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  109, or to the dynamics of turbulent flows for which large scale / small scale vortical structures interfere  136.

Nevertheless, macroscopic models offer well known advantages, namely a sound analytical framework, fast numerical schemes, the presence of a low number of parameters to be calibrated, and efficient optimization procedures. Therefore, we are convinced of the interest of keeping this point of view as dominant, while completing the models with information on the dynamics at the small scale / microscopic level. This can be achieved through several techniques, like hybrid models, homogenization, mean field games. In this project, we will focus on the aspects detailed below.

The development of adapted and efficient numerical schemes is a mandatory completion, and sometimes ingredient, of all the approaches listed below. The numerical schemes developed by the team are based on finite volumes or finite elements techniques, and constitute an important tool in the study of the considered models, providing a necessary step towards the design and implementation of the corresponding optimization algorithms, see Section 3.3.

3.3 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  147.

However, the lack of efficient and flexible numerical tools, that allow to simulate and optimize a large number of such devices on realistic configurations, is still a bottleneck for the emergence of this technology in industry. The main issue is the necessity of using 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  110. Further improvements require more advanced models, keeping into better account interactions at the microscopic scale, and adapted control techniques, see  60 and references therein.

In particular, we will focus on the following aspects:

• Junction models. We are interested in designing a general junction model both satisfying basic analytical properties guaranteeing 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  73, 98, 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  115.
• Data assimilation. In traffic flow modeling, the capability of correctly estimating and predicting the state of the system depends on the availability of rich and accurate data on the network. Up to now, the most classical sensors are fixed ones. They are composed of inductive loops (electrical wires) that are installed at different spatial positions of the network and that can measure the traffic flow, the occupancy rate (i.e. the proportion of time during which a vehicle is detected to be over the loop) and the speed (in case of a system of two distant loops). These data are useful / essential to calibrate the phenomenological relationship between flow and density which is known in the traffic literature as the Fundamental Diagram. Nowadays, thanks to the wide development of mobile internet and geolocalization techniques and its increasing adoption by the road users, smartphones have turned into perfect mobile sensors in many domains, including in traffic flow management. They can provide the research community with a large database of individual trajectory sets that are known as Floating Car Data (FCD), see 111 for a real field experiment. Classical macroscopic models, say (hyperbolic systems of) conservation laws, are not designed to take into account this new kind of microscopic data. Other formulations, like Hamilton-Jacobi partial differential equations, are most suited and have been intensively studied in the past five years (see  67, 66), with a stress on the (fixed) Eulerian framework. Up to our knowledge, there exist a few studies in the 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  80, 97 or via the Hamilton-Jacobi approach  67, 66 could allow to optimize the flow by controlling some specific vehicles corresponding to internal conditions.

4.3 Combined hormone and brachy therapies for the treatment of prostate cancer

The latest statistics published by the International Agency for Research on Cancer show that in 2018, 18.1 million new cancer cases have been identified and 9.6 million deaths have been recorded worldwide making it the second leading cause of death globally. Prostate cancer ranks third in incidence with 1.28 million cases and represents the second most commonly diagnosed male cancer.

Prostate cells need the hormone androgen to survive and function properly. For this to happen, the androgens have to bind to a protein in the prostate cells called Androgen Receptor and activate it. Since androgens act as a growth factor for the cells, one way of treating prostate cancer is through the antihormone therapy that hinder its activity. The Androgen Deprivation Therapy (ADT) aims to either reduce androgen production or to stop the androgens form working through the use of drugs. However, over time, 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 125. Indeed, it is well known that most pathological changes are associated with changes in tissue stiffness, while remaining isoechoic, and hence difficult to detect by ultrasound techniques. Based on elastic waves and heat flux reconstruction, thermoelastography shows no destructive or aggressive medical sequel, unlike 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  105, 117, we have demonstrated that Nash game approaches are efficient to tackle ill-posedness. We intend to extend the results obtained for Laplace equations in 105, and the algorithms developed in Section 3.3.5 to the following problems (of increasing difficulty):

  - Simultaneous data and parameter recovery in linear elasticity, using the 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. 133, that have the desirable convergence property in constrained problems. However, in this approach, the constraints are assumed to be known formally, by explicit expressions.

  In collaboration with 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 sub-spaces, 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 dof's. CEA disposes of such codes, but considering the computational cost of the objective functions in the complex cases, the first part of our study is focused on the construction and validation of 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 Awards

M. Binois received the ENBIS 2022 Young Statistician Award and a prix innovation et recherche appliquée by Université Côte d'Azur.

6.2 New book

A. Bayen, M.L. Delle Monache, M. Garavello, P. Goatin and B. Piccoli, Control Problems for Conservation Laws with Traffic Applications: Modeling, Analysis, and Numerical Methods, Progress in Nonlinear Differential Equations and Their Applications (Vol. 99), Springer Nature (2022). 36

7.1 New software

8.1 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], Simone Göttlich [Univ. Mannheim, Germany], Fabio Vicini [Polytechnic of Turin, Italy].

Traffic control by Connected and Automated Vehicles

We present a general multi-scale approach for modeling the interaction of controlled and automated vehicles (CAVs) with the surrounding traffic flow. The model consists of a scalar conservation law for the bulk traffic, coupled with ordinary differential equations describing the possibly interacting CAV trajectories. The coupling is realized through flux constraints at the moving bottleneck positions, inducing the formation of non-classical jump discontinuities in the traffic density. In turn, CAVs are forced to adapt their speed to the downstream traffic average velocity in congested situations. We analyze the model solutions in a Riemann-type setting, and propose an adapted finite volume scheme to compute approximate solutions for general initial data. The work paves the way to the study of general optimal control strategies for CAV velocities, aiming at improving the overall traffic flow by reducing congestion phenomena and the associated externalities. Controlling CAV desired speeds allows to act on the system to minimize any traffic density dependent cost function. More precisely, we apply Model Predictive Control (MPC) to reduce fuel consumption in congested situations.

See 24, 25, 33, 36.

Traffic flow model calibration by statistical approaches

In the framework of A. Würth's PhD thesis, we employ a Bayesian approach including a bias term to estimate first and second order model parameters, based on two traffic data sets: a set of loop detector data located on the A50 highway between Marseille and Aubagne provided by DirMED, and publicly available data from the Minnesota Department of transportation (MnDOT). In 30, we propose a Bayesian approach for parameter uncertainty quantification in macroscopic traffic flow models from cross-sectional data. A bias term is introduced and modeled as a Gaussian process to account for the traffic flow models limitations. We validate the results comparing the error metrics of both first and second order models, showing that second order models globally perform better in reconstructing traffic quantities of interest.

Besides, we proved an existence result for the associated initial-boundary value problem for general second order macroscopic models in 43.

Routing strategies in traffic flows on networks

In 42, we introduce a macroscopic differential model coupling a conservation law with a 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.

8.2 Isogeometric Discontinuous Galerkin method for compressible flows

Participants: Régis Duvigneau, Stefano Pezzano.

The co-existence of different geometrical representations in the design loop (CAD-based and mesh-based) is a real bottleneck for the application of design optimization procedures in industry, yielding a major waste of human time to convert geometrical data. Isogeometric analysis methods, which consists in using CAD bases like NURBS in a Finite-Element framework, were proposed a decade ago to facilitate interactions between geometry and simulation domains.

We investigate the extension of such methods to Discontinuous Galerkin (DG) formulations, which are better suited to hyperbolic or convection-dominated problems. Specifically, we develop a DG method for compressible Euler and Navier-Stokes equations, based on rational parametric elements, that preserves exactly the geometry of boundaries defined by NURBS, while the same rational approximation space is adopted for the solution. The following research axes are considered in this context:

• Arbitrary Eulerian-Lagrangian formulation for high-order meshes

  To enable the simulation of flows around moving or deforming bodies, an Arbitrary Eulerian-Lagrangian (ALE) formulation is proposed in the context of the isogeometric DG method. It relies on a NURBS-based grid velocity field, integrated along time over moving NURBS elements. The approach has been applied to the simulation of morphing airfoils 28.

• Geometrically exact sliding interfaces

  In the context of rotating machines (compressors, turbines, etc), computations are achieved using a rotating inner grid interfaced to an outer fixed grid. This coupling is cumbersome using classical piecewise-linear grids due to a lack of common geometrical interface. Thus, we have developed a method based on a geometrically exact sliding interface using NURBS elements, ensuring a fully conservative scheme 27.

• Isogeometric shape optimization

  We develop an optimization procedure entirely based on NURBS representations. The mesh, the shape to be optimized, as well as the flow solutions are represented by NURBS, which avoid any geometrical conversion and allows to exploit NURBS properties regarding regularity or hierarchy. The approach has also been employed in the framework of Bayesian optimization for airfoil design 28, 34.

8.3 Advanced Bayesian optimization

Participants: Mickaël Binois, Régis Duvigneau, Nicholson Collier [Argonne, USA], Mahmoud Elsawy [Atlantis team], Stéphane Lanteri [Atlantis team], Jonathan Ozik [Argonne, USA], Victor Picheny [SecondMind, GB], Henry Moss [SecondMind, GB].

Bayesian optimization of 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 35.

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 39.

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 multiobjective optimization tasks. These experiments show similar or better performance than existing methods, while being orders of magnitude faster.

Multi-fidelity modeling and optimization

To reduce the computational cost related to the use of high-fidelity simulations when evaluating the cost function, we investigate the construction of multi-fidelity auto-regressive Gaussian Process models, that can rely on different physical models (e.g. inviscid or viscous flows) or numerical accuracy (e.g. coarse or fine meshes). The objective is to construct a model that is accurate regarding the high-fidelity evaluations, but mostly based on low-fidelity simulations. Of particular interest is the definition of an efficient acquisition function, that selects both the next design point to evaluate and the corresponding fidelity level to use. This work is achieved in collaboration with SecondMind company and was the topic of Maha Ouali's internship.

8.4 Gaussian process based sequential design

Participants: Mickaël Binois, Robert Gramacy [Virginia Tech, USA], Nathan Wycoff [Virginia Tech, USA].

Besides Bayesian optimization as above, Gaussian processes are useful for a variety of other related tasks. Here we first present a method to deal with non-stationarity of the process, handling global and local scales. Secondly, we review the state of the art for high-dimensional GP modeling.

Sensitivity prewarping for local surrogate modeling

In the continual effort to improve product quality and decrease operations costs, computational modeling is increasingly being deployed to determine feasibility of product designs or configurations. Surrogate modeling of these computer experiments via local models, which induce sparsity by only considering short range interactions, can tackle huge analyses of complicated input-output relationships. However, narrowing focus to local scale means that global trends must be re-learned over and over again. In 31, we propose a framework for incorporating information from a global sensitivity analysis into the surrogate model as an input rotation and rescaling preprocessing step. We discuss the relationship between several sensitivity analysis methods based on kernel regression before describing how they give rise to a transformation of the input variables. Specifically, we perform an input warping such that the "warped simulator" is equally sensitive to all input directions, freeing local models to focus on local dynamics. Numerical experiments on observational data and benchmark test functions, including a high-dimensional computer simulator from the automotive industry, provide empirical validation.

A survey on high-dimensional Gaussian process modeling with application to Bayesian optimization

In 22 we propose a review of high-dimensional GP modeling. Extending the efficiency of Bayesian optimization (BO) to larger number of parameters has received a lot of attention over the years. Even more so has Gaussian process regression modeling in such contexts, on which most BO methods are based. A variety of structural assumptions have been tested to tame high dimension, ranging from variable selection and additive decomposition to low dimensional embeddings and beyond. Most of these approaches in turn require modifications of the acquisition function optimization strategy as well. Here we review the defining assumptions, and discuss the benefits and drawbacks of these approaches in practice.

8.5 Multi-label propagation

Participants: Mickaël Binois, Khadija Musayeva.

In 44, to prepare for the analysis of complex biological data, this 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.

8.6 Policy-based optimization

Participants: Régis Duvigneau, Jonathan Viquerat [Mines Paris-Tech].

This work concerns the development of black-box optimization methods based on single-step deep reinforcement learning (DRL) and their conceptual similarity to evolution strategy (ES) techniques 29. The connection of policy-based optimization (PBO) to evolutionary strategies (especially covariance matrix adaptation evolutionary strategy) is discussed. Relevance is assessed by benchmarking PBO against classical ES techniques on analytic functions minimization problems, and by optimizing various parametric control laws intended for the Lorenz attractor. This contribution definitely establishes PBO as a valid, versatile black-box optimization technique, and opens the way to multiple future improvements building on the inherent flexibility of the neural networks approach.

8.7 Analytical study of the Eulerian Adjoint System

Participants: Jean-Antoine Désidéri, Jacques Peter [ONERA/DAAA, Université Paris-Saclay, Châtillon].

Ordinary differential equations have been derived for the adjoint Euler equations first using the method of characteristics in 2D. For this system of partial-differential equations, the characteristic curves appear to be the streamtraces and the well-known curves of the theory applied to the flow. The differential equations satisfied along the streamtraces in 2D have then been extended and demonstrated in 3D by linear combinations of the original adjoint equations.

These findings extend their well-known counterparts for the direct system and should serve analytical and possibly numerical studies of the perfect-flow model with respect to adjoint fields or sensitivity questions. In addition to the analytical theory, the results have been illustrated by the numerical integration of the compatibility relationships for discrete 2D flow fields and dual-consistent adjoint fields over a very fine grid about an airfoil 26.

8.8 Pareto optimality and Nash games

Participants: 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].

The present work reflects our cooperation with the Information Processing and Systems Department (DTIS) of Onera Toulouse.

In the multi-objective optimization of a complex system, after the Pareto front associated with preponderant objective functions (“primary cost functions”), has been approximated, usually at a demanding computational cost, the decision to elect the final concept is still to be made since a whole set of indiscriminate Pareto-optimal solutions is available.

To complete the decision process, we had proposed a Nash game construction initiated from one such Pareto-optimal solution to target a reduction of “secondary cost functions” while quasi-maintaining the Pareto optimality of the primary cost functions. Convergence proof and first examples were given in 7.

This method has 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). The optimization was subject to functional constraints on geometry and longitudinal stability. Designs were evaluated by means of the FAST-OAD open-source software developed by ONERA and ISAE-SUPAERO, and the prioritized optimization conducted by our Nash-MGDA software (MGDA). The experiment demonstrated the efficacy of the method to greatly reduce the ascent duration, and its great efficiency in terms of computational time, permitting the numerical process to be interactive 41.

8.9 Inverse 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 106 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 45

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 107.

8.10 Combined therapies for the treatment of prostate cancer

Participants: Abderrahmane Habbal, Salma Chabbar [PhD, ACUMES and EMI, Univ. Mohammed V], Rajae Aboulaich [EMI, Univ. Mohammed V], Nabil Ismaili [PhD.MD., Univ Mohamed VI for health sciences, Casablanca], Sanae EL Mejjaoui [PhD.MD., Institute for Oncology, Avicenne Hospital, Rabat].

Prostate cancer is a 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 as summarized hereafter. Combining hormone therapy with brachytherapy allowed us to reduce the dose used from 120Gy to 80Gy. When the treatments are given at the same time, the final tumor volume is significantly reduced compared to using each therapy separately. However, starting with hormone therapy gave better results. When using intermittent hormone therapy combined with brachytherapy, we found that once the PSA level drops below the critical level, it stays at reasonable levels and therefore the hormone therapy does not reactivate. When we use continuous hormone therapy instead, the prostate is unnecessarily deprived of androgen for an almost 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 37 and 40.

9.1 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.

10.1 International initiatives

10.2 International research visitors

10.3 European initiatives

10.4 National initiatives

11.1 Promoting scientific activities

11.2 Teaching - Supervision - Juries

11.3 Popularization

12.1 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, 963-983
  HAL
• 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 Mathematik2015
  HALDOI
• 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 2021
  HALback 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, 29-68
  HALDOI
• 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 2017
  HAL
• 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, 54-63
  HALDOI
• 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, 313-318
  HALDOI
• 12 articleR.R. Duvigneau and P.P. Chandrashekar. Kriging-based optimization applied to flow control.Int. J. for Numerical Methods in Fluids69112012, 1701-1714
• 13 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
• 14 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.2013.11.034
  DOI
• 15 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
• 16 articleQ.Quentin Mercier, F.Fabrice Poirion and J.-A.Jean-Antoine Desideri. A stochastic multiple gradient descent algorithm.European Journal of Operational ResearchMay 2018, 10
  HALDOI
• 17 articleS.Souvik Roy, A.A. Borzì and A.Abderrahmane Habbal. Pedestrian motion modelled by Fokker--Planck Nash games.Royal Society open science492017, 170648
• 18 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, 21
  HALDOI
• 19 articleM.M. Twarogowska, P.P. Goatin and R.R. Duvigneau. Macroscopic modeling and simulations of room evacuation.Appl. Math. Model.38242014, 5781--5795
• 20 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
• 21 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

12.2 Publications of the year

12.3 Cited publications

• 46 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. Physics2012
  back to text
• 47 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, 963-983
  HALback to textback to text
• 48 articleG.Giovanni Alessandrini. Examples of instability in inverse boundary-value problems.Inverse Problems1341997, 887--897URL: http://dx.doi.org/10.1088/0266-5611/13/4/001
  DOIback to text
• 49 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--131
  back to text
• 50 articleP.P. Amorim, R. M.Rinaldo M. Colombo and A.A. Teixeira. On the Numerical Integration of Scalar Nonlocal Conservation Laws.ESAIM M2AN4912015, 19--37
  back to text
• 51 articleP.Paulo Amorim. On a nonlocal hyperbolic conservation law arising from a gradient constraint problem.Bull. Braz. Math. Soc. (N.S.)4342012, 599--614
  back to text
• 52 articleM.M. Annunziato and A.A. Borz\`i. A Fokker-Planck control framework for multidimensional stochastic processes.Journal of Computational and Applied Mathematics2372013, 487-507
  back to text
• 53 articleA.A. Belme, F.F. Alauzet and A.A. Dervieux. Time accurate anisotropic goal-oriented mesh adaptation for unsteady flows.J. Comput. Physics231192012, 6323--6348
  back to text
• 54 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--1803
  back to text
• 55 unpublishedE.Enrico Bertino, R.Régis Duvigneau and P.Paola Goatin. Uncertainties in traffic flow and model validation on GPS data.2015
  back to text
• 56 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--885
  back to text
• 57 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/s00211-015-0717-6
  back to textback to text
• 58 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/S0021999197957430
  DOIback to text
• 59 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--1251
  back to text
• 60 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--111
  back to text
• 61 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--1333
  back to text
• 62 articleM.M. Burger, J.J. Haskovec and M.-T.M.-T. Wolfram. Individual based and mean-field modelling of direct aggregation.Physica D2602013, 145--158
  back to text
• 63 techreportA.Alessandra Cabassi and P.Paola Goatin. Validation of traffic flow models on processed GPS data.Research Report RR-83822013
  HALback to text
• 64 unpublishedJ. ..J .A. Carrillo, S.S. Martin and M.-T.M.-T. Wolfram. A local version of the Hughes model for pedestrian flow.2015, Preprint
  back to text
• 65 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, Preprint
  back to textback to text
• 66 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--402
  back to textback to text
• 67 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, 1158-1174
  back to textback to text
• 68 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, 1150023
  back to text
• 69 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--379
  back to text
• 70 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--196
  back to text
• 71 articleR. M.Rinaldo M. Colombo and F.Francesca Marcellini. A mixed ODE-PDE model for vehicular traffic.Mathematical Methods in the Applied Sciences3872015, 1292--1302
  back to text
• 72 articleR. M.Rinaldo M. Colombo and E.E. Rossi. On the micro-macro limit in traffic flow.Rend. Semin. Mat. Univ. Padova1312014, 217--235
  back to text
• 73 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--433
  back to text
• 74 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, 1-15
  back to text
• 75 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, 6989-6994
  back to text
• 76 bookE.Emiliano Cristiani, B.Benedetto Piccoli and A.Andrea Tosin. Multiscale modeling of pedestrian dynamics.12MS&A. Modeling, Simulation and ApplicationsSpringer, Cham2014
  back to text
• 77 incollectionC. M.C. M. Dafermos. Solutions in $L^{}$ for a conservation law with memory.Analyse mathématique et applicationsMontrougeGauthier-Villars1988, 117--128
  back to text
• 78 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/s00332-013-9185-2
  DOIback to text
• 79 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--447
  back to textback to text
• 80 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--4029
  back to textback to text
• 81 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, 1
  back to text
• 82 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.2012.03.014
  back to text
• 83 techreportJ.-A.Jean-Antoine Désidéri. Révision de l'algorithme de descente à gradients multiples (MGDA) par orthogonalisation hiérarchique.8710INRIAApril 2015
  back to text
• 84 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--149
  back to text
• 85 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 Analysis2015
  back to text
• 86 articleR. J.Ronald J. DiPerna. Measure-valued solutions to conservation laws.Arch. Rational Mech. Anal.8831985, 223--270
  back to text
• 87 articleC.Christian Dogbé. Modeling crowd dynamics by the mean-field limit approach.Math. Comput. Modelling529-102010, 1506--1520
  back to text
• 88 techreportR.R. Duvigneau. A Sensitivity Equation Method for Unsteady Compressible Flows: Implementation and Verification.INRIA Research Report No 8739June 2015
  back to text
• 89 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--512
  back to textback to textback to text
• 90 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/s11538-013-9844-3
  DOIback to text
• 91 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--2523
  back to text
• 92 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--1020
  back to text
• 93 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 Zürich2014
  back to textback to text
• 94 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.2015.01.030
  DOIback to text
• 95 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)21999
  back to text
• 96 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--1247
  back to text
• 97 articleM.Mauro Garavello and B.Benedetto Piccoli. Coupling of microscopic and phase transition models at boundary.Netw. Heterog. Media832013, 649--661
  back to textback to text
• 98 bookM.Mauro Garavello and B.Benedetto Piccoli. Traffic flow on networks.1AIMS Series on Applied MathematicsConservation laws modelsAmerican Institute of Mathematical Sciences (AIMS), Springfield, MO2006
  back to text
• 99 articleP.Paola Goatin and M.Matthias Mimault. A mixed system modeling two-directional pedestrian flows.Math. Biosci. Eng.1222015, 375--392
  back to text
• 100 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/1510.04461
  back to text
• 101 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--121
  back to textback to textback to text
• 102 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--3313
  back to text
• 103 articleA.A. Griewank. Achieving logarithmic growth of temporal and spatial complexity in reverse automatic differentiation.Optimization Methods and Software11992, 35-54
  back to text
• 104 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--2163
  back to text
• 105 articleA.A. Habbal and M.Moez Kallel. Neumann-Dirichlet Nash strategies for the solution of elliptic Cauchy problems.SIAM J. Control Optim.5152013, 4066--4083
  back to textback to text
• 106 articleA.Abderrahmane Habbal, M.Moez Kallel and M.Marwa Ouni. Nash strategies for the inverse inclusion Cauchy-Stokes problem.Inverse Problems and Imaging 1342019, 36
  HALDOIback to text
• 107 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 preprint
  back to text
• 108 articleX.X. Han, P.P. Sagaut and D.D. Lucor. On sensitivity of RANS simulations to uncertain turbulent inflow conditions.Computers & Fluids612-52012
  back to textback to text
• 109 articleD.Dirk Helbing, P.Peter Molnar, I. J.Illes J Farkas and K.Kai Bolay. Self-organizing pedestrian movement.Environment and planning B2832001, 361--384
  back to text
• 110 articleD.Dirk Helbing. Traffic and related self-driven many-particle systems.Rev. Mod. Phys.7342001, 1067--1141
  back to text
• 111 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--583
  back to text
• 112 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--694
  back to text
• 113 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, 817-844
  back to text
• 114 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--4195
  back to text
• 115 articleC.Cyril Imbert and R.Régis Monneau. Flux-limited solutions for quasi-convex Hamilton--Jacobi equations on networks.arXiv preprint arXiv:1306.2428October 2014
  back to text
• 116 articleS.S. Jeon and H.H. Choi. Suboptimal feedback control of flow over a sphere.Int. J. of Heat and Fluid Flow312010
  back to text
• 117 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.2013.11.034
  DOIback to text
• 118 articleO.O.M. Knio and O.O. Le Maitre. Uncertainty propagation in CFD using polynomial chaos decomposition.Fluid Dynamics Research389September 2006, 616--640
  back to text
• 119 articleA.Alexander Kurganov and A.Anthony Polizzi. Non-Oscillatory Central Schemes for a Traffic Flow Model with Arrehenius Look-Ahead Dynamics.Netw. Heterog. Media432009, 431-451
  back to text
• 120 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--1589
  back to text
• 121 articleJ.-M.Jean-Michel Lasry and P.-L.Pierre-Louis Lions. Mean field games.Jpn. J. Math.212007, 229--260
  back to text
• 122 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--345
  back to text
• 123 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, 260-276
  back to text
• 124 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
  back to text
• 125 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.21982002
  back to text
• 126 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--294
  back to text
• 127 articleW.W.L. Oberkampf and F.F.G. Blottner. Issues in Computational Fluid Dynamics code verification and validation.AIAA Journal361998, 687--695
  back to text
• 128 bookB.Benoît Perthame. Transport equations in biology.Frontiers in MathematicsBirkhäuser Verlag, Basel2007
  back to text
• 129 articleB.Benedetto Piccoli and F.Francesco Rossi. Transport equation with nonlocal velocity in Wasserstein spaces: convergence of numerical schemes.Acta Appl. Math.1242013, 73--105
  back to text
• 130 techreportF.F. Poirion. Stochastic Multi Gradient Descent Algorithm.ONERAJuly 2014
  back to text
• 131 articleF. ..F .S. Priuli. First order mean field games in crowd dynamics.ArXiv e-printsFebruary 2014
  back to text
• 132 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 2001
  back to text
• 133 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_16
  DOIback to text
• 134 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--760
  back to text
• 135 articleP. I.Paul I. Richards. Shock waves on the highway.Operations Res.41956, 42--51
  back to text
• 136 bookP.P. Sagaut. Large Eddy Simulation for Incompressible Flows An Introduction.Springer Berlin Heidelberg2006
  back to textback to text
• 137 inproceedingsJ.J. Schaefer, T.T. West, S.S. Hosder, C.C. Rumsey, J.-R.J.-R. Carlson and W.W. Kleb. Uncertainty Quantification of Turbulence Model Closure Coefficients for Transonic Wall-Bounded Flows.22nd AIAA Computational Fluid Dynamics Conference, 22-26 June 2015, Dallas, USA.2015
  back to text
• 138 articleV.V. Schleper. A hybrid model for traffic flow and crowd dynamics with random individual properties.Math. Biosci. Eng.1222015, 393-413
  back to text
• 139 articleA.Alexandros Sopasakis and M. A.Markos A. Katsoulakis. Stochastic modeling and simulation of traffic flow: asymmetric single exclusion process with Arrhenius look-ahead dynamics.SIAM J. Appl. Math.6632006, 921--944
  back to text
• 140 articleP. R.P. R. Spalart. Detached-Eddy Simulation.Annual Review of Fluid Mechanics412009, 181-202
  back to text
• 141 inproceedingsS.Svetlana Tokareva, S.Siddhartha Mishra and C.Christoph Schwab. High Order Stochastic Finite Volume Method for the Uncertainty Quantification in Hyperbolic Conservtion Laws with Random Initial Data and Flux Coefficients.Proc. ECCOMASProc. ECCOMAS2012
  back to text
• 142 inproceedingsÉ.É. Turgeon, D.D. Pelletier and J.J. Borggaard. Sensitivity and Uncertainty Analysis for Variable Property Flows.39th AIAA Aerospace Sciences Meeting and ExhibitAIAA Paper 2001-0139Reno, NVJan. 2001
  back to textback to textback to text
• 143 bookC.Cédric Villani. Optimal transport.338Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]Old and newSpringer-Verlag, Berlin2009
  back to text
• 144 bookC.Cédric Villani. Topics in optimal transportation.58Graduate Studies in MathematicsAmerican Mathematical Society, Providence, RI2003
  back to text
• 145 techreportR.R.W. Walter and L.L. Huyse. Uncertainty analysis for fluid mechanics with applications.2002--1ICASEFebruary 2002
  back to text
• 146 articleD.D.B. Xiu and G.G.E. Karniadakis. Modeling uncertainty in flow simulations via generalized Polynomial Chaos.Journal of Computational Physics1872003, 137-167
  back to text
• 147 articleD.D. You and P.P. Moin. Active control of flow separation over an airfoil using synthetic jets.J. of Fluids and Structures242008, 1349-1357
  back to text
• 148 articleA.A. Zerbinati, A.A. Minelli, I.I. Ghazlane and J.-A.J.-A. Désidéri. Meta-Model-Assisted MGDA for Multi-Objective Functional Optimization.Computers and Fluids102http://www.sciencedirect.com/science/article/pii/S0045793014002576#2014, 116-130
  back to text