2023Activity reportProjectTeamACUMES
RNSR: 201521161R Research center Inria Centre at Université Côte d'Azur
 In partnership with:Université Côte d'Azur
 Team name: Analysis and Control of Unsteady Models for Engineering Sciences
 In collaboration with:Laboratoire JeanAlexandre Dieudonné (JAD)
 Domain:Applied Mathematics, Computation and Simulation
 Theme:Numerical schemes and simulations
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, PostDoctoral Fellow, until Aug 2023]
 Khadija Musayeva [INRIA, PostDoctoral Fellow]
 Enrico Siri [INRIA, PostDoctoral Fellow]
PhD Students
 Mustapha Bahari [UNIV MOHAMMED VI POLYTECH]
 Ilaria Ciaramaglia [INRIA, from Sep 2023]
 Agatha Joumaa [IFPEN]
 Carmen Mezquita Nieto [INRIA, from Sep 2023]
 Nathan Ricard [INRIA, from Nov 2023]
 Alexandra Wuerth [INRIA]
Interns and Apprentices
 Guillaume Coulaud [INRIA, Intern, from Apr 2023 until Sep 2023]
Administrative Assistant
 Quentin Campeon [INRIA]
Visiting Scientist
 Alessandra Rizzo [UNIV DEGLI STUDI DI MESSINA, from Mar 2023 until May 2023]
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 116, or to the dynamics of turbulent flows for which large scale / small scale vortical structures interfere 143.
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 86 for an existence result and 71, 85 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 77, 103 for some applications in traffic flow modelling, and 96, 100, 102 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 71, 85.
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 78, 91 for the derivation of LighthillWhithamRichards 129, 142 traffic flow model from FollowtheLeader and 97 for results on crowd motion models (see also 119). 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 92, and extensively used since then to prove convergence of approximate solutions and deduce existence results, see for example 98 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 59, 99. We refer, for example, to the notion of solution for nonhyperbolic systems 105, 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 99.
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 82), and proved to be successful for studying emerging selforganised flow patterns 81. The theoretical measure framework proves to be also relevant in addressing micromacro limiting procedures of mean field type 106, 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 151, 150. Indeed, as observed in 136, 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 128). 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 66, 67, 93, 127, 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 138, 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 as the one developed in 84 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, 57) 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 54, sedimentation 61, supply chains 110, conveyor belts 108, biological applications like structured populations dynamics 135, or more general problems like gradient constrained equations 56. Also, nonlocal in time terms arise in conservation laws with memory, starting from 83. 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 62, 69, 74, 107, 146. 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 62, 107 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 55 for scalar equations in one space dimension ($N=d=1$), in 75 for scalar equations in several space dimensions ($N=1$, $d\ge 1$) and in 52, 76, 80 for multidimensional systems of conservation laws. Besides, specific finite volume numerical methods have been developed recently in 52, 107 and 126.
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 68. In aerodynamics, inflow conditions like velocity modulus and direction, are subject to fluctuations 115, 134. For some engineering problems, the geometry of the boundary can also be uncertain, due to structural deformation, mechanical wear or disregard of some details 95. 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 144, 149, 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 145, 148, 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 133. 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 115, 125, 130, 153 or an entropy closure method 90, or by discretizing the probability space and extending the numerical schemes to the stochastic components 51. 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 139, 149, 152.
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 60. Besides, we plan to extend previous works on sensitivity analysis 95, 131 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 63 (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 149. 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 with respect to 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 141 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 58, 109 for the backward time integration of the adjoint variables. The sensitivity equation method (SEM) offers a promising alternative 94, 120, 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 95.
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) 143 , DetachedEddy Simulation (DES) 147 or OrganizedEddy Simulation (OES) 64, are more and more employed to analyze 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 123 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 analysis121 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 8887. Originally, we have stated a principle according to 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 have access to 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 89 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 137 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 155.
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 101. 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 53 and still completely open for systems like 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 154.
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 117. Further improvements require more advanced models, keeping into better account interactions at the microscopic scale, and adapted control techniques, see 65 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 79, 104, which limit their applicability to real world situations. HamiltonJacobi equations could be also an interesting direction of research, following the recent results obtained in 122.
 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 118 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 73, 72), 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 86, 103 or via the HamiltonJacobi approach 73, 72 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 from 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 132. 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 illposed inverse problem. In previous works 111, 124, we have demonstrated that Nash game approaches are efficient to tackle illposedness. We intend to extend the results obtained for Laplace equations in 111, 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. 140, 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 degrees of freedom. 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 New software, platforms, open data
6.1 New software
6.1.1 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
6.1.2 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
6.1.3 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.
 URL:

Contact:
Abderrahmane Habbal

Participants:
Régis Duvigneau, JeanLuc Szpyrka, David Rey, Clément Welsch, Abderrahmane Habbal
6.1.4 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. Some fixing of module names and comments is ongoing.

Contact:
Abderrahmane Habbal

Partner:
Université Côte d'Azur (UCA)
6.1.5 pinnacle

Name:
PhysicsInformed Neural Networks Computational Library and Environment

Keywords:
Neural networks, Partial differential equation, Physical simulation, Data assimilation, Inverse problem, Multiphysics modelling

Scientific Description:
Set of methods for rapid implementation of physicsinformed neural networks to solve direct and inverse problems: spacetime sampling with refinement algorithms, dense multilayer neural networks, library of physical models (mechanics, fluid, heat transfer, electromagnetics), optimisation algorithms, import/export tools for meshes and solutions.

Functional Description:
Software library for implementation of physics informed neural networks.

Contact:
Régis Duvigneau

Participants:
Régis Duvigneau, Stéphane Lanteri, Alexis Gobe, Maxime Le
7 New results
7.1 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], Alessandra Rizzo [Univ. Messina, Italy], Enrico Siri, Fabio Vicini [Polytechnic of Turin, Italy].
Traffic control by Connected and Automated Vehicles
We rely on a multiscale approach to model mixed traffic composed of a small fleet of CAVs in the bulk flow. In particular, CAVs are allowed to overtake (if on distinct lanes) or queuing (if on the same lane). Controlling CAVs desired speeds allows to act on the system to minimize the selected cost function. For the proposed control strategies, we apply both global optimization and a Model Predictive Control approach. In particular, we perform numerical tests to investigate how the CAVs number and positions impacts the result, showing that few, optimally chosen vehicles are sufficient to significantly improve the selected performance indexes, even using a decentralized control policy. Simulation results support the attractive perspective of exploiting a very small number of vehicles as endogenous control actuators to regulate traffic flow on road networks, providing a flexible alternative to traditional control methods. Moreover, we compare the impact of the proposed control strategies (decentralized, quasidecentralized, centralized). See 43.
In the aim of modeling the formation of stopandgo waves (to be controlled employing CAVs), in 46 we prove the existence of weak solutions for a class of second order traffic models with relaxation, without requiring the subcharacteristic stability condition to hold. Therefore, large oscillations may arise from small perturbations of equilibria, capturing the formation of stopandgo waves observed in reality. An analysis of the corresponding travelling waves completes the study.
Traffic flow predictions by statistical approaches
In the framework of A. Würth's PhD thesis 40, we propose a physics informed statistical framework for traffic travel time prediction. On one side, the discrepancy of the considered mathematical model is represented by a Gaussian process. On the other side, the traffic simulator is fed with boundary data predicted by a Gaussian process, forced to satisfy the mathematical equations at virtual points, resulting in a multiobjective optimization problem. This combined approach has the merit to address the shortcomings of the purely modeldriven or datadriven approaches, while leveraging their respective advantages. Indeed, models are based on physical laws, but cannot capture all the complexity of real phenomena. On the other hand, pure statistical outputs can violate basic characteristic dynamics. We validate our approach on both synthetic and real world data, showing that it delivers more reliable results compared to other methods, see 36. This approach is further extended to traffic prediction in 48, showing promising results on both synthetic and real world data.
Besides, in 49 we extend the finite volume numerical scheme proposed by Hilliges and Weidlich to second order traffic flow models consisting in $2\times 2$ systems of non strictly hyperbolic conservation laws of Temple class. The scheme is shown to satisfy some maximum principle properties on the density. We provide numerical tests illustrating the behaviour at vacuum and the gain in computational time when dealing with optimization algorithms.
Routing strategies in traffic flows on networks
In the framework of A. Joumaa's PhD thesis, 33 presents a macroscopic multiclass traffic flow model on road networks that accounts for an arbitrary number of vehicle classes with different free flow speeds. A comparison of the Eulerian and Lagrangian formulations is proposed, with the introduction of a new CourantFriedrichsLewy condition. In particular, the ${L}^{1}$error and the computational times are used to compare the performance of the two formulations and show that the Eulerian formulation outperforms the Lagrangian. The paper then extends the Eulerian formulation to traffic networks, providing a general implementation of the dynamics at junctions. We finally simulate the effect of traffic measures and policies, such as route guidance and modal shift, on total travel time and network throughput, which shows that the proposed multiclass model correctly depicts the interactions among classes and it can be used to model such behaviors in complex networks.
In 26, 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.
In 34, we address a pseudoSystem Optimum Dynamic Traffic Assignment optimization problem on road networks relying on trajectory control over a portion of the flows and limited knowledge on user response. The fractions of controlled flow moving between each origindestination couple are defined as "compliant", while the remaining portions, consisting of users free to make their own individual choices, are defined as "noncompliant". The objective is to globally improve the state of the network by controlling a varying subset of compliant traffic flows. On one hand, the selfish response of noncompliant users to changing traffic conditions is computed at each time step by updating the class related turn ratios accordingly to a discretechoice multinomial Logit model to represent users imperfect information. On the other hand, the control action is actuated by varying the flow rates over a precomputed set of routes while the coupled optimization problem takes into account an a priori fixed distribution of users at the nodes. We show how the effectiveness of the resulting finite horizon optimal control problem degrades by not considering the dynamic response of noncompliant users and how it varies according to the fraction of compliant ones. The goal of the the partial control optimization problem is to globally improve the network congestion level by rerouting a variable fraction of flows over a set of precomputed routes. The fraction of controlled users varies according to the tradeoff between the rerouting effort and the network status improvement. Results on a synthetic network are then presented and discussed in 35.
7.2 Nonlocal pedestrian flow models
Participants: Paola Goatin, Daniel Inzunza, Luis Miguel Villada [U Bìo Bìo, Chile].
In the framework of the MathAmSud project NOTION, we propose a nonlocal macroscopic pedestrian flow model for two populations with different destinations trying to avoid each other in a confined environment, where the nonlocal term accounts for anisotropic interactions, mimicking the effect of different cones of view, and the presence of walls or other obstacles in the domain. In particular, obstacles can be incorporated in the density variable, thus avoiding to include them in the vector field of preferred directions. In order to compute the solution, we propose a Finite Difference scheme that couples highorder WENO approximations for spatial discretization, a multistep TVD method for temporal discretization, and a highorder numerical derivative formula to approximate the derivatives of nonlocal terms, and in this way avoid excessive calculations. Numerical tests confirm that each population manages to evade both the presence of the obstacles and the other population. The evacuation time problem is studied, in particular, the optimal position of the obstacles is obtained using a total travel time optimization processes, see 45, 44.
7.3 MFG for twoclass traffic flows
Participants: Abderrahmane Habbal, Amal Machtalay [U Mohamed VI Polytech, Morocco] (UM6P), Imad Kissami [UM6P], Ahmed Ratnani [UM6P].
We have explored a multiclass traffic model and examined the computational feasibility of meanfield games (MFG) in obtaining approximate Nash equilibria for traffic flow games involving a large number of players. We introduced a twoclass traffic meanfield game framework, building upon classical multiclass formulations. To facilitate our analysis, we employed various numerical techniques, including highperformance computing and regularization of LGMRES solvers. By utilizing these tools, we conducted simulations at significantly larger spatial and temporal scales.
We led extensive numerical experiments considering three different scenarios involving cars and trucks, as well as three different cost functionals. Our results primarily focused on the dynamics of autonomous vehicles (AVs) in traffic, yielding results which support the effectiveness of the approach.
Moreover, we conducted original comparisons between macroscopic Nash meanfield speeds and their microscopic counterparts. These comparisons allowed us to computationally validate the $\u03f5$Nash approximation, demonstrating a slightly improved convergence rate compared to theoretical expectations.
Future directions encompass second order traffic models, the multilane case, particularly prone to noncooperative game considerations, and addressing some theoretical issues, see 113.
7.4 Learning strategies for PDEs
Participants: Guillaume Coulaud, Régis Duvigneau, Paola Goatin, Daniel Inzunza, Nathan Ricard, Maxime Le [SED Centre Inria d'Université Côte d'Azur], Adrien Bousseau [GraphDeco ProjectTeam], Guillaume Cordonnier [GraphDeco ProjectTeam], Nicolas Rosset [GraphDeco ProjectTeam].
We investigate the use of novel machine learning paradigms in the context of complex PDE systems, including the following research axes:

Interactive car design using datadriven flow model
The design of car shapes requires a delicate balance between aesthetic and performance. While fluid simulation provides the means to evaluate the aerodynamic performance of a given shape, its computational cost hinders its usage during the early explorative phases of design, when aesthetic is decided upon. We present an interactive system to assist designers in creating aerodynamic car profiles. Our system relies on a neural surrogate model, trained using a simulation database, to predict fluid flow around car shapes, providing fluid visualization and shape optimization feedback to designers as soon as they sketch a car profile. We architectured our model to support gradientbased shape optimization within a learned latent space of car profiles 31. This work is carried out in collaboration with GraphDeco ProjectTeam, in the context of Nicolas Rosset's PhD thesis.

A PINN approach for traffic state estimation and model calibration based on loop detector flow data
In 32, we analyze the performances of a Physics Informed Neural Network (PINN) strategy applied to traffic state estimation and model parameter identification in realistic situations. The traffic dynamics is modeled by a first order macroscopic traffic flow model involving two physical parameters and an auxiliary one. Besides, observations consist of (averaged) density and flow synthetic data computed at fixed space locations, simulating real loop detector measurements. We show that the proposed approach is able to give a good approximation of the underlying dynamics even with poorer information. Moreover, the precision generally improves as the number of measurement locations increases.

Multiphysics coupling using physicsinformed neural networks
PhysicsInformed Neural Networks (PINNs) have emerged as a promising paradigm for modeling complex physical phenomena, offering the potential to handle diverse scenarios to simulate coupled systems. This is a supervised or unsupervised deep learning approach that aims at learning physical laws described by partial differential equations. We consider an exploration of PINNs for multiphysics applications, by embedding the different PDE models and coupling conditions in a single learning task, through three distinct test cases: heat transfer, and conjugate heat transfer, with forced and natural convection. The investigations reveal PINNs' proficiency in accommodating parameterized resolution, addressing piecewise constant conditions, and enabling multiphysics coupling. Despite their versatility, challenges emerged, including difficulties in achieving high accuracy, error propagation near singularities, and limitations in scenarios with high Rayleigh values 42. This activity is part of Guillaume Coulaud's Master thesis and Nathan Ricard's PhD thesis.

Turbulence characterization using physicsinformed neural networks
Turbulence modeling is still a major issue in complex flow simulations, due to the limitations of turbulence models in terms of application range. Physics informed neural networks offer a promising framework to overcome this difficulty, by allowing to build simulation tools based on both PDE models and experimental data. Thus, we investigate this approach to simulate turbulent flows including data in replacement to classical turbulence closures.
The two latter activities benefit from SED support for the development of pinnacle software (6.1.5) devoted to PINNs.
7.5 Advanced Bayesian optimization
Participants: Mickaël Binois, Nicholson Collier [Argonne, USA], Régis Duvigneau, Mahmoud Elsawy [Atlantis team], Patrice Genevet [Colorado School of Mines, USA], Enzo Isnard [Atlantis team], Stéphane Lanteri [Atlantis team], Jonathan Ozik [Argonne, USA], Victor Picheny [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 50.
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 41.
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.
A game theoretic perspective on Bayesian multiobjective optimization
In 38, a book chapter, we address the question of how to efficiently solve manyobjective optimization problems in a computationally demanding blackbox simulation context. We motivate the question by applications in machine learning and engineering, and discuss specific harsh challenges in using classical Pareto approaches when the number of objectives is four or more. Then, we review solutions combining approaches from Bayesian optimization, e.g., with Gaussian processes, and concepts from game theory like Nash equilibria, KalaiSmorodinsky solutions and detail extensions like NashKalaiSmorodinsky solutions. We finally introduce the corresponding algorithms and provide some illustrating results.
7.6 Complex data analysis
Participants: Mickaël Binois, Khadija Musayeva.
In the context of the analysis of complex data sets, such as those appearing in biology, we considered two different questions. The first one is related to label learning, that is, learning missing labels from other available variables and labels. The second considers dimension reduction, to find a common set of new variables when many outputs are present.
In 39, the 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.
In 47, we propose several approaches as baselines to compute a shared active subspace for multivariate vectorvalued functions. The goal is to minimize the deviation between the function evaluations on the original space and those on the reconstructed one. This is done either by manipulating the gradients or the symmetric positive (semi)definite (SPD) matrices computed from the gradients of each component function so as to get a single structure common to all component functions. These approaches can be applied to any data irrespective of the underlying distribution unlike the existing vectorvalued approach that is constrained to a normal distribution. We test the effectiveness of these methods on five optimization problems. The experiments show that, in general, the SPDlevel methods are superior to the gradientlevel ones, and are close to the vectorvalued approach in the case of a normal distribution. Interestingly, in most cases it suffices to take the sum of the SPD matrices to identify the best shared active subspace.
7.7 Pareto optimality and Nash games
Participants: Mickaël Binois, 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].
In the multiobjective optimization of a complex system, establishing the Pareto front associated with the whole set of cost functions is usually a computationally demanding task, whose results are not always easy to analyze, while the final decision still remains to be made among Paretooptimal solutions. These observations had led us to propose a prioritized approach in which the Pareto front is calculated only for a subset of primary cost functions, those of preponderant importance, followed by an economical and decisive step in which a continuum of Nash equilibria accounting for secondary functions is calculated 7.
The method had 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) 12. These results have been presented at a Conference on “New Greener and Digital Modern Transport” (JyU., Finland, May 2023), and recently completed by Bayesian optimization and are currently in press for proceedings,
That work reflects our cooperation with the Information Processing and Systems Department (DTIS) of Onera Toulouse. It will be continued to account for additional criteria related to environmental impact and operational performance.
7.8 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 112 to tackle illposed 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 30
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 114.
7.9 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. 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. See 70 and 24.
7.10 Optimal transport and isogeometric analysis
Participants: Abderrahmane Habbal, Mustapha Bahari [U Mohamed VI Polytech, Morocco] (UM6P), Ahmed Ratnani [UM6P], Eric Sonnendrücker [Max Planck Institute].
In this work, we devise fast solvers and adaptive mesh generation procedures based on the Monge–Ampère Equation using BSplines Finite Elements, within the Isogeometric Analysis framework. Our approach ensures that the constructed mapping is a bijection, which is a major challenge in Isogeometric Analysis. First, we use standard BSplines Finite Elements to solve the Monge–Ampère Equation. An analysis of this approach shows serious limitations when dealing with high variations near the boundary. In order to solve this problem, a new formulation is derived using compatible BSplines discretization based on a discrete DeRham sequence. A new fast solver is devised in this case using the Fast Diagonalization method. Different tests are provided and show the performance of our new approach, see 23.
8 Bilateral contracts and grants with industry
8.1 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.
9 Partnerships and cooperations
9.1 International initiatives
9.1.1 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 Bio Bio
 Universidad de Concepcion
 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.
9.2 International research visitors
9.2.1 Visits of international scientists
Other international visits to the team
Alessandra Rizzo

Status:
PhD student

Institution of origin:
Università di Messina

Country:
Italy

Dates:
MarchMay, 2023 (3 months)

Context of the visit:
collaboration on second order traffic models with relaxation

Mobility program/type of mobility:
research stay
Elena Rossi

Status:
Associate Professor

Institution of origin:
Università di Modena  Reggio Emilia

Country:
Italy

Dates:
April and September, 2023 (1 week each)

Context of the visit:
collaboration on applications of conservation laws to traffic problems

Mobility program/type of mobility:
research stay
9.3 European initiatives
9.3.1 Horizon Europe
DATAHYKING
Participants: Paola Goatin, Ilaria Ciaramaglia, Carmen Mezquita Nieto.
DATAHYKING project on cordis.europa.eu

Title:
Datadriven simulation, uncertainty quantification and optimization for hyperbolic and kinetic models

Duration:
From March 1, 2023 to February 28, 2027

Partners:
 TRANSPORT & MOBILITY LEUVEN NV (TML), Belgium
 Autovie Venete S.p.A., Italy
 INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET AUTOMATIQUE (INRIA), France
 UNIVERSITE COTE D'AZUR, France
 RHEINISCHWESTFAELISCHE TECHNISCHE HOCHSCHULE AACHEN (RWTH AACHEN), Germany
 ESI GROUP (ESI SOFTWARE PAM SYSTEM INTERNATIONAL PSI), France
 Cassa di Compensazione e Garanzia s.p.a. (CC&G), Italy
 NEOVYA Mobility by Technology (NEOVYA Mobility by Technology), France
 INCICO (INICIO SPA), Italy
 SIEMENS INDUSTRY SOFTWARE NETHERLANDS BV (Siemens Industry Software Netherlands B.V.), Netherlands
 RHEINLANDPFALZISCHE TECHNISCHE UNIVERSITAT, Germany
 CENTRE DE RECHERCHE EN AERONAUTIQUE ASBL  CENAERO (CENAERO), Belgium
 UNIVERSITE DE LILLE (UNIVERSITE DE LILLE), France
 ZENSOR (ZENSOR), Belgium
 KATHOLIEKE UNIVERSITEIT LEUVEN (KU Leuven), Belgium
 UNIVERSITA DEGLI STUDI DI ROMA LA SAPIENZA (UNIROMA1), Italy
 UNIVERSITA DEGLI STUDI DI FERRARA (Unife), Italy

Inria contact:
Paola Goatin

Coordinator:
Giovanni Samaey (KU Leuven)

Summary:
Europe faces major challenges in science, society and industry, induced by the complexity of our dynamically evolving world. To tackle these challenges, mathematical models and computer simulations are indispensable, for instance to design and optimize systems using virtual prototypes. Moreover, while the big data revolution provides additional possibilities, it is currently unclear how to optimally combine simulation results with observation data into a digital. Many systems of interest consist of large numbers of particles with highly nontrivial interaction (e.g., fine dust in pollution, vehicles in mobility).
However, to date, computer simulation of such systems is usually done with highly approximate (macroscopic) models to reduce computational complexity. Facing these challenges without sacrificing the complexity of the underlying particle interactions requires a fundamentally new type of scientist that uses an interdisciplinary approach and a solid mathematical underpinning. Hence, we aim at training a new generation of modeling and simulation experts to develop virtual experimentation tools and workflows that can reliably and efficiently exploit the potential of mathematical modeling and simulation of interacting particle systems.
To this end, we create a datadriven simulation framework for kinetic models of interacting particle systems, and define a common methodology for these future modeling and simulation experts. The network focuses on (i) reliable and efficient simulation; (ii) robust consensusbased optimization, also for machine learning; (iii) multifidelity methods for uncertainty quantification and Bayesian inference; and (iv) applications in fluid flow, traffic flow, and finance, also in collaboration with industry. Moreover, the proposed EJD program will create a closely connected new generation of highly demanded European scientists, and initiate longterm partnerships to exploit synergy between academic and industrial partners.
9.3.2 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 modeling and analyzing 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.
9.4 National initiatives
9.4.1 ANR

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.

COSS  COntrol on Stratified Structures (ANR22CE400010, PI Nicolas Forcadel, INSA Rouen): The central theme of this project lies in the area of control theory and partial differential equations (in particular HamiltonJacobi equations), posed on stratified structures and networks. These equations appear very naturally in several applications. Indeed, many practical optimal control problems, such as traffic flow modeling or energy management in smartgrids networks or sealand trajectories with different dynamics, involve a state space in a stratified form (a collection of manifolds with different dimensions and associated to different dynamics). These control problems can be studied within the framework of Hamilton Jacobi equations theory; in particular, they involve admissible trajectories that have to stay in the stratified domain.
Participants: Paola Goatin.
10 Dissemination
10.1 Promoting scientific activities
10.1.1 Scientific events: organisation
General chair, scientific chair
Paola Goatin was member of the scientific committee of the following events:
 “PED2023  Conference on Pedestrian and Evacuation Dynamics”, Eindhoven (Nederlands), 2023.
 Maathrafic  “Modélisation mathématique, analyse et approximation des dynamiques routières et piétonnières”, Tours (France), June 2023.
 MTITS 2023 “8th International Conference on Models and Technologies for Intelligent Transportation Systems”, Nice (France), June 2023.
Member of the organizing committees
 Abderrahmane Habbal is member of the organizing committee of the WinterSchool on Model Reduction Methods for Control and Machine Learning February 2629, 2024 The UM6P Vanguard Center, Benguerir, Morocco.
10.1.2 Scientific events: selection
Member of the conference program committees
 Mickael Binois was part of the program committee of the Evolutionary Multiobjective Optimization (EMO) Track of the Genetic and Evolutionary Computation Conference (GECCO2023) that was held in Lisbonne as well as the NeuroExplicit AI and Expertinformed Machine Learning for Engineering and Physical Sciences workshop at ECML PKDD which was held in Turin.
 Régis Duvigneau was part of the program committee of the 3rd InriaDFKI European Summer School on AI (IDESSAI 2023).
 Paola Goatin was member of the program committee of the 26th IEEE International Conference on Intelligent Transportation Systems (IEEE ITSC 2023).
Reviewer
 Mickael Binois reviewed for the following conferences: AISTATS 2024, ICLR24, ICML 2023 and NeurIPS 2023.
 Paola Goatin reviewed for ACC 2024.
10.1.3 Journal
Member of the editorial boards
 Mickael Binois is Associate Editor of ACM Transactions on Evolutionary Learning and Optimization
 Paola Goatin is Managing Editor of Networks and Heterogeneous Media and Associate Editor of SIAM Journal on Applied Mathematics and ESAIM: Mathematical Modelling and Numerical Analysis.
Reviewer  reviewing activities
 Mickael Binois is a reviewer for the following international journals: European J. Oper. Res, J. Mach. Learn. Res, J Glob Optim, SIAMASA J Uncertain, Int J Numer Methods Eng, Struct Multidiscipl Optim, SIAM J Sci Comput, Technometrics, ACM Transactions on Evolutionary Learning and Optimization, IEEE Trans. Evol. Comput., Transactions on Machine Learning Research.
 Régis Duvigneau is a reviewer for the following international journals: Comp. & Fluids, Comp. Meth. Appl. Mech. Eng., J. Fluids & Struct., J. Opt. Soc. of America, Trans. on Elec. Comp.
 Paola Goatin reviewed for: Advances in Computational Mathematics, Comm. Math. Sci., EURO Journal on Transportation and Logistics, Journal of Differential Equations, Mathematical Methods in the Applied Sciences, SIAM J. Num. Anal., Transportation Science.
 Abderrahmane Habbal reviewed for the following international journals: J. of Math. Biology, Asian J. of Control, ARIMA, J. of Scientific Computing
10.1.4 Invited talks
 Mickael Binois : SIAM conference on Computational Science and Engineering 23, Amsterdam (Netherlands), March 2023. Symposium talk: Massively Parallel Bayesian Optimization.
 Mickael Binois and Régis Duvigneau : hybrid EOLIS workshop, January 2023. Talk: MultiFidelity Models and Optimization using Gaussian Processes.
 Mickael Binois : SIAM Conference on Optimization, Seattle (USA), June 2023. Symposium talk: Scalable Bayesian optimization for noisy problems.
 Mickael Binois : LIKE23 workshop, Bern (Switzerland), June 2023. Invited talk: Kernels for highdimensional Gaussian Process modeling
 Paola Goatin : Workshop “Control Methods in Hyperbolic Partial Differential Equations", Mathematisches Forschungsinstitut, Oberwolfach (Germany), November 2023. Invited talk: Nonlocal macroscopic models of multipopulation pedestrian flows forwalking facilities optimization.
 Paola Goatin : Hirschegg Workshop on Conservation Laws, Hirschegg (Austria), September 2023. Invited talk: Macroscopic traffic flow models for new mobility paradigms.
 Paola Goatin : Traffic and Autonomy Conference, Maiori (Italy), June 2023. Plenary talk: Traffic flow models for new mobility paradigms.
 Paola Goatin : Math 2 Product (M2P) 2023  Emerging Technologies in Computational Science for Industry, Sustainability and Innovation, Taormina (Italy), May 2023. Plenary lecture: Traffic flow models for current and future mobility challenges.
 Abderrahmane Habbal : Mathematics Würzburg Colloquium (GE), May 2023. Plenary lecture Game formulation of coupled data recovery and shape identification for Stokes problems.
 Abderrahmane Habbal : ICIAM 2023, Tokyo : August 2025, 2023. Minisymposium on Modern numerical methods for PDEconstrained optimization and control contributed talk Decentralized strategies for coupled shape and parameter inverse problems
 Abderrahmane Habbal : EuroMaghreb conference at Levico (IT) October 2023, A game theoretic viewpoint on boundary data recovery coupled to shape identification problems
10.1.5 Scientific expertise
 Paola Goatin was member of the committee of the “Fausto Saleri” prize from SIMAI for young researchers.
 Abderrahmane Habbal was sollicitated for the evaluation of a Call for Project INRAEANSES on Mathematical Obesity. November 2023.
10.1.6 Research administration
 Régis Duvigneau is head of the Scientific Committee of Platforms (cluster and immersive space) for Inria Centre at Université Côte d'Azur.
 Régis Duvigneau is member of the Scientific Committee of OPAL computing Platform at Université Côte d'Azur
 Régis Duvigneau is member of the Steering Committee of "Maison de la Simulation et Interactions" at Université Côte d'Azur.
 Paola Goatin is adjunct director of the Doctoral School of Fundamental and Applied Sciences (ED SFA) of Université Côte D’Azur.
 Paola Goatin was member of the Junior Professor Chair (JPC) hiring committee of École des Ponts ParisTech in Applied Mathematics.
 Abderrahmane Habbal was member of the hiring committee for Affiliate Professors (April, November 2023) at University Mohammed VI Polytechnic (Morocco).
 Abderrahmane Habbal is member of the local PhD committee CSD (comité de suivi doctoral) at Inria d'Université Côte d'Azur.
10.2 Teaching  Supervision  Juries
10.2.1 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, 9 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: R. Duvigneau, 3 hrs, 3rd InriaDFKI European Summer School on AI (IDESSAI 2023) , SophiaAntipolis (France), September 2023: “Physicsinformed neural networks for simulation”.
 Master: P. Goatin, projets M1 and M2, 17 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, 24 hrs, M1, Polytech Nice Sophia  Université Côte d'Azur.
 Master: A. Habbal, Stochastic Processes, 24 hrs, M1, Polytech Nice Sophia  Université Côte d'Azur.
 Master: A. Habbal, Introduction to optimization, 15 hrs, M1, Mohammed VI Polytechnic University, Morocco.
 Master: A. Habbal, Fall projects M1, 20 hrs, Polytech Nice Sophia  Université Côte d'Azur.
 Licence (L3): A. Habbal, Mathematical model of addiction, 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.
10.2.2 Supervision
 PhD defense: A. Würth, AI for road traffic modeling and management, Univ. Côte d'Azur/3IA, December 6, 2023. Supervisors: P. Goatin, M. Binois.
 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: I. Ciaramaglia, Interactions between microscopic and macroscopic models for autonomous vehicles in humandriven environments, Univ. Côte d'Azur and Univerità di Roma La Sapienza. Supervisors: P. Goatin, G. Puppo.
 PhD in progress: C. Mezquita Nieto, Modeling and optimization of multimodal transportation networks based on kinetic and hyperbolic equations, Univ. Côte d'Azur and RPTU Kaiserslautern. Supervisors: P. Goatin, A. Klar.
 PhD in progress: N. Rosset, Flow prediction from sketches, Univ. Côte d'Azur. Supervisors: A. Bousseau, G. Cordonnier, R. Duvigneau
 PhD in progress: N. Ricard, Physicsinformed neural networks for multiphysics design, Univ. Côte d'Azur. Supervisor: R. Duvigneau
 PhD in progress: M. Bahari, OptimalMass Transportation for Adaptive Mesh Generation and rRefinement. Univ. Côte d'Azur and Univ. Mohammed VI Polytechnic. Supervisors: A. Habbal, A. Ratnani.
 PhD in progress: A. Machtalay, From Mean Field Games to Agentbased Models and back Univ. Côte d'Azur and Univ. Mohammed VI Polytechnic. Supervisors: A. Habbal, A. Ratnani.
 Master Thesis project (5 months) Amal Amhamdi, Prise en compte de la dépendance en optimisation multiobjectif (Master 2, Polytech Nice Sophia  Université Côte d'Azur). Supervisor: Mickael Binois
 Master Thesis project (6 months) Guillaume Coulaud, ENSEIHT, Physics informed neural networks for multiphysics coupling, Advisor : R. Duvigneau
10.2.3 Juries
 Paola Goatin was referee of A. Hayat's Habilitation thesis “Stabilization of 1D evolution systems”, Université Paris Dauphine, February 8th, 2023.
 Paola Goatin was president of the committee of V.K. Lakshmanan's PhD thesis “Cooperative control of ecodriving trajectories for a fleet of electric connected and autonomous vehicles”, Université ParisSaclay May 23rd, 2023.
 Paola Goatin was referee of N. De Nitti's PhD thesis Analysis, control, and singular limits for hyperbolic conservation laws, FriedrichAlexanderUniversität ErlangenNürnberg, July 24th, 2023.
 Paola Goatin was member of the committee of L. Monasse's Habilitation thesis “Contributions to the simulation of hyperbolic systems in fluid and solid mechanics using computational geometry”, Université Côte d'Azur, October 12th, 2023.
 Paola Goatin was referee of C. Donadello's Habilitation thesis “Some contributions to the analysis of hyperbolic conservation laws”, Université de FrancheComté, December 21st, 2023.
11 Scientific production
11.1 Major publications
 1 articleNonlocal systems of conservation laws in several space dimensions.SIAM Journal on Numerical Analysis5222015, 963983HAL
 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 Mathematik2015HALDOI
 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 2021HALback to text
 8 articleCOOPERATION AND COMPETITION IN MULTIDISCIPLINARY OPTIMIZATION Application to the aerostructural aircraft wing shape optimization.Computational Optimization and Applications5212012, 2968HALDOI
 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 2017HAL
 10 articlePrioritized optimization by Nash games : towards an adaptive multiobjective strategy.ESAIM: Proceedings and Surveys71August 2021, 5463HALDOI
 11 articleMultiplegradient descent algorithm (MGDA) for multiobjective optimization / Algorithme de descente à gradients multiples pour l'optimisation multiobjectif.Comptes Rendus. MathématiqueTome 350Fascicule 56March 2012, 313318HALDOI
 12 reportCombining Pareto Optimality with Nash Games in MultiObjective Prioritized Optimization of an Aircraft Flight Performance.RR9490Inria  Sophia Antipolis; AcumesOctober 2022, 29HALback to text
 13 articleKrigingbased optimization applied to flow control.Int. J. for Numerical Methods in Fluids69112012, 17011714
 14 articleNeumannDirichlet Nash strategies for the solution of elliptic Cauchy problems.SIAM J. Control Optim.5152013, 40664083
 15 articleA Nashgame approach to joint image restoration and segmentation.Appl. Math. Model.3811122014, 30383053URL: http://dx.doi.org/10.1016/j.apm.2013.11.034DOI
 16 articleOn the use of secondorder derivative and metamodelbased MonteCarlo for uncertainty estimation in aerodynamics.Computers and Fluids3762010
 17 articleA stochastic multiple gradient descent algorithm.European Journal of Operational ResearchMay 2018, 10HALDOI
 18 articlePedestrian motion modelled by FokkerPlanck Nash games.Royal Society open science492017, 170648
 19 articleFinitevolume goaloriented mesh adaptation for aerodynamics using functional derivative with respect to nodal coordinates.Journal of Computational Physics313May 2016, 21HALDOI
 20 articleMacroscopic modeling and simulations of room evacuation.Appl. Math. Model.38242014, 57815795
 21 articleConstructing analysissuitable parameterization of computational domain from CAD boundary by variational harmonic method.J. Comput. Physics252November 2013
 22 articleFisherKPP with time dependent diffusion is able to model cellsheet activated and inhibited wound closure.Mathematical biosciences2922017, 3645
11.2 Publications of the year
International journals
 23 articleAdaptive isogeometric analysis using optimal transport and their fast solvers.Computer Methods in Applied Mechanics and Engineering418January 2024, 116570HALDOIback to text
 24 articleCombined Hormone and Brachy Therapies for the Treatment of Prostate Cancer.Mathematical Modelling of Natural PhenomenaNovember 2023HALDOIback to text
 25 articleA nonlocal system modeling bidirectional traffic flows.SEMA SIMAI Springer Series322023, 4966HALDOI
 26 articleNavigation system based routing strategies in traffic flows on networks.Journal of Optimization Theory and Applications1983September 2023, 930957HALDOIback to text
 27 articleInteracting moving bottlenecks in traffic flow.Networks and Heterogeneous Media1822023, 930945HALDOI
 28 articleMacroscopic traffic flow modelling: from kinematic waves to autonomous vehicles.Communications in Applied and Industrial Mathematics141January 2023, 116HALDOI
 29 articleThe initial boundary value problem for second order traffic flow models with vacuum: existence of entropy weak solutions.Nonlinear Analysis: Theory, Methods and Applications233August 2023, 113295HALDOI
 30 articleA Threeplayer Nash game for pointwise source identification in CauchyStokes problems.Journal of Computational and Applied Mathematics417January 2023, 23HALDOIback to text
 31 articleInteractive design of 2D car profiles with aerodynamic feedback.Computer Graphics Forum422February 2023HALDOIback to text
International peerreviewed conferences
 32 inproceedingsA PINN approach for traffic state estimation and model calibration based on loop detector flow data.MTITS 2023  8th International Conference on Models and Technologies for Intelligent Transportation SystemsSaintLaurentDuVar, FranceJune 2023HALback to text
 33 inproceedingsA Macroscopic Model for MultiModal Traffic Flow in Urban Networks.ITSC 2023  26th IEEE International Conference on Intelligent Transportation SystemsBilbao, Spain2023HALback to text
 34 inproceedingsAssessing the impact of noncompliant users response to SystemOptimal Dynamic Traffic Assignment.CDC 2023  62nd IEEE Conference on Decision and ControlSingapore, SingaporeDecember 2023HALback to text
 35 inproceedingsSystemOptimal Dynamic Traffic Assignment with partial users control: an analysis of different strategies.ITSC 2023  26th IEEE International Conference on Intelligent Transportation SystemsBilbao, SpainSeptember 2023HALback to text
 36 inproceedingsValidation of calibration strategies for macroscopic traffic flow models on synthetic data.IEEE PDF eXpress8th International Conference on Models and Technologies for Intelligent Transportation Systems (MTITS 2023)SaintLaurentDuVar, FranceJune 2023HALback to text
Scientific book chapters
 37 inbookThe mathematical theory of Hughes' model : a survey of results.Crowd Dynamics, Volume 4 : Analytics and Human Factors in Crowd ModelingModeling and Simulation in Science, Engineering and TechnologySpringerDecember 2023HAL
 38 inbookA game theoretic perspective on Bayesian multiobjective optimization.ManyCriteria Optimization and Decision Analysis: StateoftheArt, Present Challenges, and Future Perspectives2023HALDOIback to text
Edition (books, proceedings, special issue of a journal)
 39 proceedingsK.Khadija MusayevaM.Mickaël BinoisImproved MultiLabel Propagation for Small Data with MultiObjective Optimization.ECML PKDD 2023Lecture Notes in Computer ScienceSeptember 2023, 284300HALDOIback to text
Doctoral dissertations and habilitation theses
 40 thesisRoad traffic flow reconstruction and prediction with macroscopic models enhanced by databased statistical approaches.Université Côte d'AzurDecember 2023HALback to text
Reports & preprints
 41 miscA portfolio approach to massively parallel Bayesian optimization.2023HALback to text
 42 reportPhysicsInformed Neural Networks for Multiphysics Coupling: Application to Conjugate Heat Transfer.RR9520Université Côte d'Azur, Inria, CNRS, LJADOctober 2023HALback to text
 43 miscTraffic Control via Fleets of Connected and Automated Vehicles.2023HALback to text
 44 miscNonlocal macroscopic models of multipopulation pedestrian flows for walking facilities optimization.August 2023HALback to text
 45 miscNumerical comparison of nonlocal macroscopic models of multipopulation pedestrian flows with anisotropic kernel.2023HALback to text
 46 miscInstabilities in generic second order traffic models with relaxation.December 2023HALback to text
 47 miscShared active subspace for multivariate vectorvalued functions.December 2023HALback to text
 48 miscTraffic prediction by combining macroscopic models and Gaussian processes.December 2023HALback to text
 49 miscA cheap and easytoimplement upwind scheme for second order traffic flow models.2023HALback to text
Other scientific publications
 50 inproceedingsAdvanced Optimization Technique for Robust and Multiobjective Metasurface designs.ICMAT2023  International Conference on Materials for Advanced TechnologiesSingapore (SG), SingaporeJune 2023HALback to text
11.3 Cited publications
 51 articleA semiintrusive deterministic approach to uncertainty quantification in nonlinear fluid flow problems.J. Comput. Physics2012back to text
 52 articleNonlocal systems of conservation laws in several space dimensions.SIAM Journal on Numerical Analysis5222015, 963983HALback to textback to text
 53 articleExamples of instability in inverse boundaryvalue problems.Inverse Problems1341997, 887897URL: http://dx.doi.org/10.1088/02665611/13/4/001DOIback to text
 54 articleAn integrodifferential conservation law arising in a model of granular flow.J. Hyperbolic Differ. Equ.912012, 105131back to text
 55 articleOn the Numerical Integration of Scalar Nonlocal Conservation Laws.ESAIM M2AN4912015, 1937back to text
 56 articleOn a nonlocal hyperbolic conservation law arising from a gradient constraint problem.Bull. Braz. Math. Soc. (N.S.)4342012, 599614back to text
 57 articleA FokkerPlanck control framework for multidimensional stochastic processes.Journal of Computational and Applied Mathematics2372013, 487507back to text
 58 articleTime accurate anisotropic goaloriented mesh adaptation for unsteady flows.J. Comput. Physics231192012, 63236348back to text
 59 articleMeasure valued solutions to conservation laws motivated by traffic modelling.Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.46220702006, 17911803back to text
 60 unpublishedUncertainties in traffic flow and model validation on GPS data.2015back to text
 61 articleOn nonlocal conservation laws modelling sedimentation.Nonlinearity2432011, 855885back to text
 62 articleWellposedness of a conservation law with nonlocal flux arising in traffic flow modeling.Numer. Math.13222016, 217241URL: https://doi.org/10.1007/s0021101507176back to textback to text
 63 articleA {PDE} Sensitivity Equation Method for Optimal Aerodynamic Design.Journal of Computational Physics13621997, 366384URL: http://www.sciencedirect.com/science/article/pii/S0021999197957430DOIback to text
 64 articleAnisotropic Organised Eddy Simulation for the prediction of nonequilibrium turbulent flows around bodies.J. of Fluids and Structures2482008, 12401251back to text
 65 articleFlows on networks: recent results and perspectives.EMS Surv. Math. Sci.112014, 47111back to text
 66 articleMean field games with nonlinear mobilities in pedestrian dynamics.Discrete Contin. Dyn. Syst. Ser. B1952014, 13111333back to text
 67 articleIndividual based and meanfield modelling of direct aggregation.Physica D2602013, 145158back to text
 68 techreportValidation of traffic flow models on processed GPS data.Research Report RR83822013HALback to text
 69 unpublishedA local version of the Hughes model for pedestrian flow.2015, Preprintback to text
 70 phdthesisDevelopment and investigation of novel models in computational medicine.Université Côte d'Azur ; Université Mohamed V, Rabat (Maroc)October 2022HALback to text
 71 unpublishedA conservative scheme for nonclassical solutions to a strongly coupled PDEODE problem.2015, Preprintback to textback to text
 72 articleConvex formulations of data assimilation problems for a class of HamiltonJacobi equations.SIAM J. Control Optim.4922011, 383402back to textback to text
 73 articleLaxHopf Based Incorporation of Internal Boundary Conditions Into HamiltonJacobi Equation. Part II: Computational Methods.Automatic Control, IEEE Transactions on555May 2010, 11581174back to textback to text
 74 articleA Class Of Nonloval Models For Pedestrian Traffic.Mathematical Models and Methods in Applied Sciences22042012, 1150023back to text
 75 articleControl of the continuity equation with a non local flow.ESAIM Control Optim. Calc. Var.1722011, 353379back to text
 76 articleNonlocal crowd dynamics models for several populations.Acta Math. Sci. Ser. B Engl. Ed.3212012, 177196back to text
 77 articleA mixed ODEPDE model for vehicular traffic.Mathematical Methods in the Applied Sciences3872015, 12921302back to text
 78 articleOn the micromacro limit in traffic flow.Rend. Semin. Mat. Univ. Padova1312014, 217235back to text
 79 articleDiscussion about traffic junction modelling: conservation laws vs HamiltonJacobi equations.Discrete Contin. Dyn. Syst. Ser. S732014, 411433back to text
 80 articleExistence and uniqueness of measure solutions for a system of continuity equations with nonlocal flow.Nonlinear Differential Equations and Applications NoDEA2012, 115back to text
 81 inproceedingsHow can macroscopic models reveal selforganization in traffic flow?Decision and Control (CDC), 2012 IEEE 51st Annual Conference onDec 2012, 69896994back to text
 82 bookMultiscale modeling of pedestrian dynamics.12MS&A. Modeling, Simulation and ApplicationsSpringer, Cham2014back to text

83
incollectionSolutions in
${L}^{}$ for a conservation law with memory.Analyse mathématique et applicationsMontrougeGauthierVillars1988, 117128back to text  84 articleLargescale dynamics of meanfield games driven by local Nash equilibria.J. Nonlinear Sci.2412014, 93115URL: http://dx.doi.org/10.1007/s0033201391852DOIback to text
 85 articleA front tracking method for a strongly coupled PDEODE system with moving density constraints in traffic flow.Discrete Contin. Dyn. Syst. Ser. S732014, 435447back to textback to text
 86 articleScalar conservation laws with moving constraints arising in traffic flow modeling: an existence result.J. Differential Equations257112014, 40154029back to textback to text
 87 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, 1back to text
 88 articleMultiplegradient descent algorithm (MGDA) for multiobjective optimization.Comptes Rendus de l'Académie des Sciences Paris3502012, 313318URL: http://dx.doi.org/10.1016/j.crma.2012.03.014back to text
 89 techreportRévision de l'algorithme de descente à gradients multiples (MGDA) par orthogonalisation hiérarchique.8710INRIAApril 2015back to text
 90 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, 105149back to text
 91 articleRigorous Derivation of Nonlinear Scalar Conservation Laws from FollowtheLeader Type Models via Many Particle Limit.Archive for Rational Mechanics and Analysis2015back to text
 92 articleMeasurevalued solutions to conservation laws.Arch. Rational Mech. Anal.8,831985, 223270back to text
 93 articleModeling crowd dynamics by the meanfield limit approach.Math. Comput. Modelling529102010, 15061520back to text
 94 techreportA Sensitivity Equation Method for Unsteady Compressible Flows: Implementation and Verification.INRIA Research Report No 8739June 2015back to text
 95 articleA sensitivity equation method for fast evaluation of nearby flows and uncertainty analysis for shape parameters.Int. J. of Computational Fluid Dynamics207August 2006, 497512back to textback to textback to text
 96 articleMultiscale stochastic reactiondiffusion modeling: application to actin dynamics in filopodia.Bull. Math. Biol.7642014, 799818URL: http://dx.doi.org/10.1007/s1153801398443DOIback to text
 97 articleParticle methods for pedestrian flow models: from microscopic to nonlocal continuum models.Math. Models Methods Appl. Sci.24122014, 25032523back to text
 98 incollectionFinite volume methods.Handbook of numerical analysis, Vol. VIIHandb. Numer. Anal., VIINorthHolland, Amsterdam2000, 7131020back to text
 99 techreportConstruction of approximate entropy measure valued solutions for systems of conservation laws.201433Seminar for Applied Mathematics, ETH Zürich2014back to textback to text
 100 articleConvergence of methods for coupling of microscopic and mesoscopic reactiondiffusion simulations.J. Comput. Phys.2892015, 117URL: http://dx.doi.org/10.1016/j.jcp.2015.01.030DOIback to text
 101 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)21999back to text
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