2023Activity reportProject-TeamACUMES

3.1 Research directions

The project develops along the following two axes:

• modeling complex systems through novel (unconventional) PDE systems, accounting for multi-scale 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 up-to-date 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 multi-scale phenomena and uncertainties

Dynamical models consisting of evolutionary PDEs, mainly of hyperbolic type, appear classically in the applications studied by the previous Project-Team Opale (compressible flows, traffic, cell-dynamics, 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 self-organizing 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.

3.3 Optimization and control algorithms for systems governed by PDEs

The non-classical models described above are developed in the perspective of design improvement for real-life 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.

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 micro-devices, like pulsating jets or vibrating membranes, that can modify vortices generated by vehicles. Thanks to flow non-linearities, 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 high-order 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. Adjoint-based (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 small-scale actuators, we investigate the use of isogeometric analysis methods, which allow to account exactly for CAD-based geometries in a high-order 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 self-organization 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 well-posedness 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. Hamilton-Jacobi 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 Hamilton-Jacobi 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 time-Lagrangian as well as space-Lagrangian 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 micro-macro models  86, 103 or via the Hamilton-Jacobi 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, castration-resistant cells that are able to sustain growth in a low androgen environment emerge. The castration-resistant 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 long-term 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 non-invasive 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 X-ray and comparables techniques, making it a potentially prominent choice for patients.

  Physical principles of thermoelastography originally rely on dynamical structural responses of tissues, but as a first approach, we only consider static responses of linear elastic structures.

  The mathematical formulation of the thermoelasticity reconstruction is based on data completion and material identification, making it a harsh ill-posed inverse problem. In previous works  111, 124, we have demonstrated that Nash game approaches are efficient to tackle ill-posedness. 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 so-called Kohn and Vogelius functional (ongoing work, some promising results obtained).

  - Data recovery in coupled heat-thermoelasticity systems.

  - Data recovery in linear thermoelasticity under stochastic heat flux, where the imposed flux is stochastic.

  - Data recovery in coupled heat-thermoelasticity systems under stochastic heat flux, formulated as an incomplete information Nash game.

  - Application to robust identification of cracks.

• Constraint elimination in Quasi-Newton methods. In single-objective 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 quasi-Newton'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 Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm is well-known and commonly employed. In this method, the Hessian is corrected at each new iteration by rank-one matrices defined from several evaluations of the gradient only. The BFGS method has "super-linear convergence".

  For constrained problems, certain authors have developed so-called 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 ONERA-Meudon, 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 test-case, although in low dimensions, demonstrate the feasibility of this approach. On-going research is on theorizing this method, and testing cases of higher dimensions.

• Multi-objective optimization for nanotechnologies. Our team takes part in a larger collaboration with CEA/LETI (Grenoble), initiated by the Inria Project-Team 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 test-cases of increasing complexity:

  - infrared micro-source;

  - micro-photoacoustic cell;

  - nanophotonic device.

  These cases involve from a few geometric parameters to be optimized to a functional minimization subject to a finite-element 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 meta-models, typically of RBF-type. Multi-objective optimization will be carried out subsequently by MGDA, and possibly Nash games.

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 hormono-radio 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).

6.1 New software

7.1 Macroscopic traffic flow models on networks

Participants: Mickaël Binois, Paola Goatin, Alexandra Würth, Chiara Daini [KOPERNIC Project-Team, 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 multi-scale 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, quasi-decentralized, centralized). See 43.

In the aim of modeling the formation of stop-and-go 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 sub-characteristic stability condition to hold. Therefore, large oscillations may arise from small perturbations of equilibria, capturing the formation of stop-and-go 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 multi-objective optimization problem. This combined approach has the merit to address the shortcomings of the purely model-driven or data-driven 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 multi-class 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 Courant-Friedrichs-Lewy 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 multi-class 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 Hamilton-Jacobi equation to respectively model the nonlinear transportation process and the strategic choices of users. Furthermore, the model is adapted to the multi-population 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 pseudo-System 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 origin-destination couple are defined as "compliant", while the remaining portions, consisting of users free to make their own individual choices, are defined as "non-compliant". 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 non-compliant users to changing traffic conditions is computed at each time step by updating the class related turn ratios accordingly to a discrete-choice 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 non-compliant 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 pre-computed routes. The fraction of controlled users varies according to the trade-off 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 multi-step TVD method for temporal discretization, and a high-order 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 two-class traffic flows

Participants: Abderrahmane Habbal, Amal Machtalay [U Mohamed VI Polytech, Morocco] (UM6P), Imad Kissami [UM6P], Ahmed Ratnani [UM6P].

We have explored a multi-class traffic model and examined the computational feasibility of mean-field games (MFG) in obtaining approximate Nash equilibria for traffic flow games involving a large number of players. We introduced a two-class traffic mean-field game framework, building upon classical multi-class formulations. To facilitate our analysis, we employed various numerical techniques, including high-performance 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 mean-field speeds and their microscopic counterparts. These comparisons allowed us to computationally validate the $ϵ-$Nash approximation, demonstrating a slightly improved convergence rate compared to theoretical expectations.

Future directions encompass second order traffic models, the multi-lane case, particularly prone to non-cooperative 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 Project-Team], Guillaume Cordonnier [GraphDeco Project-Team], Nicolas Rosset [GraphDeco Project-Team].

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 data-driven 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 gradient-based shape optimization within a learned latent space of car profiles 31. This work is carried out in collaboration with GraphDeco Project-Team, 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 physics-informed neural networks

  Physics-Informed 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 physics-informed 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 nano-photonic devices

In collaboration with Atlantis Project-Team, we consider the optimization of optical meta-surface devices, which are able to alter light properties by operating at nano-scale. In the context of Maxwell equations, modified to account for nano-scale 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 time-accurate 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 multi-objective 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 one-at-a-time. For expensive-to-evaluate black-boxes, batch versions of Bayesian optimization have been proposed. They work by building a surrogate model of the black-box 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 black-box 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 trade-off and a portfolio allocation. We compare the approach with related methods on deterministic and noisy functions, for mono- and multi-objective optimization tasks. These experiments show similar or better performance than existing methods, while being orders of magnitude faster.

A game theoretic perspective on Bayesian multi-objective optimization

In 38, a book chapter, we address the question of how to efficiently solve many-objective optimization problems in a computationally demanding black-box 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, Kalai-Smorodinsky solutions and detail extensions like Nash-Kalai-Smorodinsky 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 multi-label learning from small number of labelled data. We demonstrate that the straightforward binary-relevance extension of the interpolated label propagation algorithm, the harmonic function, is a competitive learning method with respect to many widely-used 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 game-theory 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 vector-valued 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 vector-valued 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 SPD-level methods are superior to the gradient-level ones, and are close to the vector-valued 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, Jean-Antoine 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 multi-objective 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 Pareto-optimal 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 multi-objective optimization of the flight performance of an Airbus-A320-type aircraft in terms of take-off fuel mass and operational empty weight (primary cost functions) concurrently with ascent-to-cruise 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 Cauchy-Stokes 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 ill-posed Cauchy-Stokes 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 Cauchy-Stokes problem is ill-posed for both the sources and missing data reconstructions, and designing stable and efficient algorithms is challenging. We reformulate the problem as a three-player Nash game. Thanks to a source identifiability result derived for the Cauchy-Stokes 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 Kohn-Vogelius 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 test-cases.The obtained results corroborate the efficiency of our 3-player Nash game approach to solve parameter or shape identification for Cauchy-Stokes problems 30

The second direction is dedicated to the solution of the data completion problem for non-linear flows. We consider two kinds of non linearities leading to either a non Newtonian Stokes flow or to Navier-Stokes equations. Our recent numerical results show that it is possible to perform a one-shot 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 hormone-dependent cancer characterized by two types of cancer cells, androgen-dependent cancer cells and androgen-resistant 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 non-existent 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 B-Splines 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 B-Splines 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 B-Splines 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.

8.1 Bilateral contracts with industry

• Mycophyto (2020-...): this research contract involving Université Côte d'Azur is financing the post-doctoral contract of Khadija Musayeva. The goal is to develop prediction algorithms based on environmental data.

  Participants: Mickaël Binois, Khadija Musayeva.

9.1 International initiatives

9.2 International research visitors

9.3 European initiatives

9.4 National initiatives

10.1 Promoting scientific activities

10.2 Teaching - Supervision - Juries

11.1 Major publications

• 1 articleA.Aekta Aggarwal, R. M.Rinaldo M. Colombo and P.Paola Goatin. Nonlocal systems of conservation laws in several space dimensions.SIAM Journal on Numerical Analysis5222015, 963-983HAL
• 2 articleB.Boris Andreianov, P.Paola Goatin and N.Nicolas Seguin. Finite volume schemes for locally constrained conservation laws.Numer. Math.1154With supplementary material available online2010, 609--645
• 3 articleS.Sebastien Blandin and P.Paola Goatin. Well-posedness of a conservation law with non-local flux arising in traffic flow modeling.Numerische Mathematik2015HALDOI
• 4 articleR. M.Rinaldo M. Colombo and P.Paola Goatin. A well posed conservation law with a variable unilateral constraint.J. Differential Equations23422007, 654--675
• 5 articleM. L.M. L. Delle Monache and P.Paola Goatin. Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result.J. Differential Equations257112014, 4015--4029
• 6 articleM. L.M. L. Delle Monache, J.J. Reilly, S.S. Samaranayake, W.W. Krichene, P.P. Goatin and A.Alexandre Bayen. A PDE-ODE model for a junction with ramp buffer.SIAM J. Appl. Math.7412014, 22--39
• 7 inproceedingsJ.-A.Jean-Antoine Désidéri. Adaptation by Nash games in gradient-based multi-objective/multi-disciplinary optimization.JANO13 - Mathematical Control and Numerical Applications372Springer Proceedings in Mathematics & Statistics SeriesKhouribga, MoroccoFebruary 2021HALback to text
• 8 articleJ.-A.Jean-Antoine Désidéri. COOPERATION AND COMPETITION IN MULTIDISCIPLINARY OPTIMIZATION Application to the aero-structural aircraft wing shape optimization.Computational Optimization and Applications5212012, 29-68HALDOI
• 9 inbookJ.-A.Jean-Antoine Desideri and R.Régis Duvigneau. Parametric optimization of pulsating jets in unsteady flow by Multiple-Gradient Descent Algorithm (MGDA).Numerical Methods for Differential Equations, Optimization, and Technological Problems, Modeling, Simulation and Optimization for Science and TechnologyJanuary 2017HAL
• 10 articleJ.-A.Jean-Antoine Désidéri and R.Régis Duvigneau. Prioritized optimization by Nash games : towards an adaptive multi-objective strategy.ESAIM: Proceedings and Surveys71August 2021, 54-63HALDOI
• 11 articleJ.-A.Jean-Antoine Désidéri. Multiple-gradient descent algorithm (MGDA) for multiobjective optimization / Algorithme de descente à gradients multiples pour l'optimisation multiobjectif.Comptes Rendus. MathématiqueTome 350Fascicule 5-6March 2012, 313-318HALDOI
• 12 reportJ.-A.Jean-Antoine Désidéri, J.Julien Wintz, N.Nathalie Bartoli, C.Christophe David and S.Sébastien Defoort. Combining Pareto Optimality with Nash Games in Multi-Objective Prioritized Optimization of an Aircraft Flight Performance.RR-9490Inria - Sophia Antipolis; AcumesOctober 2022, 29HALback to text
• 13 articleR.R. Duvigneau and P.P. Chandrashekar. Kriging-based optimization applied to flow control.Int. J. for Numerical Methods in Fluids69112012, 1701-1714
• 14 articleA.A. Habbal and M.M. Kallel. Neumann-Dirichlet Nash strategies for the solution of elliptic Cauchy problems.SIAM J. Control Optim.5152013, 4066--4083
• 15 articleM.Moez Kallel, R.Rajae Aboulaich, A.Abderrahmane Habbal and M.Maher Moakher. A Nash-game approach to joint image restoration and segmentation.Appl. Math. Model.3811-122014, 3038--3053URL: http://dx.doi.org/10.1016/j.apm.2013.11.034DOI
• 16 articleM.M. Martinelli and R.R. Duvigneau. On the use of second-order derivative and metamodel-based Monte-Carlo for uncertainty estimation in aerodynamics.Computers and Fluids3762010
• 17 articleQ.Quentin Mercier, F.Fabrice Poirion and J.-A.Jean-Antoine Desideri. A stochastic multiple gradient descent algorithm.European Journal of Operational ResearchMay 2018, 10HALDOI
• 18 articleS.Souvik Roy, A.A. Borzì and A.Abderrahmane Habbal. Pedestrian motion modelled by Fokker--Planck Nash games.Royal Society open science492017, 170648
• 19 articleG.Giovanni Todarello, F.Floris Vonck, S.Sébastien Bourasseau, J.Jacques Peter and J.-A.Jean-Antoine Desideri. Finite-volume goal-oriented mesh adaptation for aerodynamics using functional derivative with respect to nodal coordinates.Journal of Computational Physics313May 2016, 21HALDOI
• 20 articleM.M. Twarogowska, P.P. Goatin and R.R. Duvigneau. Macroscopic modeling and simulations of room evacuation.Appl. Math. Model.38242014, 5781--5795
• 21 articleG.G. Xu, B.B. Mourrain, A.A. Galligo and R.R. Duvigneau. Constructing analysis-suitable parameterization of computational domain from CAD boundary by variational harmonic method.J. Comput. Physics252November 2013
• 22 articleB.Boutheina Yahyaoui, M.Mekki Ayadi and A.Abderrahmane Habbal. Fisher-KPP with time dependent diffusion is able to model cell-sheet activated and inhibited wound closure.Mathematical biosciences2922017, 36--45

11.2 Publications of the year

11.3 Cited publications

• 51 articleR.R. Abgrall and P. M.P. M. Congedo. A semi-intrusive deterministic approach to uncertainty quantification in non-linear fluid flow problems.J. Comput. Physics2012back to text
• 52 articleA.Aekta Aggarwal, R. M.Rinaldo M. Colombo and P.Paola Goatin. Nonlocal systems of conservation laws in several space dimensions.SIAM Journal on Numerical Analysis5222015, 963-983HALback to textback to text
• 53 articleG.Giovanni Alessandrini. Examples of instability in inverse boundary-value problems.Inverse Problems1341997, 887--897URL: http://dx.doi.org/10.1088/0266-5611/13/4/001DOIback to text
• 54 articleD.Debora Amadori and W.Wen Shen. An integro-differential conservation law arising in a model of granular flow.J. Hyperbolic Differ. Equ.912012, 105--131back to text
• 55 articleP.P. Amorim, R. M.Rinaldo M. Colombo and A.A. Teixeira. On the Numerical Integration of Scalar Nonlocal Conservation Laws.ESAIM M2AN4912015, 19--37back to text
• 56 articleP.Paulo Amorim. On a nonlocal hyperbolic conservation law arising from a gradient constraint problem.Bull. Braz. Math. Soc. (N.S.)4342012, 599--614back to text
• 57 articleM.M. Annunziato and A.A. Borz\`i. A Fokker-Planck control framework for multidimensional stochastic processes.Journal of Computational and Applied Mathematics2372013, 487-507back to text
• 58 articleA.A. Belme, F.F. Alauzet and A.A. Dervieux. Time accurate anisotropic goal-oriented mesh adaptation for unsteady flows.J. Comput. Physics231192012, 6323--6348back to text
• 59 articleS.Sylvie Benzoni-Gavage, R. M.Rinaldo M. Colombo and P.Piotr Gwiazda. Measure valued solutions to conservation laws motivated by traffic modelling.Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.46220702006, 1791--1803back to text
• 60 unpublishedE.Enrico Bertino, R.Régis Duvigneau and P.Paola Goatin. Uncertainties in traffic flow and model validation on GPS data.2015back to text
• 61 articleF.F. Betancourt, R.R. Bürger, K. H.K. H. Karlsen and E. M.E. M. Tory. On nonlocal conservation laws modelling sedimentation.Nonlinearity2432011, 855--885back to text
• 62 articleS.Sebastien Blandin and P.Paola Goatin. Well-posedness of a conservation law with non-local flux arising in traffic flow modeling.Numer. Math.13222016, 217--241URL: https://doi.org/10.1007/s00211-015-0717-6back to textback to text
• 63 articleJ.Jeff Borggaard and J.John Burns. A {PDE} Sensitivity Equation Method for Optimal Aerodynamic Design.Journal of Computational Physics13621997, 366--384URL: http://www.sciencedirect.com/science/article/pii/S0021999197957430DOIback to text
• 64 articleR.R. Bourguet, M.M. Brazza, G.G. Harran and R.R. El Akoury. Anisotropic Organised Eddy Simulation for the prediction of non-equilibrium turbulent flows around bodies.J. of Fluids and Structures2482008, 1240--1251back to text
• 65 articleA.Alberto Bressan, S.Sunċica Ċanić, M.Mauro Garavello, M.Michael Herty and B.Benedetto Piccoli. Flows on networks: recent results and perspectives.EMS Surv. Math. Sci.112014, 47--111back to text
• 66 articleM.Martin Burger, M.Marco Di Francesco, P. A.Peter A. Markowich and M.-T.Marie-Therese Wolfram. Mean field games with nonlinear mobilities in pedestrian dynamics.Discrete Contin. Dyn. Syst. Ser. B1952014, 1311--1333back to text
• 67 articleM.M. Burger, J.J. Haskovec and M.-T.M.-T. Wolfram. Individual based and mean-field modelling of direct aggregation.Physica D2602013, 145--158back to text
• 68 techreportA.Alessandra Cabassi and P.Paola Goatin. Validation of traffic flow models on processed GPS data.Research Report RR-83822013HALback to text
• 69 unpublishedJ. ..J .A. Carrillo, S.S. Martin and M.-T.M.-T. Wolfram. A local version of the Hughes model for pedestrian flow.2015, Preprintback to text
• 70 phdthesisS.Salma Chabbar. Development and investigation of novel models in computational medicine.Université Côte d'Azur ; Université Mohamed V, Rabat (Maroc)October 2022HALback to text
• 71 unpublishedC.Christophe Chalons, M. L.Maria Laura Delle Monache and P.Paola Goatin. A conservative scheme for non-classical solutions to a strongly coupled PDE-ODE problem.2015, Preprintback to textback to text
• 72 articleC. G.Christian G. Claudel and A. M.Alexandre M. Bayen. Convex formulations of data assimilation problems for a class of Hamilton-Jacobi equations.SIAM J. Control Optim.4922011, 383--402back to textback to text
• 73 articleC.C.G. Claudel and A. M.Alexandre M. Bayen. Lax-Hopf Based Incorporation of Internal Boundary Conditions Into Hamilton-Jacobi Equation. Part II: Computational Methods.Automatic Control, IEEE Transactions on555May 2010, 1158-1174back to textback to text
• 74 articleR. M.Rinaldo M. Colombo, M.Mauro Garavello and M.Magali Lécureux-Mercier. A Class Of Nonloval Models For Pedestrian Traffic.Mathematical Models and Methods in Applied Sciences22042012, 1150023back to text
• 75 articleR. M.Rinaldo M. Colombo, M.Michael Herty and M.Magali Mercier. Control of the continuity equation with a non local flow.ESAIM Control Optim. Calc. Var.1722011, 353--379back to text
• 76 articleR. M.Rinaldo M. Colombo and M.Magali Lécureux-Mercier. Nonlocal crowd dynamics models for several populations.Acta Math. Sci. Ser. B Engl. Ed.3212012, 177--196back to text
• 77 articleR. M.Rinaldo M. Colombo and F.Francesca Marcellini. A mixed ODE-PDE model for vehicular traffic.Mathematical Methods in the Applied Sciences3872015, 1292--1302back to text
• 78 articleR. M.Rinaldo M. Colombo and E.E. Rossi. On the micro-macro limit in traffic flow.Rend. Semin. Mat. Univ. Padova1312014, 217--235back to text
• 79 articleG.Guillaume Costeseque and J.-P.Jean-Patrick Lebacque. Discussion about traffic junction modelling: conservation laws vs Hamilton-Jacobi equations.Discrete Contin. Dyn. Syst. Ser. S732014, 411--433back to text
• 80 articleG.Gianluca Crippa and M.Magali Lécureux-Mercier. Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow.Nonlinear Differential Equations and Applications NoDEA2012, 1-15back to text
• 81 inproceedingsE.E. Cristiani, B.B. Piccoli and A.A. Tosin. How can macroscopic models reveal self-organization in traffic flow?Decision and Control (CDC), 2012 IEEE 51st Annual Conference onDec 2012, 6989-6994back to text
• 82 bookE.Emiliano Cristiani, B.Benedetto Piccoli and A.Andrea Tosin. Multiscale modeling of pedestrian dynamics.12MS&A. Modeling, Simulation and ApplicationsSpringer, Cham2014back to text
• 83 incollectionC. M.C. M. Dafermos. Solutions in $L^{}$ for a conservation law with memory.Analyse mathématique et applicationsMontrougeGauthier-Villars1988, 117--128back to text
• 84 articleP.Pierre Degond, J.-G.Jian-Guo Liu and C.Christian Ringhofer. Large-scale dynamics of mean-field games driven by local Nash equilibria.J. Nonlinear Sci.2412014, 93--115URL: http://dx.doi.org/10.1007/s00332-013-9185-2DOIback to text
• 85 articleM. L.Maria Laura Delle Monache and P.Paola Goatin. A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow.Discrete Contin. Dyn. Syst. Ser. S732014, 435--447back to textback to text
• 86 articleM. L.M. L. Delle Monache and P.P. Goatin. Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result.J. Differential Equations257112014, 4015--4029back to textback to text
• 87 inbookJ.-A.J.-A. Désidéri. Multiple-Gradient Descent Algorithm (\em MGDA) for Pareto-Front 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 JubileesSpringer-Verlag2014, 1back to text
• 88 articleJ.-A.Jean-Antoine Désidéri. Multiple-gradient descent algorithm (MGDA) for multiobjective optimization.Comptes Rendus de l'Académie des Sciences Paris3502012, 313-318URL: http://dx.doi.org/10.1016/j.crma.2012.03.014back to text
• 89 techreportJ.-A.Jean-Antoine Désidéri. Révision de l'algorithme de descente à gradients multiples (MGDA) par orthogonalisation hiérarchique.8710INRIAApril 2015back to text
• 90 incollectionB.Bruno Després, G.Gaël Poëtte and D.Didier Lucor. Robust uncertainty propagation in systems of conservation laws with the entropy closure method.Uncertainty quantification in computational fluid dynamics92Lect. Notes Comput. Sci. Eng.Springer, Heidelberg2013, 105--149back to text
• 91 articleM.M. Di Francesco and M. ..M .D. Rosini. Rigorous Derivation of Nonlinear Scalar Conservation Laws from Follow-the-Leader Type Models via Many Particle Limit.Archive for Rational Mechanics and Analysis2015back to text
• 92 articleR. J.Ronald J. DiPerna. Measure-valued solutions to conservation laws.Arch. Rational Mech. Anal.8,831985, 223--270back to text
• 93 articleC.Christian Dogbé. Modeling crowd dynamics by the mean-field limit approach.Math. Comput. Modelling529-102010, 1506--1520back to text
• 94 techreportR.R. Duvigneau. A Sensitivity Equation Method for Unsteady Compressible Flows: Implementation and Verification.INRIA Research Report No 8739June 2015back to text
• 95 articleR.R. Duvigneau and D.D. Pelletier. A sensitivity equation method for fast evaluation of nearby flows and uncertainty analysis for shape parameters.Int. J. of Computational Fluid Dynamics207August 2006, 497--512back to textback to textback to text
• 96 articleR.Radek Erban, M. B.Mark B. Flegg and G. A.Garegin A. Papoian. Multiscale stochastic reaction-diffusion modeling: application to actin dynamics in filopodia.Bull. Math. Biol.7642014, 799--818URL: http://dx.doi.org/10.1007/s11538-013-9844-3DOIback to text
• 97 articleR.R. Etikyala, S.S. Göttlich, A.A. Klar and S.S. Tiwari. Particle methods for pedestrian flow models: from microscopic to nonlocal continuum models.Math. Models Methods Appl. Sci.24122014, 2503--2523back to text
• 98 incollectionR.Robert Eymard, T.Thierry Gallouët and R.Raphaèle Herbin. Finite volume methods.Handbook of numerical analysis, Vol. VIIHandb. Numer. Anal., VIINorth-Holland, Amsterdam2000, 713--1020back to text
• 99 techreportU.Ulrik Fjordholm, R.Roger Kappeli, S.Siddhartha Mishra and E.Eitan Tadmor. Construction of approximate entropy measure valued solutions for systems of conservation laws.2014-33Seminar for Applied Mathematics, ETH Zürich2014back to textback to text
• 100 articleM. B.Mark B. Flegg, S.Stefan Hellander and R.Radek Erban. Convergence of methods for coupling of microscopic and mesoscopic reaction-diffusion simulations.J. Comput. Phys.2892015, 1--17URL: http://dx.doi.org/10.1016/j.jcp.2015.01.030DOIback to text
• 101 inproceedingsF.F. Fleuret and D.D. Geman. Graded learning for object detection.Proceedings of the workshop on Statistical and Computational Theories of Vision of the IEEE international conference on Computer Vision and Pattern Recognition (CVPR/SCTV)21999back to text
• 102 articleB.Benjamin Franz, M. B.Mark B. Flegg, S. J.S. Jonathan Chapman and R.Radek Erban. Multiscale reaction-diffusion algorithms: PDE-assisted Brownian dynamics.SIAM J. Appl. Math.7332013, 1224--1247back to text
• 103 articleM.Mauro Garavello and B.Benedetto Piccoli. Coupling of microscopic and phase transition models at boundary.Netw. Heterog. Media832013, 649--661back to textback to text
• 104 bookM.Mauro Garavello and B.Benedetto Piccoli. Traffic flow on networks.1AIMS Series on Applied MathematicsConservation laws modelsAmerican Institute of Mathematical Sciences (AIMS), Springfield, MO2006back to text
• 105 articleP.Paola Goatin and M.Matthias Mimault. A mixed system modeling two-directional pedestrian flows.Math. Biosci. Eng.1222015, 375--392back to text
• 106 unpublishedP.Paola Goatin and F.Francesco Rossi. A traffic flow model with non-smooth metric interaction: well-posedness and micro-macro limit.2015, PreprintURL: http://arxiv.org/abs/1510.04461back to text
• 107 articleP.Paola Goatin and S.Sheila Scialanga. Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity.Netw. Heterog. Media1112016, 107--121back to textback to textback to text
• 108 articleS.Simone Göttlich, S.Simon Hoher, P.Patrick Schindler, V.Veronika Schleper and A.Alexander Verl. Modeling, simulation and validation of material flow on conveyor belts.Applied Mathematical Modelling38132014, 3295--3313back to text
• 109 articleA.A. Griewank. Achieving logarithmic growth of temporal and spatial complexity in reverse automatic differentiation.Optimization Methods and Software11992, 35-54back to text
• 110 articleM.Michael Gröschel, A.Alexander Keimer, G.Günter Leugering and Z.Zhiqiang Wang. Regularity theory and adjoint-based optimality conditions for a nonlinear transport equation with nonlocal velocity.SIAM J. Control Optim.5242014, 2141--2163back to text
• 111 articleA.A. Habbal and M.Moez Kallel. Neumann-Dirichlet Nash strategies for the solution of elliptic Cauchy problems.SIAM J. Control Optim.5152013, 4066--4083back to textback to text
• 112 articleA.Abderrahmane Habbal, M.Moez Kallel and M.Marwa Ouni. Nash strategies for the inverse inclusion Cauchy-Stokes problem.Inverse Problems and Imaging 1342019, 36HALDOIback to text
• 113 unpublishedA.Abderrahmane Habbal, I.Imad Kissami, A.Amal Machtalay and A.Ahmed Ratnani. Computational investigations of a two-class traffic flow model : mean-field and microscopic dynamics.December 2021, working paper or preprintHALback to text
• 114 unpublishedA.Abderrahmane Habbal and M.Marwa Ouni. Coupled data recovery and shape identification : Nash games for the nonlinear Cauchy-Stokes case.December 2021, working paper or preprintback to text
• 115 articleX.X. Han, P.P. Sagaut and D.D. Lucor. On sensitivity of RANS simulations to uncertain turbulent inflow conditions.Computers & Fluids612-52012back to textback to text
• 116 articleD.Dirk Helbing, P.Peter Molnar, I. J.Illes J Farkas and K.Kai Bolay. Self-organizing pedestrian movement.Environment and planning B2832001, 361--384back to text
• 117 articleD.Dirk Helbing. Traffic and related self-driven many-particle systems.Rev. Mod. Phys.7342001, 1067--1141back to text
• 118 articleJ. C.Juan C Herrera, D. B.Daniel B Work, R.Ryan Herring, X. J.Xuegang Jeff Ban, Q.Quinn Jacobson and A. M.Alexandre M. Bayen. Evaluation of traffic data obtained via GPS-enabled mobile phones: The Mobile Century field experiment.Transportation Research Part C: Emerging Technologies1842010, 568--583back to text
• 119 articleS. P.Serge P. Hoogendoorn, F. L.Femke L.M. van Wageningen-Kessels, W.Winnie Daamen and D. C.Dorine C. Duives. Continuum modelling of pedestrian flows: From microscopic principles to self-organised macroscopic phenomena.Physica A: Statistical Mechanics and its Applications41602014, 684--694back to text
• 120 articleH.H. Hristova, S.S. Etienne, D.D. Pelletier and J.J. Borggaard. A continuous sensitivity equation method for time-dependent incompressible laminar flows.Int. J. for Numerical Methods in Fluids502004, 817-844back to text
• 121 articleT.T.J.R. Hughes, J.J.A. Cottrell and Y.Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement.Computer Methods in Applied Mechanics and Engineering1942005, 4135--4195back to text
• 122 articleC.Cyril Imbert and R.Régis Monneau. Flux-limited solutions for quasi-convex Hamilton--Jacobi equations on networks.arXiv preprint arXiv:1306.2428October 2014back to text
• 123 articleS.S. Jeon and H.H. Choi. Suboptimal feedback control of flow over a sphere.Int. J. of Heat and Fluid Flow312010back to text
• 124 articleM.Moez Kallel, R.Rajae Aboulaich, A.Abderrahmane Habbal and M.Maher Moakher. A Nash-game approach to joint image restoration and segmentation.Appl. Math. Model.3811-122014, 3038--3053URL: http://dx.doi.org/10.1016/j.apm.2013.11.034DOIback to text
• 125 articleO.O.M. Knio and O.O. Le Maitre. Uncertainty propagation in CFD using polynomial chaos decomposition.Fluid Dynamics Research389September 2006, 616--640back to text
• 126 articleA.Alexander Kurganov and A.Anthony Polizzi. Non-Oscillatory Central Schemes for a Traffic Flow Model with Arrehenius Look-Ahead Dynamics.Netw. Heterog. Media432009, 431-451back to text
• 127 articleA.Aime Lachapelle and M.-T.Marie-Therese Wolfram. On a mean field game approach modeling congestion and aversion in pedestrian crowds.Transportation Research Part B: Methodological45102011, 1572--1589back to text
• 128 articleJ.-M.Jean-Michel Lasry and P.-L.Pierre-Louis Lions. Mean field games.Jpn. J. Math.212007, 229--260back to text
• 129 articleM. J.Michael J. Lighthill and G. B.Gerald B. Whitham. On kinematic waves. II. A theory of traffic flow on long crowded roads.Proc. Roy. Soc. London. Ser. A.2291955, 317--345back to text
• 130 articleG.G. Lin, C.-H.C.-H. Su and G.G.E. Karniadakis. Predicting shock dynamics in the presence of uncertainties.Journal of Computational Physics2172006, 260-276back to text
• 131 articleM.M. Martinelli and R.R. Duvigneau. On the use of second-order derivative and metamodel-based Monte-Carlo for uncertainty estimation in aerodynamics.Computers and Fluids3762010back to text
• 132 articleC.C.R. Merritt, F.F. Forsberg, J.J. Liu and F.F. Kallel. In-vivo elastography in animal models: Feasibility studies, (abstract). J. Ultrasound Med.21982002back to text
• 133 articleS.Siddhartha Mishra, C.Christoph Schwab and J.Jonas Sukys. Multi-level Monte Carlo finite volume methods for uncertainty quantification in nonlinear systems of balance laws.Lecture Notes in Computational Science and Engineering922013, 225--294back to text
• 134 articleW.W.L. Oberkampf and F.F.G. Blottner. Issues in Computational Fluid Dynamics code verification and validation.AIAA Journal361998, 687--695back to text
• 135 bookB.Benoît Perthame. Transport equations in biology.Frontiers in MathematicsBirkhäuser Verlag, Basel2007back to text
• 136 articleB.Benedetto Piccoli and F.Francesco Rossi. Transport equation with nonlocal velocity in Wasserstein spaces: convergence of numerical schemes.Acta Appl. Math.1242013, 73--105back to text
• 137 techreportF.F. Poirion. Stochastic Multi Gradient Descent Algorithm.ONERAJuly 2014back to text
• 138 articleF. ..F .S. Priuli. First order mean field games in crowd dynamics.ArXiv e-printsFebruary 2014back to text
• 139 inproceedingsM.M.M. Putko, P.P.A. Newman, A.A.C. Taylor and L.L.L. Green. Approach for uncertainty propagation and robust design in CFD using sensitivity derivatives.15th AIAA Computational Fluid Dynamics ConferenceAIAA Paper 2001-2528Anaheim, CAJune 2001back to text
• 140 incollectionC.Chunhong Qi, K.KyleA. Gallivan and P.-A.P.-A. Absil. Riemannian BFGS Algorithm with Applications.Recent Advances in Optimization and its Applications in EngineeringSpringer Berlin Heidelberg2010, 183-192URL: http://dx.doi.org/10.1007/978-3-642-12598-0_16DOIback to text
• 141 articleJ.J. Reilly, W.W. Krichene, M. L.M. L. Delle Monache, S.S. Samaranayake, P.P. Goatin and A. M.Alexandre M. Bayen. Adjoint-based optimization on a network of discretized scalar conservation law PDEs with applications to coordinated ramp metering.J. Optim. Theory Appl.16722015, 733--760back to text
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