Keywords
Computer Science and Digital Science
 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
Other Research Topics and Application Domains
 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]
 Régis Duvigneau [INRIA, Senior Researcher, HDR]
 JeanAntoine Désidéri [INRIA, Emeritus, HDR]
Faculty Member
 Abderrahmane Habbal [UNIV COTE D'AZUR, Associate Professor, HDR]
PostDoctoral Fellows
 Daniel Eduardo Inzunza Herrera [INRIA]
 Khadija Musayeva [UNIV COTE D'AZUR]
 Enrico Siri [INRIA, from Oct 2022]
PhD Students
 Mustapha Bahari [UNIV MOHAMMED VI POLYTECH, from Sep 2022]
 Salma CHABBAR [Univ. Côte d'Azur and Ecole Mohammadia d'ingénieurs (EMI, Rabat)]
 Agatha Joumaa [IFPEN, from Nov 2022]
 Yessennia Carolina Martinez Martinez [ANID, from Sep 2022]
 Alexandra Wuerth [INRIA]
Administrative Assistant
 Montserrat Argente [INRIA]
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 109, or to the dynamics of turbulent flows for which large scale / small scale vortical structures interfere 136.
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 65, 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 71, 97 for some applications in traffic flow modelling, and 90, 94, 96 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 65, 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 72, 85 for the derivation of LighthillWhithamRichards 122, 135 traffic flow model from FollowtheLeader and 91 for results on crowd motion models (see also 112). 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 54, 93. We refer, for example, to the notion of solution for nonhyperbolic systems 99, 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 93.
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 76), and proved to be successful for studying emerging selforganised flow patterns 75. The theoretical measure framework proves to be also relevant in addressing micromacro limiting procedures of mean field type 100, 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 144, 143. Indeed, as observed in 129, 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 61, 62, 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 131, 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, 52) 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 49, sedimentation 56, supply chains 104, conveyor belts 102, biological applications like structured populations dynamics 128, or more general problems like gradient constrained equations 51. 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 57, 64, 68, 101, 139. 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 57, 101 and lane formation in crossing pedestrian flows.
General analytical results on nonlocal conservation laws, proving existence and possibly uniqueness of solutions of the Cauchy problem for (1), can be found in 50 for scalar equations in one space dimension ($N=d=1$), in 69 for scalar equations in several space dimensions ($N=1$, $d\ge 1$) and in 47, 70, 74 for multidimensional systems of conservation laws. Besides, specific finite volume numerical methods have been developed recently in 47, 101 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  boundary value problems, regularity of solutions 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 63. In aerodynamics, inflow conditions like velocity modulus and direction, are subject to fluctuations 108, 127. 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 137, 142, 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 138, 141, 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 108, 118, 123, 146 or an entropy closure method 84, or by discretizing the probability space and extending the numerical schemes to the stochastic components 46. 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 132, 142, 145.
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 55. 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 58 (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 142. 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 134 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 53, 103 for the backward time integration of the adjoint variables. The sensitivity equation method (SEM) offers a promising alternative 88, 113, 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) 136 , DetachedEddy Simulation (DES) 140 or OrganizedEddy Simulation (OES) 59, 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 Integration of ComputerAided Design and analysis for shape optimization
A major difficulty in shape optimization is related to the multiplicity of geometrical representations handled during the design process. From highorder ComputerAided Design (CAD) objects to discrete meshbased descriptions, several geometrical transformations have to be performed, that considerably impact the accuracy, the robustness and the complexity of the design loop. This is even more critical when multiphysics applications are targeted, including moving bodies.
To overcome this difficulty, we intend to investigate isogeometric analysis114 methods, which propose to use the same CAD representations for the computational domain and the physical solutions yielding geometrically exact simulations. In particular, hyperbolic systems and compressible aerodynamics are targeted.
3.3.3 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 130 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 148.
3.3.4 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 95. Our proposal is designed to tackle derivativefree, expensive models, hence requiring very few model evaluations to converge to the solution.
3.3.5 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 48 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 147.
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 110. Further improvements require more advanced models, keeping into better account interactions at the microscopic scale, and adapted control techniques, see 60 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 73, 98, 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 111 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 67, 66), 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, 97 or via the HamiltonJacobi approach 67, 66 could allow to optimize the flow by controlling some specific vehicles corresponding to internal conditions.
4.3 Combined hormone and brachy therapies for the treatment of prostate cancer
The latest statistics published by the International Agency for Research on Cancer show that in 2018, 18.1 million new cancer cases have been identified and 9.6 million deaths have been recorded worldwide making it the second leading cause of death globally. Prostate cancer ranks third in incidence with 1.28 million cases and represents the second most commonly diagnosed male cancer.
Prostate cells need the hormone androgen to survive and function properly. For this to happen, the androgens have to bind to a protein in the prostate cells called Androgen Receptor and activate it. Since androgens act as a growth factor for the cells, one way of treating prostate cancer is through the antihormone therapy that hinder its activity. The Androgen Deprivation Therapy (ADT) aims to either reduce androgen production or to stop the androgens form working through the use of drugs. However, over time, castrationresistant cells that are able to sustain growth in a low androgen environment emerge. The castrationresistant cells can either be androgen independent or androgen repressed meaning that they have a negative growth rate when the androgen is abundant in the prostate. In order to delay the development of castration resistance and reduce its occurrence, the Intermittent Androgen Deprivation Therapy is used.
On the other hand, brachytherapy is an effective radiation therapy used in the treatment of prostate cancer by placing a sealed radiation source inside the prostate gland. It can be delivered in high dose rates (HDR) or low dose rates (LDR) depending on the radioactive source used and the duration of treatment.
In the HDR brachytherapy the source is placed temporarily in the prostate for a few minutes to deliver high dose radiation while for the LDR brachytherapy low radiations dose are delivered from radioactive sources permanently placed in the prostate. The radioactivity of the source decays over time, therefore its presence in the prostate does not cause any longterm concern as its radioactivity disappears eventually. In practice, brachytherapy is prescribed either as monotherapy, often for localized tumors, or combined with another therapy such as external beam radiation therapy for which the total dose prescribed is divided between internal and external radiation. Brachytherapy can also be prescribed in combination with hormone therapy.
However, in the existing literature there is currently no mathematical model that explores this combination of treatments. Our aim is to develop a computational model based on partial differential equations to assess the effectiveness of combining androgen deprivation therapy with brachytherapy in the treatment of prostate cancer. The resulting simulations can be used to explore potential unconventional therapeutic strategies.
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.

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 105, 117, we have demonstrated that Nash game approaches are efficient to tackle illposedness. We intend to extend the results obtained for Laplace equations in 105, and the algorithms developed in Section 3.3.5 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. 133, 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 (now Atlantis), 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 efficiency, thus reducing energy consumption and pollutant emission.
Regarding the impact on health care, our research activity and preliminary results on hormonoradio therapies for prostate cancer show that combining hormone therapy with brachytherapy allowed us to reduce the radiative dose used from 120Gy to 80Gy. When the treatments are given at the same time, the final tumor volume is significantly reduced compared to using each therapy separately. The outcomes for public health in terms of financial cost and limitations of undesired side effects is of very high potential.
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 received the ENBIS 2022 Young Statistician Award and a prix innovation et recherche appliquée by Université Côte d'Azur.
6.2 New book
A. Bayen, M.L. Delle Monache, M. Garavello, P. Goatin and B. Piccoli, Control Problems for Conservation Laws with Traffic Applications: Modeling, Analysis, and Numerical Methods, Progress in Nonlinear Differential Equations and Their Applications (Vol. 99), Springer Nature (2022). 36
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:
 Publications:

Contact:
JeanAntoine Désidéri

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:

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:

Contact:
Abderrahmane Habbal

Participants:
Régis Duvigneau, JeanLuc Szpyrka, David Rey, Clément Welsch, Abderrahmane Habbal
7.1.4 RoadNetwork

Keywords:
Road traffic, Road network, Python, Numerical simulations

Functional Description:
Python library dedicated to create, manipulate and simulate ODE traffic equations on networks

Release Contributions:
First version Needs some fixing of module names and comments

Contact:
Abderrahmane Habbal

Partner:
Université Côte d'Azur (UCA)
8 New results
8.1 Macroscopic traffic flow models on networks
Participants: Mickaël Binois, Paola Goatin, Alexandra Würth, Chiara Daini [KOPERNIC ProjectTeam, INRIA Paris], Maria Laura Delle Monache [UC Berkeley, USA], Antonella Ferrara [Univ. Pavia, Italy], Adriano Festa [Polytechnic of Turin, Italy], Simone Göttlich [Univ. Mannheim, Germany], Fabio Vicini [Polytechnic of Turin, Italy].
Traffic control by Connected and Automated Vehicles
We present a general multiscale approach for modeling the interaction of controlled and automated vehicles (CAVs) with the surrounding traffic flow. The model consists of a scalar conservation law for the bulk traffic, coupled with ordinary differential equations describing the possibly interacting CAV trajectories. The coupling is realized through flux constraints at the moving bottleneck positions, inducing the formation of nonclassical jump discontinuities in the traffic density. In turn, CAVs are forced to adapt their speed to the downstream traffic average velocity in congested situations. We analyze the model solutions in a Riemanntype setting, and propose an adapted finite volume scheme to compute approximate solutions for general initial data. The work paves the way to the study of general optimal control strategies for CAV velocities, aiming at improving the overall traffic flow by reducing congestion phenomena and the associated externalities. Controlling CAV desired speeds allows to act on the system to minimize any traffic density dependent cost function. More precisely, we apply Model Predictive Control (MPC) to reduce fuel consumption in congested situations.
Traffic flow model calibration by statistical approaches
In the framework of A. Würth's PhD thesis, we employ 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 publicly available data from the Minnesota Department of transportation (MnDOT). In 30, we propose a Bayesian approach for parameter uncertainty quantification in macroscopic traffic flow models from crosssectional data. A bias term is introduced and modeled as a Gaussian process to account for the traffic flow models limitations. We validate the results comparing the error metrics of both first and second order models, showing that second order models globally perform better in reconstructing traffic quantities of interest.
Besides, we proved an existence result for the associated initialboundary value problem for general second order macroscopic models in 43.
Routing strategies in traffic flows on networks
In 42, we introduce a macroscopic differential model coupling a conservation law with a HamiltonJacobi equation to respectively model the nonlinear transportation process and the strategic choices of users. Furthermore, the model is adapted to the multipopulation case, where every population differs in the level of traffic information about the system. This enables us to study the impact of navigation choices on traffic flows on road networks.
8.2 Isogeometric Discontinuous Galerkin method for compressible flows
Participants: Régis Duvigneau, Stefano Pezzano.
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:

Arbitrary EulerianLagrangian formulation for highorder meshes
To enable the simulation of flows around moving or deforming 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 approach has been applied to the simulation of morphing airfoils 28.

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 cumbersome using classical piecewiselinear grids due to a lack of common geometrical interface. Thus, we have developed a method based on a geometrically exact sliding interface using NURBS elements, ensuring a fully conservative scheme 27.

Isogeometric shape optimization
We develop an optimization procedure entirely based on NURBS representations. The mesh, the shape to be optimized, as well as the flow solutions are represented by NURBS, which avoid any geometrical conversion and allows to exploit NURBS properties regarding regularity or hierarchy. The approach has also been employed in the framework of Bayesian optimization for airfoil design 28, 34.
8.3 Advanced Bayesian optimization
Participants: Mickaël Binois, Régis Duvigneau, Nicholson Collier [Argonne, USA], Mahmoud Elsawy [Atlantis team], Stéphane Lanteri [Atlantis team], Jonathan Ozik [Argonne, USA], Victor Picheny [SecondMind, GB], Henry Moss [SecondMind, GB].
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 context 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 35.
Massively parallel Bayesian optimization
Motivated by a large scale multiobjective optimization problem for which thousands of evaluations can be conducted in parallel, we develop an efficient approach to tackle this issue in 39.
One way to reduce the time of conducting optimization studies is to evaluate designs in parallel rather than just oneatatime. For expensivetoevaluate blackboxes, batch versions of Bayesian optimization have been proposed. They work by building a surrogate model of the blackbox that can be used to select the designs to evaluate efficiently via an infill criterion. Still, with higher levels of parallelization becoming available, the strategies that work for a few tens of parallel evaluations become limiting, in particular due to the complexity of selecting more evaluations. It is even more crucial when the blackbox is noisy, necessitating more evaluations as well as repeating experiments. Here we propose a scalable strategy that can keep up with massive batching natively, focused on the exploration/exploitation tradeoff and a portfolio allocation. We compare the approach with related methods on deterministic and noisy functions, for mono and multiobjective optimization tasks. These experiments show similar or better performance than existing methods, while being orders of magnitude faster.
Multifidelity modeling and optimization
To reduce the computational cost related to the use of highfidelity simulations when evaluating the cost function, we investigate the construction of multifidelity autoregressive Gaussian Process models, that can rely on different physical models (e.g. inviscid or viscous flows) or numerical accuracy (e.g. coarse or fine meshes). The objective is to construct a model that is accurate regarding the highfidelity evaluations, but mostly based on lowfidelity simulations. Of particular interest is the definition of an efficient acquisition function, that selects both the next design point to evaluate and the corresponding fidelity level to use. This work is achieved in collaboration with SecondMind company and was the topic of Maha Ouali's internship.
8.4 Gaussian process based sequential design
Participants: Mickaël Binois, Robert Gramacy [Virginia Tech, USA], Nathan Wycoff [Virginia Tech, USA].
Besides Bayesian optimization as above, Gaussian processes are useful for a variety of other related tasks. Here we first present a method to deal with nonstationarity of the process, handling global and local scales. Secondly, we review the state of the art for highdimensional GP modeling.
Sensitivity prewarping for local surrogate modeling
In the continual effort to improve product quality and decrease operations costs, computational modeling is increasingly being deployed to determine feasibility of product designs or configurations. Surrogate modeling of these computer experiments via local models, which induce sparsity by only considering short range interactions, can tackle huge analyses of complicated inputoutput relationships. However, narrowing focus to local scale means that global trends must be relearned over and over again. In 31, we propose a framework for incorporating information from a global sensitivity analysis into the surrogate model as an input rotation and rescaling preprocessing step. We discuss the relationship between several sensitivity analysis methods based on kernel regression before describing how they give rise to a transformation of the input variables. Specifically, we perform an input warping such that the "warped simulator" is equally sensitive to all input directions, freeing local models to focus on local dynamics. Numerical experiments on observational data and benchmark test functions, including a highdimensional computer simulator from the automotive industry, provide empirical validation.
A survey on highdimensional Gaussian process modeling with application to Bayesian optimization
In 22 we propose a review of highdimensional GP modeling. Extending the efficiency of Bayesian optimization (BO) to larger number of parameters has received a lot of attention over the years. Even more so has Gaussian process regression modeling in such contexts, on which most BO methods are based. A variety of structural assumptions have been tested to tame high dimension, ranging from variable selection and additive decomposition to low dimensional embeddings and beyond. Most of these approaches in turn require modifications of the acquisition function optimization strategy as well. Here we review the defining assumptions, and discuss the benefits and drawbacks of these approaches in practice.
8.5 Multilabel propagation
Participants: Mickaël Binois, Khadija Musayeva.
In 44, to prepare for the analysis of complex biological data, this work focuses on multilabel learning from small number of labelled data. We demonstrate that the straightforward binaryrelevance extension of the interpolated label propagation algorithm, the harmonic function, is a competitive learning method with respect to many widelyused evaluation measures. This is achieved mainly by a new transition matrix that better captures the underlying manifold structure. Furthermore, we show that when there exists label dependence, we can use the outputs of a competitive learning method as part of the input to the harmonic function to provide improved results over those of the original model. Finally, since we are using multiple metrics to thoroughly evaluate the performance of the algorithm, we propose to use the gametheory based method of Kalai and Smorodinsky to output a single compromise solution. This method can be applied to any learning model irrespective of the number of evaluation measures used.
8.6 Policybased optimization
Participants: Régis Duvigneau, Jonathan Viquerat [Mines ParisTech].
This work concerns the development of blackbox optimization methods based on singlestep deep reinforcement learning (DRL) and their conceptual similarity to evolution strategy (ES) techniques 29. The connection of policybased optimization (PBO) to evolutionary strategies (especially covariance matrix adaptation evolutionary strategy) is discussed. Relevance is assessed by benchmarking PBO against classical ES techniques on analytic functions minimization problems, and by optimizing various parametric control laws intended for the Lorenz attractor. This contribution definitely establishes PBO as a valid, versatile blackbox optimization technique, and opens the way to multiple future improvements building on the inherent flexibility of the neural networks approach.
8.7 Analytical study of the Eulerian Adjoint System
Participants: JeanAntoine Désidéri, Jacques Peter [ONERA/DAAA, Université ParisSaclay, Châtillon].
Ordinary differential equations have been derived for the adjoint Euler equations first using the method of characteristics in 2D. For this system of partialdifferential equations, the characteristic curves appear to be the streamtraces and the wellknown curves of the theory applied to the flow. The differential equations satisfied along the streamtraces in 2D have then been extended and demonstrated in 3D by linear combinations of the original adjoint equations.
These findings extend their wellknown counterparts for the direct system and should serve analytical and possibly numerical studies of the perfectflow model with respect to adjoint fields or sensitivity questions. In addition to the analytical theory, the results have been illustrated by the numerical integration of the compatibility relationships for discrete 2D flow fields and dualconsistent adjoint fields over a very fine grid about an airfoil 26.
8.8 Pareto optimality and Nash games
Participants: JeanAntoine Désidéri, Sébastien Defoort [ONERA/DTIS, Université de Toulouse], Nathalie Bartoli [ONERA/DTIS, Université de Toulouse], Christophe David [ONERA/DTIS, Université de Toulouse], Julien Wintz [SED, INRIA Sophia Antipolis].
The present work reflects our cooperation with the Information Processing and Systems Department (DTIS) of Onera Toulouse.
In the multiobjective optimization of a complex system, after the Pareto front associated with preponderant objective functions (“primary cost functions”), has been approximated, usually at a demanding computational cost, the decision to elect the final concept is still to be made since a whole set of indiscriminate Paretooptimal solutions is available.
To complete the decision process, we had proposed a Nash game construction initiated from one such Paretooptimal solution to target a reduction of “secondary cost functions” while quasimaintaining the Pareto optimality of the primary cost functions. Convergence proof and first examples were given in 7.
This method has been applied to the multiobjective optimization of the flight performance of an AirbusA320type aircraft in terms of takeoff fuel mass and operational empty weight (primary cost functions) concurrently with ascenttocruise altitude duration (secondary). The optimization was subject to functional constraints on geometry and longitudinal stability. Designs were evaluated by means of the FASTOAD opensource software developed by ONERA and ISAESUPAERO, and the prioritized optimization conducted by our NashMGDA software (MGDA). The experiment demonstrated the efficacy of the method to greatly reduce the ascent duration, and its great efficiency in terms of computational time, permitting the numerical process to be interactive 41.
8.9 Inverse CauchyStokes problems solved as Nash games
Participants: Abderrahmane Habbal, Marwa Ouni [PhD, LAMSIN, Univ. Tunis Al Manar], Moez Kallel [LAMSIN, Univ. Tunis Al Manar].
We extend in two directions our results published in 106 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 first two 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 45
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 where such an approach is applied to solve inverse problems for nonlinear systems 107.
8.10 Combined therapies for the treatment of prostate cancer
Participants: Abderrahmane Habbal, Salma Chabbar [PhD, ACUMES and EMI, Univ. Mohammed V], Rajae Aboulaich [EMI, Univ. Mohammed V], Nabil Ismaili [PhD.MD., Univ Mohamed VI for health sciences, Casablanca], Sanae EL Mejjaoui [PhD.MD., Institute for Oncology, Avicenne Hospital, Rabat].
Prostate cancer is a hormonedependent cancer characterized by two types of cancer cells, androgendependent cancer cells and androgenresistant ones. The objective of this work is to present a novel mathematical model for the treatment of prostate cancer under combined hormone therapy and brachytherapy. Using a system of partial differential equations, we quantify and study the evolution of the different cell densities involved in prostate cancer and their responses to the two treatments. The numerical simulations are carried out on FreeFem++ using a 2D finite element method. Numerical simulations of tumor growth under different therapeutic strategies are explored and discussed as summarized hereafter. Combining hormone therapy with brachytherapy allowed us to reduce the dose used from 120Gy to 80Gy. When the treatments are given at the same time, the final tumor volume is significantly reduced compared to using each therapy separately. However, starting with hormone therapy gave better results. When using intermittent hormone therapy combined with brachytherapy, we found that once the PSA level drops below the critical level, it stays at reasonable levels and therefore the hormone therapy does not reactivate. When we use continuous hormone therapy instead, the prostate is unnecessarily deprived of androgen for an almost nonexistent reduction in tumor volume compared to intermittent deprivation. The use of hormone therapy over a short period of time is therefore sufficient to give good results. The results also showed that the dose used in the combined treatments affects the tumor relapse.
9 Bilateral contracts and grants with industry
9.1 Bilateral contracts with industry

Mycophyto (2020...): this research contract involving Université Côte d'Azur is financing the postdoctoral contract of Khadija Musayeva. The goal is to develop prediction algorithms based on environmental data.
Participants: Mickaël Binois, Khadija Musayeva.
10 Partnerships and cooperations
10.1 International initiatives
10.1.1 Associate Teams in the framework of an Inria International Lab or in the framework of an Inria International Program
NOLOCO

Title:
Efficient numerical schemes for nonlocal transport phenomena

Duration:
2018 > 2022

Coordinator:
Luis Miguel Villada Osorio (lvillada@ubiobio.cl)

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.
See also: project web site
Participants: Régis Duvigneau, Paola Goatin, Daniel Inzunza.
10.1.2 STIC/MATH/CLIMAT AmSud projects
NOTION

Title:
nonlocal conservation laws for engineering, biological and epidemiological applications: theory and numerics

Program:
MATHAmSud

Duration:
January 1, 2022 – December 31, 2023

Local supervisor:
Paola Goatin

Partners:
 Universidad del Biobio
 Bürger (Chili)
 Universidad de Cordoba

Inria contact:
Paola Goatin

Summary:
Conservation laws with flux function depending on integral evaluations of the conserved quantities arise in several models describing engineering, biological and epidemiological applications. The presence of nonlocal terms makes the classical techniques developed for hyperbolic systems of conservation laws inapplicable as such, thus requiring the development of novel analytical and numerical tools. Moreover, the presence of integral terms has a huge impact on the cost of numerical simulations, motivating the design of efficient approximation schemes. This project aims to tackle the above mentioned analytical and numerical challenges, focusing on engineering applications (sedimentation, traffic, population dynamics, etc) and biological and epidemiological phenomenon.
10.1.3 Participation in other International Programs
FACCTS (France And Chicago Collaborating in The Sciences)
Participants: Mickael Binois, Alexandra Wuerth.

Title:
Developing Next Generation Tools for Largescale Computational Science

Partner Institutions:
 University of Chicago (USA)
 Argonne National Laboratory (USA)

Date/Duration
: 20222024

Additionnal info/keywords:
It is increasingly recognized that computational models play a critical role in developing new insights across scientific domains. A central element in the computational model development and application workflow is running in silico experiments with the model, what we refer to broadly as model exploration (ME), in order to iteratively implement and understand its capabilities, establish its trustworthiness and apply it to the problems of interest. However, as high performance computing (HPC) resources become more powerful and ubiquitous, and computational models increase in complexity to exploit those advances, sophisticated and iterative statistical ME algorithms are needed to efficiently characterize the model behaviors. In this proposal we seek to extend the complementary expertise of our France and Chicagobased research groups to further develop HPCoriented Bayesian optimization algorithms and HPC workflow methods, with the goal of developing next generation tools for lowering barriers to largescale ME approaches across scientific domains.
10.2 International research visitors
10.2.1 Visits of international scientists
Other international visits to the team
Robert B. Gramacy

Status
Professor

Institution of origin:
Virginia Polytechnic Institute and State University

Country:
United States

Dates:
June 2022

Context of the visit:
Invitation by the CIROQUO consortium

Mobility program/type of mobility:
Research stay and tutorial on deep Gaussian processes
Harold Contreras

Status
PhD student

Institution of origin:
Universidad de Concepcion

Country:
Chile

Dates:
June  July 2022

Context of the visit:
Associated Team NOLOCO

Mobility program/type of mobility:
research stay
Luis Miguel Villada

Status
Associate Professor

Institution of origin:
Universidad de Concepcion

Country:
Chile

Dates:
June  July 2022

Context of the visit:
Associated Team NOLOCO

Mobility program/type of mobility:
research stay
10.3 European initiatives
10.3.1 Other european programs/initiatives
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 website

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.
Participants: Paola Goatin.
10.4 National initiatives
10.4.1 ANR

Project OPERA (20192022): 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).
Participants: Régis Duvigneau.

Institute 3IA Côte d'Azur : The 3IA Côte d'Azur is one of the four "Interdisciplinary Institutes of Artificial Intelligence" that were created in France in 2019. Its ambition is to create an innovative ecosystem that is influential at the local, national and international levels, and a focal point of excellence for research, education and the world of AI.
ACUMES is involved with the project “Data driven traffic management” in the axis AI for smart and secure territories (20202024), for which P. Goatin is chair holder. This project aims at contributing to the transition to intelligent mobility management practices through an efficient use of available resources and information, fostering data collection and provision. We focus on improving traffic flow on road networks by using advanced mathematical models and statistical techniques leveraging the information recovered by real data.
Participants: Paola Goatin, Daniel Inzunza, Alexandra Würth.
11 Dissemination
11.1 Promoting scientific activities
11.1.1 Scientific events: organisation
General chair, scientific chair
 P. Goatin was member of the scientific committee of CEMRACS 2022 “Transport in Physics, Biology and Urban traffic”, CIRM (Marseille, France), JulyAugust 2022.
 A. Habbal was member of the program committee for CARI 2022 (African Conference on Computer Science and Applied Mathematics 2022) Cameroon and Tunisia, from October 04 to 07 2022.
Member of the organizing committees
 M. Binois and R. Duvigneau were part of the program commitee of the Artificial Intelligence for Waves  AI4Wave workshop, October 2022.
 A. Habbal was coorganizer of the Annual meeting of the GdR Mathematics of Optimization and Applications MOA, 1114 October 2022, Nice.
11.1.2 Scientific events: selection
Reviewer
 M. Binois reviewed for the following conferences: AISTATS 2023, ICML 2022 and NeurIPS 2022
 P. Goatin reviewed for ECC 2022
11.1.3 Journal
Member of the editorial boards
 M. Binois is Associate Editor of ACM Transactions on Evolutionary Learning and Optimization
 P. Goatin is Associate Editor of Networks and Heterogeneous Media, SIAM Journal on Applied Mathematics and ESAIM: Mathematical Modelling and Numerical Analysis.
Reviewer  reviewing activities
 M. Binois is a reviewer for the following international journals: AMOS, CMAME, IJDS, JCGS, JOGO, JRSSC, JSTP, JStat, JUQ, KNOSYS, MACH, Operations Research, Optics Letters, PLOS One, Technometrics and TELO.
 R. Duvigneau is reviewer for the following international journals: J. of Heat and Fluid Flow, Comp. Meth. Applied Mech. Eng., J. of Comp. Physics, Comp. and Fluids, Int. J. Num. Meth. Eng.
 P. Goatin reviewed for: Nonlinear Analysis, SIAM Journal on Control and Optimization.
 A. Habbal reviewed for Journal of Scientific Computing, Trends in Computer Science and Information Technology
11.1.4 Invited talks
 M. Binois: Michigan State University statistics colloquium, online, February 2022. Invited talk: Scaling up multiobjective Bayesian optimization.
 M. Binois: Dagstuhl seminar: Theory of Randomized Optimization Heuristics, Germany, February 2022. Talk: A brief introduction to Bayesian optimization.
 M. Binois: SIAM Conference on Uncertainty Quantification, Atlanta (USA), April 2022. Symposium Talk: Leveraging Replication in Sequential Design Tasks.
 M. Binois: ENBIS 22 annual conference, Trondheim (Norway), June 2022. Plenary Talk: Sequential Learning of Active Subspaces.
 P. Goatin: ECCOMAS 2022  8th European Congress on Computational Methods in Applied Sciences and Engineering, Oslo (Norway), June 2022. Semiplenary lecture: Multiscale models for mixed humandriven and autonomous vehicles.
 P. Goatin: NorPDE 2022  2nd Norwegian meeting on PDEs, Bergen (Norway), June 2022. Talk: Conservation laws with moving constraints arising in traffic modeling.
 P. Goatin: ECC 2022  European Control Conference, London (United Kingdom), July 2022. Invited session: “Datadriven modelling and control for future traffic systems”. Talk: Centralized traffic control via small fleets of connected and automated vehicles.
11.1.5 Scientific expertise
 R. Duvigneau evaluated some projects for the Italian Ministry for Universities and Research.
 P. Goatin evaluated a project for ÖAW (Austrian Academy of Science) program New Frontiers Group.
 P. Goatin evaluated a project for University of Genoa.
 P. Goatin was member of the advisory board of DISMA Excellence Project of Politecnico di Torino (20182022).
 P. Goatin was member of the committee of the SMAIGAMNI PhD award (2021 and 2022).
 A. Habbal was reviewer for an ANR project (generic call CE46  Numerical models, simulation, applications).
11.1.6 Research administration
 R. Duvigneau is head of the Scientific Committee of Platforms (cluster and immersive space) for Inria Centre at Université Côte d'Azur.
 R. Duvigneau is member of the Scientific Committee of OPAL computing Platform at Université Côte d'Azur
 R. Duvigneau is member of the Steering Committee of "Maison de la Simulation et Interactions" 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 was member of the Full Professor hiring committee of INSA Rouen and INP de Bordeaux in Applied Mathematics.
 A. Habbal was member of hiring committee of an Associate Professor in Mathematics of Optimization and Learning, Université du Littoral Côte d’Opale.
11.2 Teaching  Supervision  Juries
11.2.1 Teaching
 Master: M. Binois, Optimisation bayésienne, 9 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, 18 hrs, M2, Mohammed VI Polytechnic University, Morocco.
 Master: J.A. Désidéri, Multidisciplinary Optimization, ISAE Supaéro (Toulouse), 5 hrs.
 Master: R. Duvigneau, Advanced Optimization, 28 hrs, M2, Polytech Nice Sophia  Université Côte d'Azur.
 Advanced course: P. Goatin, 6 hrs, XLVII Summer School on Mathematical Physiscs, Ravello (Italy), September 2020: “Macroscopic traffic flow models on networks”.
 Master: P. Goatin, projets M2, 7 hrs, Polytech Nice Sophia  Université Côte d'Azur.
 Master: P. Goatin, Optimization, 24 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, 18 hrs, M1, Polytech Nice Sophia  Université Côte d'Azur.
 Master: A. Habbal, Numerical methods for PDEs, 18 hrs, M1, Polytech Nice Sophia  Université Côte d'Azur.
 Master: A. Habbal, Introduction to 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.
 Licence (L1): A. Habbal, Mathematics reinforcement, 36 hrs, Polytech Nice Sophia  Université Côte d'Azur.
 Master Thesis project (6 months) Wissale Khoudraji, Mathematics of Obesity (Master 2, Polytechnic Univ. Mohamed VI, Morocco). Supervisor : A. Habbal
 Master Thesis project (6 months) Mustapha Bouchaara, Game Theory applied to medical care policies, (Master 2, Polytechnic Univ. Mohamed VI, Morocco). Supervisor : A. Habbal
 Master Thesis project (3 months) Maha Ouali, ENSTA, Multifidelity Gaussian Process models. Advisor : R. Duvigneau and M. Binois
11.2.2 Supervision
 PhD defense : S. Chabbar, Development and investigation of novel models in computational medicine, Rabat, 22 september 2022. Supervisors : A. Habbal, R. Aboulaich
 PhD in progress: A. Joumaa, Pseudorealtime optimization of the environmental performance of urban mobility using macroscopic and multimodal modeling approaches, Univ. Côte d’Azur/IFPEN. Supervisors: P. Goatin, G. De Nunzio.
 PhD in progress: A. Würth, AI for road traffic modeling and management, Univ. Côte d’Azur/3IA. Supervisors: P. Goatin, M. Binois.
 PhD in progress: N. Rosset, Flow prediction from sketches, Univ. Côte d’Azur. Supervisors: A. Bousseau, G. Cordonnier, R. Duvigneau
 PhD in progress: A. Machtalay, From meanfield games to agentbased models (and back) through Markov Chain aggregation, Univ. Côte d’Azur. and UM6P (Morocco) Supervisors : A. Habbal, A. Ratnani
 PhD in progress: Mustapha Bahari, Optimal Mass Transportation for adaptive mesh generation and rrefinement using Isogeometric analysis, Univ. Côte d’Azur. and UM6P (Morocco) Supervisors : A. Habbal, A. Ratnani
11.2.3 Juries
 P. Goatin was referee of L. Guan's PhD thesis Optimal boundary traffic control and state estimation with disturbances, Université Grenoble Alpes, September 15th, 2022.
 P. Goatin was member of the committee of E. Pinsard's PhD thesis “Modélisation du mouvement de foules denses: phénoménologie et couplage de modèles”, Université ParisSaclay, December 8th, 2022.
11.3 Popularization
11.3.1 Articles and contents
 R. Duvigneau contributed to the article "L'art du mélange est une science", Science & Vie, December 2022.
11.3.2 Interventions
 M. Binois did the Chiche training by Claude Vadel.
 P. Goatin spoke at the roundtable on “Mobilité et transports : défis actuels” held at the Forum Emploi Math in Paris (October 2022).
12 Scientific production
12.1 Major publications
 1 articleNonlocal systems of conservation laws in several space dimensions.SIAM Journal on Numerical Analysis5222015, 963983
 2 articleFinite volume schemes for locally constrained conservation laws.Numer. Math.1154With supplementary material available online2010, 609645
 3 articleWellposedness of a conservation law with nonlocal flux arising in traffic flow modeling.Numerische Mathematik2015
 4 articleA well posed conservation law with a variable unilateral constraint.J. Differential Equations23422007, 654675
 5 articleScalar conservation laws with moving constraints arising in traffic flow modeling: an existence result.J. Differential Equations257112014, 40154029
 6 articleA PDEODE model for a junction with ramp buffer.SIAM J. Appl. Math.7412014, 2239
 7 inproceedingsAdaptation by Nash games in gradientbased multiobjective/multidisciplinary optimization.JANO13  Mathematical Control and Numerical Applications372Springer Proceedings in Mathematics & Statistics SeriesKhouribga, MoroccoFebruary 2021
 8 articleCOOPERATION AND COMPETITION IN MULTIDISCIPLINARY OPTIMIZATION Application to the aerostructural aircraft wing shape optimization.Computational Optimization and Applications5212012, 2968
 9 inbookParametric optimization of pulsating jets in unsteady flow by MultipleGradient Descent Algorithm (MGDA).Numerical Methods for Differential Equations, Optimization, and Technological Problems, Modeling, Simulation and Optimization for Science and TechnologyJanuary 2017
 10 articlePrioritized optimization by Nash games : towards an adaptive multiobjective strategy.ESAIM: Proceedings and Surveys71August 2021, 5463
 11 articleMultiplegradient descent algorithm (MGDA) for multiobjective optimization / Algorithme de descente à gradients multiples pour l'optimisation multiobjectif.Comptes Rendus. MathématiqueTome 350Fascicule 56March 2012, 313318
 12 articleKrigingbased optimization applied to flow control.Int. J. for Numerical Methods in Fluids69112012, 17011714
 13 articleNeumannDirichlet Nash strategies for the solution of elliptic Cauchy problems.SIAM J. Control Optim.5152013, 40664083
 14 articleA Nashgame approach to joint image restoration and segmentation.Appl. Math. Model.3811122014, 30383053URL: http://dx.doi.org/10.1016/j.apm.2013.11.034
 15 articleOn the use of secondorder derivative and metamodelbased MonteCarlo for uncertainty estimation in aerodynamics.Computers and Fluids3762010
 16 articleA stochastic multiple gradient descent algorithm.European Journal of Operational ResearchMay 2018, 10
 17 articlePedestrian motion modelled by FokkerPlanck Nash games.Royal Society open science492017, 170648
 18 articleFinitevolume goaloriented mesh adaptation for aerodynamics using functional derivative with respect to nodal coordinates.Journal of Computational Physics313May 2016, 21
 19 articleMacroscopic modeling and simulations of room evacuation.Appl. Math. Model.38242014, 57815795
 20 articleConstructing analysissuitable parameterization of computational domain from CAD boundary by variational harmonic method.J. Comput. Physics252November 2013
 21 articleFisherKPP 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
 22 articleA survey on highdimensional Gaussian process modeling with application to Bayesian optimization.ACM Transactions on Evolutionary Learning and Optimization222022
 23 articleA nonlocal system modeling bidirectional traffic flows.SEMA SIMAI Springer Series2022
 24 articleMultiScale Model Based Hierarchical Control of Freeway Traffic via Platoons of Connected and Automated Vehicles.IEEE Open Journal of Intelligent Transportation Systems2022
 25 articleInteracting moving bottlenecks in traffic flow.Networks and Heterogeneous Media2022
 26 articleOrdinary differential equations for the adjoint Euler equations.Physics of Fluids348August 2022, 086113
 27 articleA fullyconservative sliding grid algorithm for compressible flows using an Isogeometric Discontinuous Galerkin scheme.Computer Methods in Applied Mechanics and Engineering39515May 2022
 28 articleGeometrically Consistent Aerodynamic Optimization using an Isogeometric Discontinuous Galerkin Method.Computers & Mathematics with Applications128December 2022
 29 articlePolicybased optimization: singlestep policy gradient method seen as an evolution strategy.Neural Computing and ApplicationsSeptember 2022
 30 articleData driven uncertainty quantification in macroscopic traffic flow models.Advances in Computational Mathematics482022
 31 articleSensitivity Prewarping for Local Surrogate Modeling.Technometrics6442022, 535
International peerreviewed conferences
 32 inproceedingsAdvanced numerical modeling methods for the characterization and optimization of metasurfaces.3rd URSI Atlantic / AsiaPacific Radio Science Meeting  2022Gran Canaria, SpainIEEEJuly 2022, 14
 33 inproceedingsCentralized Traffic Control via Small Fleets of Connected and Automated Vehicles.2022 European Control Conference (ECC)London, United Kingdom2022, 371376
 34 inproceedingsCADConsistent Aerodynamic Design via the Isogeometric Paradigm.56th 3AF International Conference on Applied AerodynamicsToulouse, FranceMarch 2022
Conferences without proceedings
 35 inproceedingsAdvanced Numerical Modeling Methods for the Characterization and Optimization of Metasurfaces.ATAPRASC 2022  3rd URSI Atlantic and Asia Pacific Radio Science MeetingGran Canaria, SpainIEEEMay 2022, 14
Scientific books
 36 bookControl Problems for Conservation Laws with Traffic Applications: Modeling, Analysis, and Numerical Methods.PNLDESC  99Progress in Nonlinear Differential Equations and Their Applications. Subseries in ControlBirkhäuser2022
Doctoral dissertations and habilitation theses
 37 thesisDevelopment and investigation of novel models in computational medicine.Université Côte d'Azur; Université Mohamed V, Rabat (Maroc)October 2022
Reports & preprints
 38 miscAdaptive isogeometric analysis using optimal transport.October 2022
 39 miscA portfolio approach to massively parallel Bayesian optimization.May 2022
 40 miscCombined Hormone and Brachy Therapies for the Treatment of Prostate Cancer.October 2022
 41 reportCombining Pareto Optimality with Nash Games in MultiObjective Prioritized Optimization of an Aircraft Flight Performance.RR9490Inria  Sophia Antipolis; AcumesOctober 2022, 29
 42 miscNavigation system based routing strategies in traffic flows on networks.July 2022
 43 miscThe initial boundary value problem for second order traffic flow models with vacuum: existence of entropy weak solutions.October 2022
 44 miscRevisiting MultiLabel Propagation: the Case of Small Data.December 2022
 45 miscA Threeplayer Nash game for pointwise source identification in CauchyStokes problems.January 2023, 114613
12.3 Cited publications
 46 articleA semiintrusive deterministic approach to uncertainty quantification in nonlinear fluid flow problems.J. Comput. Physics2012
 47 articleNonlocal systems of conservation laws in several space dimensions.SIAM Journal on Numerical Analysis5222015, 963983
 48 articleExamples of instability in inverse boundaryvalue problems.Inverse Problems1341997, 887897URL: http://dx.doi.org/10.1088/02665611/13/4/001
 49 articleAn integrodifferential conservation law arising in a model of granular flow.J. Hyperbolic Differ. Equ.912012, 105131
 50 articleOn the Numerical Integration of Scalar Nonlocal Conservation Laws.ESAIM M2AN4912015, 1937
 51 articleOn a nonlocal hyperbolic conservation law arising from a gradient constraint problem.Bull. Braz. Math. Soc. (N.S.)4342012, 599614
 52 articleA FokkerPlanck control framework for multidimensional stochastic processes.Journal of Computational and Applied Mathematics2372013, 487507
 53 articleTime accurate anisotropic goaloriented mesh adaptation for unsteady flows.J. Comput. Physics231192012, 63236348
 54 articleMeasure valued solutions to conservation laws motivated by traffic modelling.Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.46220702006, 17911803
 55 unpublishedUncertainties in traffic flow and model validation on GPS data.2015
 56 articleOn nonlocal conservation laws modelling sedimentation.Nonlinearity2432011, 855885
 57 articleWellposedness of a conservation law with nonlocal flux arising in traffic flow modeling.Numer. Math.13222016, 217241URL: https://doi.org/10.1007/s0021101507176
 58 articleA {PDE} Sensitivity Equation Method for Optimal Aerodynamic Design.Journal of Computational Physics13621997, 366384URL: http://www.sciencedirect.com/science/article/pii/S0021999197957430
 59 articleAnisotropic Organised Eddy Simulation for the prediction of nonequilibrium turbulent flows around bodies.J. of Fluids and Structures2482008, 12401251
 60 articleFlows on networks: recent results and perspectives.EMS Surv. Math. Sci.112014, 47111
 61 articleMean field games with nonlinear mobilities in pedestrian dynamics.Discrete Contin. Dyn. Syst. Ser. B1952014, 13111333
 62 articleIndividual based and meanfield modelling of direct aggregation.Physica D2602013, 145158
 63 techreportValidation of traffic flow models on processed GPS data.Research Report RR83822013
 64 unpublishedA local version of the Hughes model for pedestrian flow.2015, Preprint
 65 unpublishedA conservative scheme for nonclassical solutions to a strongly coupled PDEODE problem.2015, Preprint
 66 articleConvex formulations of data assimilation problems for a class of HamiltonJacobi equations.SIAM J. Control Optim.4922011, 383402
 67 articleLaxHopf Based Incorporation of Internal Boundary Conditions Into HamiltonJacobi Equation. Part II: Computational Methods.Automatic Control, IEEE Transactions on555May 2010, 11581174
 68 articleA Class Of Nonloval Models For Pedestrian Traffic.Mathematical Models and Methods in Applied Sciences22042012, 1150023
 69 articleControl of the continuity equation with a non local flow.ESAIM Control Optim. Calc. Var.1722011, 353379
 70 articleNonlocal crowd dynamics models for several populations.Acta Math. Sci. Ser. B Engl. Ed.3212012, 177196
 71 articleA mixed ODEPDE model for vehicular traffic.Mathematical Methods in the Applied Sciences3872015, 12921302
 72 articleOn the micromacro limit in traffic flow.Rend. Semin. Mat. Univ. Padova1312014, 217235
 73 articleDiscussion about traffic junction modelling: conservation laws vs HamiltonJacobi equations.Discrete Contin. Dyn. Syst. Ser. S732014, 411433
 74 articleExistence and uniqueness of measure solutions for a system of continuity equations with nonlocal flow.Nonlinear Differential Equations and Applications NoDEA2012, 115
 75 inproceedingsHow can macroscopic models reveal selforganization in traffic flow?Decision and Control (CDC), 2012 IEEE 51st Annual Conference onDec 2012, 69896994
 76 bookMultiscale modeling of pedestrian dynamics.12MS&A. Modeling, Simulation and ApplicationsSpringer, Cham2014

77
incollectionSolutions in
${L}^{}$ for a conservation law with memory.Analyse mathématique et applicationsMontrougeGauthierVillars1988, 117128  78 articleLargescale dynamics of meanfield games driven by local Nash equilibria.J. Nonlinear Sci.2412014, 93115URL: http://dx.doi.org/10.1007/s0033201391852
 79 articleA front tracking method for a strongly coupled PDEODE system with moving density constraints in traffic flow.Discrete Contin. Dyn. Syst. Ser. S732014, 435447
 80 articleScalar conservation laws with moving constraints arising in traffic flow modeling: an existence result.J. Differential Equations257112014, 40154029
 81 inbookMultipleGradient 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 articleMultiplegradient 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 techreportRévision de l'algorithme de descente à gradients multiples (MGDA) par orthogonalisation hiérarchique.8710INRIAApril 2015
 84 incollectionRobust 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 articleRigorous Derivation of Nonlinear Scalar Conservation Laws from FollowtheLeader Type Models via Many Particle Limit.Archive for Rational Mechanics and Analysis2015
 86 articleMeasurevalued solutions to conservation laws.Arch. Rational Mech. Anal.8831985, 223270
 87 articleModeling crowd dynamics by the meanfield limit approach.Math. Comput. Modelling529102010, 15061520
 88 techreportA Sensitivity Equation Method for Unsteady Compressible Flows: Implementation and Verification.INRIA Research Report No 8739June 2015
 89 articleA sensitivity equation method for fast evaluation of nearby flows and uncertainty analysis for shape parameters.Int. J. of Computational Fluid Dynamics207August 2006, 497512
 90 articleMultiscale stochastic reactiondiffusion modeling: application to actin dynamics in filopodia.Bull. Math. Biol.7642014, 799818URL: http://dx.doi.org/10.1007/s1153801398443
 91 articleParticle methods for pedestrian flow models: from microscopic to nonlocal continuum models.Math. Models Methods Appl. Sci.24122014, 25032523
 92 incollectionFinite volume methods.Handbook of numerical analysis, Vol. VIIHandb. Numer. Anal., VIINorthHolland, Amsterdam2000, 7131020
 93 techreportConstruction of approximate entropy measure valued solutions for systems of conservation laws.201433Seminar for Applied Mathematics, ETH Zürich2014
 94 articleConvergence 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
 95 inproceedingsGraded 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)21999
 96 articleMultiscale reactiondiffusion algorithms: PDEassisted Brownian dynamics.SIAM J. Appl. Math.7332013, 12241247
 97 articleCoupling of microscopic and phase transition models at boundary.Netw. Heterog. Media832013, 649661
 98 bookTraffic flow on networks.1AIMS Series on Applied MathematicsConservation laws modelsAmerican Institute of Mathematical Sciences (AIMS), Springfield, MO2006
 99 articleA mixed system modeling twodirectional pedestrian flows.Math. Biosci. Eng.1222015, 375392
 100 unpublishedA traffic flow model with nonsmooth metric interaction: wellposedness and micromacro limit.2015, PreprintURL: http://arxiv.org/abs/1510.04461
 101 articleWellposedness and finite volume approximations of the LWR traffic flow model with nonlocal velocity.Netw. Heterog. Media1112016, 107121
 102 articleModeling, simulation and validation of material flow on conveyor belts.Applied Mathematical Modelling38132014, 32953313
 103 articleAchieving logarithmic growth of temporal and spatial complexity in reverse automatic differentiation.Optimization Methods and Software11992, 3554
 104 articleRegularity theory and adjointbased optimality conditions for a nonlinear transport equation with nonlocal velocity.SIAM J. Control Optim.5242014, 21412163
 105 articleNeumannDirichlet Nash strategies for the solution of elliptic Cauchy problems.SIAM J. Control Optim.5152013, 40664083
 106 articleNash strategies for the inverse inclusion CauchyStokes problem.Inverse Problems and Imaging 1342019, 36
 107 unpublishedCoupled data recovery and shape identification : Nash games for the nonlinear CauchyStokes case.December 2021, working paper or preprint
 108 articleOn sensitivity of RANS simulations to uncertain turbulent inflow conditions.Computers & Fluids61252012
 109 articleSelforganizing pedestrian movement.Environment and planning B2832001, 361384
 110 articleTraffic and related selfdriven manyparticle systems.Rev. Mod. Phys.7342001, 10671141
 111 articleEvaluation of traffic data obtained via GPSenabled mobile phones: The Mobile Century field experiment.Transportation Research Part C: Emerging Technologies1842010, 568583
 112 articleContinuum modelling of pedestrian flows: From microscopic principles to selforganised macroscopic phenomena.Physica A: Statistical Mechanics and its Applications41602014, 684694
 113 articleA continuous sensitivity equation method for timedependent incompressible laminar flows.Int. J. for Numerical Methods in Fluids502004, 817844
 114 articleIsogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement.Computer Methods in Applied Mechanics and Engineering1942005, 41354195
 115 articleFluxlimited solutions for quasiconvex HamiltonJacobi equations on networks.arXiv preprint arXiv:1306.2428October 2014
 116 articleSuboptimal feedback control of flow over a sphere.Int. J. of Heat and Fluid Flow312010
 117 articleA 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 articleUncertainty propagation in CFD using polynomial chaos decomposition.Fluid Dynamics Research389September 2006, 616640
 119 articleNonOscillatory Central Schemes for a Traffic Flow Model with Arrehenius LookAhead Dynamics.Netw. Heterog. Media432009, 431451
 120 articleOn a mean field game approach modeling congestion and aversion in pedestrian crowds.Transportation Research Part B: Methodological45102011, 15721589
 121 articleMean field games.Jpn. J. Math.212007, 229260
 122 articleOn kinematic waves. II. A theory of traffic flow on long crowded roads.Proc. Roy. Soc. London. Ser. A.2291955, 317345
 123 articlePredicting shock dynamics in the presence of uncertainties.Journal of Computational Physics2172006, 260276
 124 articleOn the use of secondorder derivative and metamodelbased MonteCarlo for uncertainty estimation in aerodynamics.Computers and Fluids3762010
 125 articleInvivo elastography in animal models: Feasibility studies, (abstract). J. Ultrasound Med.21982002
 126 articleMultilevel Monte Carlo finite volume methods for uncertainty quantification in nonlinear systems of balance laws.Lecture Notes in Computational Science and Engineering922013, 225294
 127 articleIssues in Computational Fluid Dynamics code verification and validation.AIAA Journal361998, 687695
 128 bookTransport equations in biology.Frontiers in MathematicsBirkhäuser Verlag, Basel2007
 129 articleTransport equation with nonlocal velocity in Wasserstein spaces: convergence of numerical schemes.Acta Appl. Math.1242013, 73105
 130 techreportStochastic Multi Gradient Descent Algorithm.ONERAJuly 2014
 131 articleFirst order mean field games in crowd dynamics.ArXiv eprintsFebruary 2014
 132 inproceedingsApproach for uncertainty propagation and robust design in CFD using sensitivity derivatives.15th AIAA Computational Fluid Dynamics ConferenceAIAA Paper 20012528Anaheim, CAJune 2001
 133 incollectionRiemannian BFGS Algorithm with Applications.Recent Advances in Optimization and its Applications in EngineeringSpringer Berlin Heidelberg2010, 183192URL: http://dx.doi.org/10.1007/9783642125980_16
 134 articleAdjointbased optimization on a network of discretized scalar conservation law PDEs with applications to coordinated ramp metering.J. Optim. Theory Appl.16722015, 733760
 135 articleShock waves on the highway.Operations Res.41956, 4251
 136 bookLarge Eddy Simulation for Incompressible Flows An Introduction.Springer Berlin Heidelberg2006
 137 inproceedingsUncertainty Quantification of Turbulence Model Closure Coefficients for Transonic WallBounded Flows.22nd AIAA Computational Fluid Dynamics Conference, 2226 June 2015, Dallas, USA.2015
 138 articleA hybrid model for traffic flow and crowd dynamics with random individual properties.Math. Biosci. Eng.1222015, 393413
 139 articleStochastic modeling and simulation of traffic flow: asymmetric single exclusion process with Arrhenius lookahead dynamics.SIAM J. Appl. Math.6632006, 921944
 140 articleDetachedEddy Simulation.Annual Review of Fluid Mechanics412009, 181202
 141 inproceedingsHigh Order Stochastic Finite Volume Method for the Uncertainty Quantification in Hyperbolic Conservtion Laws with Random Initial Data and Flux Coefficients.Proc. ECCOMASProc. ECCOMAS2012
 142 inproceedingsSensitivity and Uncertainty Analysis for Variable Property Flows.39th AIAA Aerospace Sciences Meeting and ExhibitAIAA Paper 20010139Reno, NVJan. 2001
 143 bookOptimal transport.338Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]Old and newSpringerVerlag, Berlin2009
 144 bookTopics in optimal transportation.58Graduate Studies in MathematicsAmerican Mathematical Society, Providence, RI2003
 145 techreportUncertainty analysis for fluid mechanics with applications.20021ICASEFebruary 2002
 146 articleModeling uncertainty in flow simulations via generalized Polynomial Chaos.Journal of Computational Physics1872003, 137167
 147 articleActive control of flow separation over an airfoil using synthetic jets.J. of Fluids and Structures242008, 13491357
 148 articleMetaModelAssisted MGDA for MultiObjective Functional Optimization.Computers and Fluids102http://www.sciencedirect.com/science/article/pii/S0045793014002576#2014, 116130