zemzemi, Mickaël Binois.
Fractional flow reserve (FFR) has proved its efficiency in improving patients diagnosis. From both economical and clinical viewpoints, a realistic simulation of vascular blood flow inside the coronary arteries could be a better alternative to the invasive FFR. In this view, we consider a 2D reconstructed left coronary tree with two artificial lesions of different degrees. We use a generalized fluid model with a Carreau law and use a coupled multidomain method to implement Windkessel boundary conditions at the outlets. We introduce our methodology to quantify the virtual FFR, and lead several numerical experiments. We compare FFR results from Navier Stokes versus generalized flow model, and for Windkessel versus traction free outlets boundary conditions or mixed outlets boundary conditions. We also investigate some sources of uncertainty that the FFR index might encounter during the invasive procedure, in particular the arbitrary position of the distal sensor. The computational FFR results show that the degree of stenosis is not enough to classify a lesion, while there is a good agreement between Navier Stokes and the non Newtonian flow model adopted in classifying coronary lesions. Furthermore, we highlight that the lack of standardization while making FFR measurement might be misleading regarding the significance of stenosis 19.
9 Bilateral contracts and grants with industry
9.1 Bilateral contracts with industry
 Etic Data (20192020): Acumes has set up a 12 months research and development contract with the company Etic Data on "Predictive modeling and proactive driving of customers behaviour in massive data BtoC context".
 Mycophyto (2020...): this research contract involving Université Côte d'Azur is financing the postdoctoral contract of Khadija Musayeva.
10 Partnerships and cooperations
10.1 International initiatives
10.1.1 Inria International Labs
 Acronym: NOLOCO (Inria Chile)
 Title: Efficient numerical schemes for nonlocal transport phenomena
 Duration: 2018  2020
 Coordinator: Paola Goatin

Partners:
 Department of Mathematics, Universidad del BioBio (Chile): Prof. Luis Miguel Villada Osorio
 Center for Research in Mathematical Engineering (CI2MA), Universidad de Concepcion (Chile): Prof. Raimund Burger
 Laboratoire de Mathématiques Université de Versailles St. Quentin (France): Prof. Christophe Chalons
 Inria contact: Paola Goatin

Summary:
This project tackles theoretical and numerical issues arising in the mathematical study of conservation laws with nonlocal flux functions. These equations include in a variety of applications, ranging from traffic flows to industrial processes and biology, and are intended to model macroscopically the action of nonlocal interactions occurring at the microscopic level.
The team, bilocated in France and Chile, has complementary skills covering the analysis, numerical approximation and optimization of nonlinear hyperbolic systems of conservation laws, and their application to the modeling of vehicular and pedestrian traffic flows, sedimentation and other industrial problems.
Based on the members' expertise and on the preliminary results obtained by the team, the project will focus on the following aspects:  The development of efficient, highorder finite volume numerical schemes for the computation of approximate solutions of nonlocal equations.  The sensitivity analysis of the solutions on model parameters or initial conditions
The impact of the project is therefore twofold: while addressing major mathematical advances in the theory and numerical approximation of highly nonstandard problems, it puts the basis for innovative tools to handle modern applications in engineering sciences.
10.1.2 Inria international partners
Declared Inria international partners
 Acronym: ORESTE
 Title: Optimal REroute Strategies for Traffic managEment
 Duration: 2018  2022
 Coordinator: Paola Goatin

Partners:
 University of California Berkeley (United States)  Electrical Engineering and Computer Science (EECS) (EECS)  Prof. Alexandre M. Bayen
 Inria contact: Paola Goatin

Summary:
The rapidly changing transportation ecosystem opens new challenges in traffic modeling and optimization approaches. We will focus in particular on the two following aspects:
Route choice apps impact. The vast use of personal route choice systems through phone applications or other devices is modifying the traditional flow of networks, requiring new models for accounting of the guidance impact. Indeed, routing apps have changed traffic patterns in the US and Europe, leading to new congestion patterns where previously no traffic was observed. Over the last decade, GPS enabled smart phones and connected personal navigation devices have disrupted the mobility landscape. Initially, the availability of traffic information led to better guidance of a small portion of motorists in the system. But as the majority of the driving public started to use apps, the systematic broadcasting of “selfish” best routes led to the worsening of traffic in numerous places, ultimately leading to the first lawsuit against one specific company in particular (Waze) accused to be the cause of these problems. This is just the beginning of an evolution, which, if not controlled and regulated, will progressively asphyxiate urban landscapes (already nearly hundreds of occurrences of this phenomenon are noticed by the popular media, which indicates the presence of probably thousands of such issues in the US alone). Traffic managers are typically not equipped to fix these problems, and typically do not fund this research, as in order to be able to regulate and fix the problem, fundamental science needs to be advanced, modeling and game theory in particular, so remediation can happen (for which the traffic managers are equipped). In this project, we will mainly focus on the development and study of new macroscopic dynamical models to describe the aforementioned phenomena, and we will explore control strategies to mitigate their impact.
Autonomous vehicles. Besides, the foreseen deployment of connected and autonomous vehicles (CAVs) opens new perspectives both in traffic modeling and control. Indeed, CAVs are expected to modify the classical macroscopic traffic dynamics due to their peculiar motion laws, which are more uniform than human drivers’ and follow different rules. Besides, due to their extended information on neighboring traffic conditions, the resulting dynamics would have a nonlocal character, justifying the use of rapidly developing nonlocal models. In any case, the different behavior of autonomous vehicles requires the design of new multiclass models capable of accounting for different vehicle classes characteristics and mutual interactions. Moreover, CAVs could be used as endogenous variable speed limiters, thus providing new action points to control traffic flow. Preliminary results show that the presence of few controlled vehicles can positively affect traffic conditions. In this setting, the interaction of AVs with the surrounding traffic can be described by strongly coupled PDEODE systems, which have been largely studied by the ACUMES team. Yet, the study of CAVs impact in realistic situations requires further research, in particular towards model validation, for which the Berkeley team will provide the necessary data.
Informal international partners
 University of Milano Bicocca, Mathematics and Applications (M. Garavello: https://
sites. )google. com/ site/ maurogaravello/  University of Rutgers  Camden, Department of Mathematical Science (B. Piccoli: https://
piccoli. )camden. rutgers. edu/  University of Texas Arlington (S. Roy, https://
mentis. )uta. edu/ explore/ profile/ souvikroy
10.1.3 Participation in other international programs
PHC Procope
 Program: Program Hubert Curien Procope (Germany)
 Title: Nonlocal conservation laws for engineering applications
 Duration: January 2019  December 2020
 Coordinator: P. Goatin and S. Göttlich (Univ. Mannheim)

Partners:
 University of Mannheim (Germany)
 Inria contact: Paola Goatin
 Summary: This project tackles theoretical and numerical issues arising in the mathematical study of conservation laws with nonlocal flux functions. These equations appear in a variety of applications, ranging from traffic flows to industrial processes and biology, and are intended to model macroscopically the action of nonlocal interactions occurring at the microscopic level. The team, bilocated in France and Germany, has complementary skills covering the analysis, numerical approximation and optimization of nonlinear hyperbolic systems of conservation laws, and their application to the modeling of vehicular and pedestrian traffic flows, manufacturing systems and other industrial problems. Based on the members expertise and on the preliminary results obtained by both teams, the project will focus on the following interconnected aspects: The treatment of boundary conditions, both from the analytical and the numerical point of views, in order to provide a sound basis to address specific problems arising in the applications. The development of efficient, highorder finite volume numerical schemes for the computation of approximate solutions of nonlocal equations. The investigation of optimal control problems with corresponding optimality systems and the design of appropriate and adaptive optimization algorithms. Targeted applications include vehicular traffic (mainly in connection with vehicletovehicle communication and consumption/pollution estimation), crowd motion (in connection with safe building evacuation procedures), and manufacturing systems (intelligent production). The impact of the project is therefore twofold: while addressing major mathematical advances in the theory and numerical approximation of highly nonstandard problems, it puts the basis for innovative tools to handle modern applications in engineering sciences.
PHC Utique
 Program: Program Hubert Curien PHC Utique (Tunisia)
 Project acronym: NAMReD
 Project title: Novel Algorithms and Models for Data Reconstruction
 Duration: January 2018  December 2020
 Coordinator: A. Habbal and M. Kallel (Univ. Tunis al Manar)
 Summary: The project goal is the design of new and efficient algorithms tailored for data reconstruction involving illposed problems. We rely on an original use of game theory and pKirchhoff methods. We apply these approaches for missing data recovery and image restoration.
10.2 European initiatives
10.2.1 Collaborations in European programs, except FP7 and H2020
Program: COST
 Project acronym: CA18232
 Project title: Mathematical models for interacting dynamics on networks
 Duration: October 2019  September 2023
 Coordinator: University of Ljubljana (Prof. Marjeta Kramar Fijavz)

Partners:
see https://
www. cost. eu/ actions/ CA18232/ #tabsName:parties  Inria contact: Paola Goatin

Summary:
Many physical, biological, chemical, financial or even social phenomena can be described by dynamical systems. It is quite common that the dynamics arises as a compound effect of the interaction between subsystems in which case we speak about coupled systems. This Action shall study such interactions in particular cases from three points of view:
 the abstract approach to the theory behind these systems,
 applications of the abstract theory to coupled structures like networks, neighbouring domains divided by permeable membranes, possibly nonhomogeneous simplicial complexes, etc.,
 modelling reallife situations within this framework.
The purpose of this Action is to bring together leading groups in Europe working on a range of issues connected with modelling and analysing mathematical models for dynamical systems on networks. It aims to develop a semigroup approach to various (non)linear dynamical systems on networks as well as numerical methods based on modern variational methods and applying them to road traffic, biological systems, and further reallife models. The Action also explores the possibility of estimating solutions and long time behaviour of these systems by collecting basic combinatorial information about underlying networks.
10.3 National initiatives
10.3.1 ANR

Project OPERA (20192021): Adaptive planar optics
This project is composed of Inria teams ATLANTIS, ACUMES and HIEPACS, CNRS CRHEA lab. and company NAPA. Its objective is the characterization and design of new metasurfaces for optics (opera web site).
11 Dissemination
11.1 Promoting scientific activities
11.1.1 Scientific events: organisation
General chair, scientific chair
 P. Goatin is member of the scientific committee of the annual seminar CEAGAMNI “Numerical fluidmechanics”.
 A. Habbal was member of the scientific committee of the CARI 2020 Colloque Africain sur la Recherche en Informatique et Mathématiques Appliquées Thies, Senegal, October 2020.
Member of the organizing committees
 P. Goatin was member of the organizing committee of the IPAM (UCLA) long program on “Mathematical Challenges and Opportunities for Autonomous Vehicles”, Los Angeles (USA), fall 2020
 P. Goatin was member of the organizing committee of the International online program “Fifty Years of Kruzhkov Entropy Solutions, and Beyond”, fall 2020.
11.1.2 Scientific events: selection
Reviewer
 M. Binois reviewed for the following conferences: AISTATS 2020, ICLR 2021, ICML 2020, IJCAIPRICAI 2020 and NeurIPS 2020.
 R. Duvigneau reviewed for the AIAA Aviation Forum 2021.
 P. Goatin was Associate Editor for the 23rd Intelligent Transportation Systems Conference (IEEE ITSC 2020).
 A. Habbal reviewed for the CARI 2020 conference
11.1.3 Journal
Member of the editorial boards
 P. Goatin is Managing Editor of Networks and Heterogeneous Media.
 P. Goatin is Associate Editor of SIAM Journal on Applied Mathematics.
Reviewer  reviewing activities
 M. Binois is a reviewer for the following international journals: Annals of Applied Statistics, Journal of Applied Statistics, Computers & Operations Research, Information and Inference, Journal of Mechanical Engineering Science (Part. C), Journal of Uncertainty Quantification, Mathematical Programming, Optimization and Engineering, Structural and Multidisciplinary Optimization, and Technometrics.
 R. Duvigneau is reviewer for the following international journals: Computers and Fluids, International Journal for Numerical Methods in Fluids, Journal of Fluid and Structures, Computer Methods for Applied Mechanics Engineering, Computer Aided Geometric Design, Applied Mathematics and Mechanics, Engineering Optimization, Ocean Engineering.
 P. Goatin reviewed for the following international journals: Annales de l'Institut Henri Poincaré / Analyse non lineaire, IEEE Transactions on Automatic Control, Vietnam Journal of Mathematics.
 A. Habbal was reviewer for the following international journals : Journal of Scientific Computing, Journal of Dynamical and Control Systems, Revue Africaine de la Recherche en Informatique et Mathématiques Appliquées, Systems and Control Letters, AMS Mathematical Reviews, Numerische Mathematics
11.1.4 Invited talks
 M. Binois: Université Mohammed VI Polytechnique, Ben Guérir (Morocco), January 2020. Invited talk: Scaling up multiobjective Bayesian optimization.
 M. Binois: Dagstuhl Seminar 20031  Scalability in Multiobjective Optimization, Dagstuhl (Germany), January 2020. Talk: Scaling up multiobjective Bayesian optimization.

P. Goatin: IPAM (UCLA) long program on “Mathematical Challenges and Opportunities for Autonomous Vehicles” (November 2020, online)
Workshop “Large Scale Autonomy: Connectivity and Mobility Networks”.
Talk: A multipopulation traffic flow model on networks accounting for vehicle automation.

P. Goatin: SophI.A Summit (November 2020, online)
Invited talk: “Data driven traffic management”.

P. Goatin: Thematic trimester “Fifty Years of Kruzhkov Entropy Solutions, and Beyond” (November 2020, online)
Talk: Entropy conditions in traffic flow applications.

P. Goatin: IPAM (UCLA) long program on “Mathematical Challenges and Opportunities for Autonomous Vehicles” (September 2020, online)
Tutorial lecture: Macroscopic models for Autonomous Vehicles.
11.1.5 Scientific expertise
 P. Goatin is member of the advisory board of DISMA Excellence Project of Politecnico di Torino (20182022).
 P. Goatin was proposal reviewer for FONDECYT (Chile).
 A. Habbal is member of and reviewer for the CNRS MODCOV Project https://
modcov19. math. cnrs. fr/
11.1.6 Research administration
 R. Duvigneau is head of the Scientific Committee of Platforms (cluster and immersive space) at Inria Sophia Antipolis Méditerranée.
 R. Duvigneau is member of the Scientific Committee of OPAL computing Platform at Université Côte d'Azur.
 P. Goatin is member of the board of the Doctoral School of Fundamental and Applied Sciences (ED SFA) of Université Côte D’Azur.
 P. Goatin is member of the GAMNISMAI committee.
 P. Goatin was member of the Full Professor hiring committee of Université de FrancheComté in Applied Mathematics (PR).
 A. Habbal is founding member of the African scholarly Society on Digital Sciences (ASDS)
11.2 Teaching  Supervision  Juries
11.2.1 Teaching
 Master: M. Binois, Optimisation bayésienne, 6 hrs, M2, Polytech Nice Sophia  Université Côte d'Azur.
 Master: M. Binois, Optimization, 24 hrs, M1, Polytech Nice Sophia  Université Côte d'Azur.
 Master: M. Binois, Bayesian optimization, 9 hrs, M2, Mohammed VI Polytechnic University, Morocco.
 Master: R. Duvigneau, Advanced Optimization, 28 hrs, M2, Polytech Nice Sophia  Université Côte d'Azur.
 Master: A. Habbal, Numerical Methods for Partial Differential Equations, 66 hrs, M1, Polytech Nice Sophia  Université Côte d'Azur.
 Master: J.A. Désidéri, Multidisciplinary Optimization, 22.5 hrs, joint Institut Supérieur de l'Aéronautique et de l'Espace (ISAE Supaéro, "Complex Systems") and M2 (Mathematics), Toulouse.
 Master: A. Habbal, Optimization, 66 hrs, M1, Polytech Nice Sophia  Université Côte d'Azur.
 Master: A. Habbal, Stochastic Processes, 24 hrs, M1, Polytech Nice Sophia  Université Côte d'Azur.
 Master: A. Habbal, Combinatorial optimization, 15 hrs, M1, Mohammed VI Polytechnic University, Morocco.
 Licence (L3): A. Habbal, Implement and Experiment PSO, 48 hrs, L3 Semester Project, Polytech Nice Sophia  Université Côte d'Azur.
11.2.2 Supervision
 PhD in progress: S. Pezzano, Isogeometric analysis with moving grids, Univ. Côte d'Azur. Supervisor: R. Duvigneau.
 PhD in progress: S. Chabbar, Modeling and simulation of tumor growth ; the case of prostate cancer, Jan 2019, Supervisors: A. Habbal, Rajae Aboulaich (LERMA, EMI, Rabat), A. Ratnani (UM6P, Benguerir, Morocco).
 PhD in progress: Marwa Ouni, Solving inverses problems in fluid mechanics with game strategies, October 2016, Supervisors: A. Habbal, Moez Kallel (LAMSIN, ENIT, Tunis).
11.2.3 Juries
 J.A. Désidéri was reviewer of O. Montonen's Ph.D. thesis "On Multiobjective Optimization from the Nonsmooth Perspective", University of Turku (Finland).
 J.A. Désidéri was reviewer of N. Garland's Ph.D. thesis "Introspective metamodelling for the analysis of simulated physical phenomena", École Nationale Supérieure des Mines (SaintÉtienne).
 R. Duvigneau was reviewer of Mickaël Rivier's PhD thesis “Lowcost methods for constrained multiobjective optimization under uncertainty”, Ecole Polytechnique, September 30th, 2020.
 P. Goatin was member of the committee of S. Mollier's PhD thesis “Twodimensional models for largescale traffic networks”, Université de Grenoble, February 13th, 2020.
 P. Goatin was referee of M. Menci's PhD thesis “Analytical foundations of a class of hybrid models with applications to collective dynamics”, Università Campo BioMedico di Roma, March 2020.
 P. Goatin was external member of the committee of N. Salehi's PhD thesis “Realistic onramp coupling conditions for macroscopic highway network models”, Temple University, May 28th, 2020.
 P. Goatin was president of the committee of B. Guelmame's PhD thesis “On a Hamiltonian regularization and regularity of entropy solutions of some nonlinear hyperbolic equations”, Univ. Côte d'Azur, September 23rd, 2020.
11.3 Popularization
11.3.1 Articles and contents
 M.L. Delle Monache and P. Goatin, Traffic management via Autonomous Vehicles, ECCOMAS Newsletter, July 2020, 1015.
12 Scientific production
12.1 Major publications
 1 article'Nonlocal systems of conservation laws in several space dimensions'.SIAM Journal on Numerical Analysis5222015, 963983
 2 article'Finite volume schemes for locally constrained conservation laws'.Numer. Math.1154With supplementary material available online2010, 609645
 3 article 'Wellposedness of a conservation law with nonlocal flux arising in traffic flow modeling'. Numerische Mathematik 2015
 4 article'A well posed conservation law with a variable unilateral constraint'.J. Differential Equations23422007, 654675
 5 article'Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result'.J. Differential Equations257112014, 40154029
 6 article'A PDEODE model for a junction with ramp buffer'.SIAM J. Appl. Math.7412014, 2239
 7 article'Krigingbased optimization applied to flow control'.Int. J. for Numerical Methods in Fluids69112012, 17011714
 8 article'NeumannDirichlet Nash strategies for the solution of elliptic Cauchy problems'.SIAM J. Control Optim.5152013, 40664083
 9 article'A Nashgame approach to joint image restoration and segmentation'.Appl. Math. Model.3811122014, 30383053URL: http://dx.doi.org/10.1016/j.apm.2013.11.034
 10 article 'On the use of secondorder derivative and metamodelbased MonteCarlo for uncertainty estimation in aerodynamics'. Computers and Fluids 37 6 2010
 11 article'Pedestrian motion modelled by FokkerPlanck Nash games'.Royal Society open science492017, 170648
 12 article'Macroscopic modeling and simulations of room evacuation'.Appl. Math. Model.38242014, 57815795
 13 article 'Constructing analysissuitable parameterization of computational domain from CAD boundary by variational harmonic method'. J. Comput. Physics 252 November 2013
 14 article'FisherKPP with time dependent diffusion is able to model cellsheet activated and inhibited wound closure'.Mathematical biosciences2922017, 3645
12.2 Publications of the year
International journals
 15 article'Uncertainty quantification in a macroscopic traffic flow model calibrated on GPS data'.Mathematical Biosciences and Engineering1722020, 15111533
 16 article'The KalaiSmorodinski solution for manyobjective Bayesian optimization'.Journal of Machine Learning Research211502020, 142
 17 article'Multiscale population dynamics in reproductive biology: singular perturbation reduction in deterministic and stochastic models'.ESAIM: Proceedings and Surveys672020, 7299
 18 article'A nonlocal pedestrian flow model accounting for anisotropic interactions and domain boundaries'.Mathematical Biosciences and Engineering1752020, 58835906
 19 article 'Virtual FFR quantified with a generalized flow model using Windkessel boundary conditions ; Application to a patientspecific coronary tree.'. Computational and Mathematical Methods in Medicine 2020
 20 article'A nonlocal traffic flow model for 1to1 junctions'.European Journal of Applied Mathematics3162020, 10291049
 21 article 'MicroMacro limit of a nonlocal generalized AwRascle type model'. SIAM Journal on Applied Mathematics 2020
 22 article 'LagrangianAntidiffusive Remap schemes for nonlocal multiclass traffic flow models'. Computational and Applied Mathematics 39 60 2020
 23 article 'A threephase fundamental diagram from threedimensional traffic data'. Axioms 10 1 2021
 24 article 'CADconsistent adaptive refinement using a NURBSbased Discontinuous Galerkin method'. International Journal for Numerical Methods in Fluids 92 9 September 2020
 25 article'Numerical optimization methods for metasurfaces'.Laser and Photonics Reviews1410October 2020, 1900445
 26 article 'A modified sensitivity equation method for the Euler equations in presence of shocks'. Numerical Methods for Partial Differential Equations 36 4 July 2020
 27 article'A multiscale model for traffic regulation via autonomous vehicles'.Journal of Differential Equations26972020, 60886124
 28 article'Comparative study of macroscopic traffic flow models at road junctions'.Networks and Heterogeneous Media1522020, 261279
 29 article 'A macroscopic traffic flow model accounting for bounded acceleration'. SIAM Journal on Applied Mathematics 2020
 30 article'A macroscopic traffic flow model with finite buffers on networks: Wellposedness by means of HamiltonJacobi equations'.Communications in Mathematical Sciences1862020, 15691604
 31 article 'A NURBSbased Discontinuous Galerkin method for conservation laws with highorder moving meshes'. Journal of Computational Physics January 2021
 32 article'A macroscopic model for platooning in highway traffic'.SIAM Journal on Applied Mathematics8012020, 639656
 33 article 'Traffic Control via Platoons of Intelligent Vehicles for Saving Fuel Consumption in Freeway Systems'. IEEE Control Systems Letters 2020
 34 article'Wellposedness of a nonlocal model for material flow on conveyor belts'.ESAIM: Mathematical Modelling and Numerical Analysis5422020, 679704
 35 article'A NonNested Infilling Strategy for MultiFidelity based Efficient Global Optimization'.International Journal for Uncertainty Quantification111January 2021, 130
International peerreviewed conferences
 36 inproceedings 'A NURBSbased Discontinuous Galerkin Framework for Compressible Aerodynamics'. AIAA Aviation 2020 Forum Proceedings of AIAA Aviation 2020 Forum Reno, United States June 2020
Conferences without proceedings
 37 inproceedings 'Statistical Learning Optimization for Highly Efficient Metasurface Designs'. SIAM Conference on Computational Science and Engineering 2021 Texas, United States March 2021
 38 inproceedings 'Determination of pointforces via extended boundary measurements using a game strategy approach'. CARI 2020  Colloque Africain sur la Recherche en Informatique et Mathématiques Appliquées CARI 2020  Colloque Africain sur la Recherche en Informatique et en Mathématiques Appliquées Thiès, Senegal October 2020
Reports & preprints
 39 report 'Prioritized multiobjective optimization of a sandwich panel'. INRIA Sophia Antipolis  Méditerranée (France) September 2020
 40 misc 'Evaluating Gaussian Process Metamodels and Sequential Designs for Noisy Level Set Estimation'. March 2020
 41 misc 'Shape sensitivity analysis in aerodynamics using an isogeometric Discontinuous Galerkin method'. September 2020
12.3 Cited publications
 42 article 'A semiintrusive deterministic approach to uncertainty quantification in nonlinear fluid flow problems'. J. Comput. Physics 2012
 43 article'Nonlocal systems of conservation laws in several space dimensions'.SIAM Journal on Numerical Analysis5222015, 963983
 44 article'Examples of instability in inverse boundaryvalue problems'.Inverse Problems1341997, 887897URL: http://dx.doi.org/10.1088/02665611/13/4/001
 45 article'Modeling actin cable contraction'.Comput. Math. Appl.6432012, 310321URL: http://dx.doi.org/10.1016/j.camwa.2012.02.041
 46 article'A Mathematical Model for Dorsal Closure'.Journal of Theoretical Biology2681January 2011, 105119URL: http://hal.inria.fr/inria00544350/en
 47 article'An integrodifferential conservation law arising in a model of granular flow'.J. Hyperbolic Differ. Equ.912012, 105131
 48 article'On the Numerical Integration of Scalar Nonlocal Conservation Laws'.ESAIM M2AN4912015, 1937
 49 article'On a nonlocal hyperbolic conservation law arising from a gradient constraint problem'.Bull. Braz. Math. Soc. (N.S.)4342012, 599614
 50 article'A FokkerPlanck control framework for multidimensional stochastic processes'.Journal of Computational and Applied Mathematics2372013, 487507
 51 article'Time accurate anisotropic goaloriented mesh adaptation for unsteady flows'.J. Comput. Physics231192012, 63236348
 52 article'Measure valued solutions to conservation laws motivated by traffic modelling'.Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.46220702006, 17911803
 53 unpublished 'Uncertainties in traffic flow and model validation on GPS data'. 2015
 54 article'On nonlocal conservation laws modelling sedimentation'.Nonlinearity2432011, 855885
 55 unpublished'hetGP: Heteroskedastic Gaussian Process Modeling and Sequential Design in R'.December 2019, working paper or preprint
 56 article'Wellposedness of a conservation law with nonlocal flux arising in traffic flow modeling'.Numer. Math.13222016, 217241URL: https://doi.org/10.1007/s0021101507176

57
article'A
Sensitivity Equation Method for Optimal Aerodynamic Design'.Journal of Computational Physics13621997, 366384URL: http://www.sciencedirect.com/science/article/pii/S0021999197957430  58 article'Anisotropic Organised Eddy Simulation for the prediction of nonequilibrium turbulent flows around bodies'.J. of Fluids and Structures2482008, 12401251
 59 article'Flows on networks: recent results and perspectives'.EMS Surv. Math. Sci.112014, 47111
 60 article'Mean field games with nonlinear mobilities in pedestrian dynamics'.Discrete Contin. Dyn. Syst. Ser. B1952014, 13111333
 61 article'Individual based and meanfield modelling of direct aggregation'.Physica D2602013, 145158
 62 techreport 'Validation of traffic flow models on processed GPS data'. Research Report RR8382 2013
 63 unpublished'A local version of the Hughes model for pedestrian flow'.2015, Preprint
 64 unpublished'A conservative scheme for nonclassical solutions to a strongly coupled PDEODE problem'.2015, Preprint
 65 article'Convex formulations of data assimilation problems for a class of HamiltonJacobi equations'.SIAM J. Control Optim.4922011, 383402
 66 article'LaxHopf Based Incorporation of Internal Boundary Conditions Into HamiltonJacobi Equation. Part II: Computational Methods'.Automatic Control, IEEE Transactions on555May 2010, 11581174
 67 article'A Class Of Nonloval Models For Pedestrian Traffic'.Mathematical Models and Methods in Applied Sciences22042012, 1150023
 68 article'Control of the continuity equation with a non local flow'.ESAIM Control Optim. Calc. Var.1722011, 353379
 69 article'Nonlocal crowd dynamics models for several populations'.Acta Math. Sci. Ser. B Engl. Ed.3212012, 177196
 70 article'A mixed ODEPDE model for vehicular traffic'.Mathematical Methods in the Applied Sciences3872015, 12921302
 71 article'On the micromacro limit in traffic flow'.Rend. Semin. Mat. Univ. Padova1312014, 217235
 72 article'Discussion about traffic junction modelling: conservation laws vs HamiltonJacobi equations'.Discrete Contin. Dyn. Syst. Ser. S732014, 411433
 73 article'Existence and uniqueness of measure solutions for a system of continuity equations with nonlocal flow'.Nonlinear Differential Equations and Applications NoDEA2012, 115
 74 inproceedings'How can macroscopic models reveal selforganization in traffic flow?'Decision and Control (CDC), 2012 IEEE 51st Annual Conference onDec 2012, 69896994
 75 book 'Multiscale modeling of pedestrian dynamics'. 12 MS&A. Modeling, Simulation and Applications Springer, Cham 2014
 76 article 'Qu'estce que la FFR? Comment l'utiliser?' Réalités Cardiologiques Janvier/Février 2013

77
incollection'Solutions in
${L}^{}$ for a conservation law with memory'.Analyse mathématique et applicationsMontrougeGauthierVillars1988, 117128  78 article'Largescale dynamics of meanfield games driven by local Nash equilibria'.J. Nonlinear Sci.2412014, 93115URL: http://dx.doi.org/10.1007/s0033201391852
 79 article'A front tracking method for a strongly coupled PDEODE system with moving density constraints in traffic flow'.Discrete Contin. Dyn. Syst. Ser. S732014, 435447
 80 article'Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result'.J. Differential Equations257112014, 40154029
 81 inbook'MultipleGradient Descent Algorithm (\em MGDA) for ParetoFront Identification'.34Numerical Methods for Differential Equations, Optimization, and Technological ProblemsModeling, Simulation and Optimization for Science and Technology, Fitzgibbon, W.; Kuznetsov, Y.A.; Neittaanmäki, P.; Pironneau, O. Eds.J. Périaux and R. Glowinski JubileesSpringerVerlag2014, 1
 82 article'Multiplegradient descent algorithm (MGDA) for multiobjective optimization'.Comptes Rendus de l'Académie des Sciences Paris3502012, 313318URL: http://dx.doi.org/10.1016/j.crma.2012.03.014
 83 techreport 'Révision de l'algorithme de descente à gradients multiples (MGDA) par orthogonalisation hiérarchique'. 8710 INRIA April 2015
 84 incollection'Robust uncertainty propagation in systems of conservation laws with the entropy closure method'.Uncertainty quantification in computational fluid dynamics92Lect. Notes Comput. Sci. Eng.Springer, Heidelberg2013, 105149
 85 article 'Rigorous Derivation of Nonlinear Scalar Conservation Laws from FollowtheLeader Type Models via Many Particle Limit'. Archive for Rational Mechanics and Analysis 2015
 86 article'Measurevalued solutions to conservation laws'.Arch. Rational Mech. Anal.8831985, 223270
 87 article'Modeling crowd dynamics by the meanfield limit approach'.Math. Comput. Modelling529102010, 15061520
 88 techreport 'A Sensitivity Equation Method for Unsteady Compressible Flows: Implementation and Verification'. INRIA Research Report No 8739 June 2015
 89 article'A sensitivity equation method for fast evaluation of nearby flows and uncertainty analysis for shape parameters'.Int. J. of Computational Fluid Dynamics207August 2006, 497512
 90 article'Multiscale stochastic reactiondiffusion modeling: application to actin dynamics in filopodia'.Bull. Math. Biol.7642014, 799818URL: http://dx.doi.org/10.1007/s1153801398443
 91 article'Particle methods for pedestrian flow models: from microscopic to nonlocal continuum models'.Math. Models Methods Appl. Sci.24122014, 25032523
 92 incollection'Finite volume methods'.Handbook of numerical analysis, Vol. VIIHandb. Numer. Anal., VIINorthHolland, Amsterdam2000, 7131020
 93 article' Multiple rows of cells behind an epithelial wound edge extend cryptic lamellipodia to collectively drive cellsheet movement'. Journal of Cell Science118Pt 12005, 5163
 94 techreport 'Construction of approximate entropy measure valued solutions for systems of conservation laws'. 201433 Seminar for Applied Mathematics, ETH Zürich 2014
 95 article'Convergence of methods for coupling of microscopic and mesoscopic reactiondiffusion simulations'.J. Comput. Phys.2892015, 117URL: http://dx.doi.org/10.1016/j.jcp.2015.01.030
 96 inproceedings 'Graded learning for object detection'. Proceedings of the workshop on Statistical and Computational Theories of Vision of the IEEE international conference on Computer Vision and Pattern Recognition (CVPR/SCTV) 2 1999
 97 article'Multiscale reactiondiffusion algorithms: PDEassisted Brownian dynamics'.SIAM J. Appl. Math.7332013, 12241247
 98 article'Coupling of microscopic and phase transition models at boundary'.Netw. Heterog. Media832013, 649661
 99 book 'Traffic flow on networks'. 1 AIMS Series on Applied Mathematics Conservation laws models American Institute of Mathematical Sciences (AIMS), Springfield, MO 2006
 100 article'A mixed system modeling twodirectional pedestrian flows'.Math. Biosci. Eng.1222015, 375392
 101 unpublished'A traffic flow model with nonsmooth metric interaction: wellposedness and micromacro limit'.2015, PreprintURL: http://arxiv.org/abs/1510.04461
 102 article'Wellposedness and finite volume approximations of the LWR traffic flow model with nonlocal velocity'.Netw. Heterog. Media1112016, 107121
 103 article'Modeling, simulation and validation of material flow on conveyor belts'.Applied Mathematical Modelling38132014, 32953313
 104 article'Achieving logarithmic growth of temporal and spatial complexity in reverse automatic differentiation'.Optimization Methods and Software11992, 3554
 105 article'Regularity theory and adjointbased optimality conditions for a nonlinear transport equation with nonlocal velocity'.SIAM J. Control Optim.5242014, 21412163
 106 article'Assessing the ability of the 2D FisherKPP equation to model cellsheet wound closure'.Math. Biosci.2522014, 4559URL: http://dx.doi.org/10.1016/j.mbs.2014.03.009
 107 article'NeumannDirichlet Nash strategies for the solution of elliptic Cauchy problems'.SIAM J. Control Optim.5152013, 40664083
 108 article'Nash strategies for the inverse inclusion CauchyStokes problem'.Inverse Problems and Imaging 1342019, 36
 109 article 'On sensitivity of RANS simulations to uncertain turbulent inflow conditions'. Computers & Fluids 61 25 2012
 110 article'Selforganizing pedestrian movement'.Environment and planning B2832001, 361384
 111 article'Traffic and related selfdriven manyparticle systems'.Rev. Mod. Phys.7342001, 10671141
 112 article'Evaluation of traffic data obtained via GPSenabled mobile phones: The Mobile Century field experiment'.Transportation Research Part C: Emerging Technologies1842010, 568583
 113 article'Continuum modelling of pedestrian flows: From microscopic principles to selforganised macroscopic phenomena'.Physica A: Statistical Mechanics and its Applications41602014, 684694
 114 article'A continuous sensitivity equation method for timedependent incompressible laminar flows'.Int. J. for Numerical Methods in Fluids502004, 817844
 115 article 'Fluxlimited solutions for quasiconvex HamiltonJacobi equations on networks'. arXiv preprint arXiv:1306.2428 October 2014
 116 article 'Suboptimal feedback control of flow over a sphere'. Int. J. of Heat and Fluid Flow 31 2010
 117 article'A Nashgame approach to joint image restoration and segmentation'.Appl. Math. Model.3811122014, 30383053URL: http://dx.doi.org/10.1016/j.apm.2013.11.034
 118 article'Uncertainty propagation in CFD using polynomial chaos decomposition'.Fluid Dynamics Research389September 2006, 616640
 119 article'NonOscillatory Central Schemes for a Traffic Flow Model with Arrehenius LookAhead Dynamics'.Netw. Heterog. Media432009, 431451
 120 article'On a mean field game approach modeling congestion and aversion in pedestrian crowds'.Transportation Research Part B: Methodological45102011, 15721589
 121 article'Mean field games'.Jpn. J. Math.212007, 229260
 122 article'On kinematic waves. II. A theory of traffic flow on long crowded roads'.Proc. Roy. Soc. London. Ser. A.2291955, 317345
 123 article'Predicting shock dynamics in the presence of uncertainties'.Journal of Computational Physics2172006, 260276
 124 article 'On the use of secondorder derivative and metamodelbased MonteCarlo for uncertainty estimation in aerodynamics'. Computers and Fluids 37 6 2010
 125 article 'Invivo elastography in animal models: Feasibility studies, (abstract)'. J. Ultrasound Med. 21 98 2002
 126 article'Multilevel Monte Carlo finite volume methods for uncertainty quantification in nonlinear systems of balance laws'.Lecture Notes in Computational Science and Engineering922013, 225294
 127 article'“Virtual”(computed) fractional flow reserve: current challenges and limitations'.JACC: Cardiovascular Interventions882015, 10091017
 128 article'Issues in Computational Fluid Dynamics code verification and validation'.AIAA Journal361998, 687695
 129 book 'Transport equations in biology'. Frontiers in Mathematics Birkhäuser Verlag, Basel 2007
 130 article'Transport equation with nonlocal velocity in Wasserstein spaces: convergence of numerical schemes'.Acta Appl. Math.1242013, 73105
 131 techreport 'Stochastic Multi Gradient Descent Algorithm'. ONERA July 2014
 132 article'Collective migration of an epithelial monolayer in response to a model wound'.Proceedings of the National Academy of Sciences104412007, 1598815993
 133 article 'First order mean field games in crowd dynamics'. ArXiv eprints February 2014
 134 inproceedings 'Approach for uncertainty propagation and robust design in CFD using sensitivity derivatives'. 15th AIAA Computational Fluid Dynamics Conference AIAA Paper 20012528 Anaheim, CA June 2001
 135 incollection'Riemannian BFGS Algorithm with Applications'.Recent Advances in Optimization and its Applications in EngineeringSpringer Berlin Heidelberg2010, 183192URL: http://dx.doi.org/10.1007/9783642125980_16
 136 article'Adjointbased optimization on a network of discretized scalar conservation law PDEs with applications to coordinated ramp metering'.J. Optim. Theory Appl.16722015, 733760
 137 article'Shock waves on the highway'.Operations Res.41956, 4251
 138 book 'Large Eddy Simulation for Incompressible Flows An Introduction'. Springer Berlin Heidelberg 2006
 139 inproceedings 'Uncertainty Quantification of Turbulence Model Closure Coefficients for Transonic WallBounded Flows'. 22nd AIAA Computational Fluid Dynamics Conference, 2226 June 2015, Dallas, USA. 2015
 140 article'A hybrid model for traffic flow and crowd dynamics with random individual properties'.Math. Biosci. Eng.1222015, 393413
 141 article'Stochastic modeling and simulation of traffic flow: asymmetric single exclusion process with Arrhenius lookahead dynamics'.SIAM J. Appl. Math.6632006, 921944
 142 article'DetachedEddy Simulation'.Annual Review of Fluid Mechanics412009, 181202
 143 inproceedings 'High Order Stochastic Finite Volume Method for the Uncertainty Quantification in Hyperbolic Conservtion Laws with Random Initial Data and Flux Coefficients'. Proc. ECCOMAS Proc. ECCOMAS 2012
 144 article'Fractional flow reserve calculation from 3dimensional quantitative coronary angiography and TIMI frame count: a fast computer model to quantify the functional significance of moderately obstructed coronary arteries'.JACC: Cardiovascular Interventions772014, 768777
 145 inproceedings 'Sensitivity and Uncertainty Analysis for Variable Property Flows'. 39th AIAA Aerospace Sciences Meeting and Exhibit AIAA Paper 20010139 Reno, NV Jan. 2001
 146 book 'Optimal transport'. 338 Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Old and new SpringerVerlag, Berlin 2009
 147 book 'Topics in optimal transportation'. 58 Graduate Studies in Mathematics American Mathematical Society, Providence, RI 2003
 148 techreport 'Uncertainty analysis for fluid mechanics with applications'. 20021 ICASE February 2002
 149 unpublished'Sequential Learning of Active Subspaces'.November 2019, https://arxiv.org/abs/1907.11572  working paper or preprint
 150 article'Modeling uncertainty in flow simulations via generalized Polynomial Chaos'.Journal of Computational Physics1872003, 137167
 151 article'Active control of flow separation over an airfoil using synthetic jets'.J. of Fluids and Structures242008, 13491357
 152 article'MetaModelAssisted MGDA for MultiObjective Functional Optimization'.Computers and Fluids102http://www.sciencedirect.com/science/article/pii/S0045793014002576#2014, 116130
 153 article'Fractional flow reserve versus angiography for guidance of PCI in patients with multivessel coronary artery disease (FAME): 5year followup of a randomised controlled trial'.The Lancet386100062015, 18531860