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 turns out to be inadequate. We refer for example to self-organizing phenomena observed in pedestrian flows 110, or to the dynamics of turbulent flows for which large scale / small scale vortical structures interfere 137.

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 148.

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 111. Further improvements require more advanced models, keeping into better account interactions at the microscopic scale, and adapted control techniques, see 62 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 75, 100, 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 116.
• Data assimilation. In traffic flow modeling, the capability of correctly estimating and predicting the state of the system depends on the availability of rich and accurate data on the network. Up to now, the most classical sensors are fixed ones. They are composed of inductive loops (electrical wires) that are installed at different spatial positions of the network and that can measure the traffic flow, the occupancy rate (i.e. the proportion of time during which a vehicle is detected to be over the loop) and the speed (in case of a system of two distant loops). These data are useful / essential to calibrate the phenomenological relationship between flow and density which is known in the traffic literature as the Fundamental Diagram. Nowadays, thanks to the wide development of mobile internet and geolocalization techniques and its increasing adoption by the road users, smartphones have turned into perfect mobile sensors in many domains, including in traffic flow management. They can provide the research community with a large database of individual trajectory sets that are known as Floating Car Data (FCD), see 112 for a real field experiment. Classical macroscopic models, say (hyperbolic systems of) conservation laws, are not designed to take into account this new kind of microscopic data. Other formulations, like Hamilton-Jacobi partial differential equations, are most suited and have been intensively studied in the past five years (see 69, 68), 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 82, 99 or via the Hamilton-Jacobi approach 69, 68 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 3 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 126. 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 107, 118, we have demonstrated that Nash game approaches are efficient to tackle ill-posedness. We intend to extend the results obtained for Laplace equations in 107, 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. 134, 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 (Radial Basis Functions). Multi-objective optimization will be carried out subsequently by MGDA, and possibly Nash games.

5.1 Impact of research results

Acumes's research activity in traffic modeling and control is intended to improve road network efficiency, thus reducing energy consumption and pollutant emission.

The research activities related to isogeometric analysis and physics-informed neural networks (PINNs) aim at facilitating the use of numerical simulations and design optimization in engineering, yielding a gain of efficiency, for instance in transportation industry (cars, aircrafts) or energy industry (air conditioning, turbines).

The research conducted with the startup Mycophyto aims at reducing the use of chemical fertilizers and phytopharmaceutical products by developing natural biostimulants (mycorrhyzal fungi). It started with the arrival of Khadija Musayeva in October 2020.

The research conducted in ANR NEMATIC aims at exploring the ability to control the growth of filamentous fungi, which have the potential for the production of biofuels and biosourced chemicals. Investigations started with the arrival of Nicolas Fricker in January 2023.

The research conducted with company Altair on code OpenRadioss aims at improving the resolution of multimaterial flows in presence of large density ratios, with a view to security applications such as shaped-charge detonation. It started with the arrival of Alexandre Vieira in October 2024.

The research conducted with Konstantin Brenner during Amal Chakib's M2 internship on shallow water equations with moving obstacles has important applications for flooding events and impacts on infrastructures.

With the increasing demands of modern applications such as electric and hybrid vehicles and renewable energy storage, the limitations of current commercial batteries using liquid or gel electrolytes have become critical. We initiated a research activity on solid state batteries. Their better understanding and hence optimizing is expected to strikingly improve industrial properties, such as higher energy densities, longer lifespans, cost-effectiveness, low maintenance, and enhanced safety.

7.1 New software

8.1 Macroscopic traffic flow models on networks

Participants: Eric Andoni, Paola Goatin, Chiara Daini [KOPERNIC Project-Team, Inria Paris], Maria Laura Delle Monache [UC Berkeley, USA], Antonella Ferrara [Univ. Pavia, Italy], Agatha Joumaa, Carmen Nieto, Benedetto Piccoli [Rutgers U, USA], Alessandra Rizzo [Univ. Messina, Italy], Enrico Siri.

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 29.

In the same light, in 44 we propose a new model for multi-lane traffic with moving bottlenecks, e.g., autonomous vehicles (AV). It consists of a system of balance laws for traffic in each lane, coupled in the source terms for lane changing, and fully coupled to ODEs for the AVs' trajectories. More precisely, each AV solves a controlled equation depending on the traffic density, while the PDE on the corresponding lane has a flux constraint at the AV's location. We prove existence of entropy weak solutions, and we characterize the limiting behavior for the source term converging to zero (without AVs), corresponding to a scalar conservation law for the total density. The convergence in the presence of AVs is more delicate and we show that the limit does not satisfy an entropic equation for the total density as in the original coupled ODE-PDE model. Finally, we illustrate our results via numerical simulations.

In the aim of modeling the formation of stop-and-go waves (to be controlled employing CAVs), in 32 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.

In 30, we study the boundary stabilization of Generic Second Order Macroscopic traffic models in Lagrangian coordinates. These consist in $2\times 2$ nonlinear hyperbolic systems of balance equations with a relaxation-type source term. We provide the existence of weak solutions of the Initial Boundary Value problem for generic relaxation terms. In particular, we do not require the sub-characteristic stability condition to hold, so that equilibria are unstable and perturbations may lead to the formation of large oscillations, modeling the appearance and persistence of stop-and-go waves. Moreover, since the largest eigenvalue of the system is null, the boundaries are characteristic, and the available results on boundary controllability do not apply. Therefore, we perform a detailed analysis of the Wave Front Tracking approximate solutions to show that weak solutions can be steered to the corresponding equilibrium state by prescribing the equilibrium speed at the right boundary. This corresponds to controlling the speed of one vehicle to stabilize the upstream traffic flow. The result is illustrated through a numerical example.

Routing strategies in traffic flows on networks

The research focused on the development of a general multi-class macroscopic traffic model that can be applied to large-scale real-world networks. The study investigated the impact of a Stackelberg control strategy on a “compliant” fraction of the traffic flow, defining their tendency to comply as a function of the difference in travel costs between the routes independently chosen by users and those proposed by the controller. The optimization process also accounts for the independent reaction of the non-controlled fraction of users, framing the problem as a Stackelberg game. Furthermore, the study explored the effects of the control strategy on the system at different penetration rates, defined as the percentage of users targeted by the controller. It also considered varying levels of user willingness to comply with the proposed routing strategy. The analysis aimed to understand how these two factors — penetration rate and user compliance — interact and influence the overall system efficiency, traffic flow, and congestion patterns.

To implement this model, we developed the Python library PyLate (7.1.4). This library enables the rapid setup of macroscopic traffic simulations on synthetic networks and, by leveraging the osmnx library, allows for the seamless import of real-world road networks from OpenStreetMap.

Hyperbolic-parabolic models for the management of traffic generated pollution

In 41, vehicular traffic flows through a merge regulated by traffic lights and produces pollutants that diffuse in the surrounding region. This situation motivates a general hyperbolic-parabolic system, whose well-posedness and stability are here proved in L1. Roads are allowed to be also 2–dimensional. The effects of stop & go waves are comprised, leading to measure source terms in the parabolic equation. The traffic lights, as well as inflows and outflows, can be regulated to minimize the presence of pollutant in given regions.

8.2 Nonlocal pedestrian flow models

Participants: Paola Goatin, Ilaria Ciaramaglia, Harols Deivi Contreras [Universidad San Sebastian, Chile], Daniel Inzunza [Universidad San Sebastian, Chile], Gabriella Puppo [Univ. Roma La Sapienza, Italy], Elena Rossi [Univ. Modena - Reggio Emilia, Italy], Luis-Miguel Villada [Universidad del Bio Bio, Chile].

In the framework of Ilaria Ciaramaglia 's PhD thesis, 27 provides the well-posedness of weak entropy solutions of a scalar non-local traffic flow model with time delay. Existence is obtained by convergence of finite volume approximate solutions constructed by Lax-Friedrich and Hilliges-Weidlich schemes, while the L1 stability with respect to the initial data and the delay parameter relies on a Kruzkov-type doubling of variable technique. Numerical tests are provided to illustrate the efficiency of the proposed schemes, as well as the solution dependence on the delay and look-ahead parameters.

In 28, we propose and study a nonlocal system of balance laws, which models the traffic dynamics on a two-lane and two-way road where drivers have a preferred lane (the lane on their right) and the other one is used only for overtaking. In this model, the convective part is intended to describe the intralane dynamics of vehicles: the flux function includes local and nonlocal terms, namely, the velocity function in each lane depends locally on the density of the class of vehicles traveling on their preferred lane and in a nonlocal form on the density of the class of vehicles overtaking in the opposite direction. The source terms are intended to describe the coupling between the two lanes: the overtaking and return criteria depend on weighted means of the downstream traffic density of the class of vehicles traveling in their preferred lane and of the class of vehicles traveling in the opposite direction on the same lane. We construct approximate solutions using a finite volume scheme and we prove existence of weak solutions by means of compactness estimates. We also show some numerical simulations to describe the behavior of the numerical solutions in different situations and to illustrate some features of model.

In 45, we consider a class of multi-population pedestrian models consisting in a system of nonlocal conservation laws coupled in the nonlocal components and describing several groups of pedestrians moving towards their respective targets while trying to avoid each other and the obstacles limiting the walking domain. Specifically, the nonlocal operators account for interactions occurring at the microscopic level as a reaction to the presence of other individuals or obstacles along the preferred path. In particular, the presence of obstacles is implemented in the nonlocal terms of the equations and not as classical boundary conditions. This allows to rewrite domain shape optimization problems as PDE-constrained problems. In this paper, we investigate the well-posedness of such optimization problems by proving the stability of solutions with respect to the positions and shapes of the obstacles. A differentiability result in the linear case is also provided. These properties are illustrated with a numerical example. See also 36.

8.3 Mean Field Games

Participants: Abderrahmane Habbal, Amal Machtalay [U Mohamed VI Polytech, Morocco] (UM6P), Imad Kissami [UM6P], Ahmed Ratnani [UM6P], Meryeme Jahid [UM6P], Lahcen Maniar [Univ. Cadi Ayyad, Marrakech, Morocco], S.E. Chorfi [Univ. Cadi Ayyad, Marrakech, Morocco].

• Two-class Traffic Flows 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 108.

• Degenerate mean-Field Game Systems We investigate inverse backward-in-time problems for a class of second-order degenerate Mean-Field Game (MFG) systems.

  More precisely, given the final datum at time $t=T$ of a solution to the one-dimensional mean-field game system with a degenerate diffusion coefficient, we aim to determine the intermediate states, at some $t=t_0$ for any $0\le t_0<T$, i.e., the value function and the mean distribution at intermediate times, respectively.

  We prove conditional stability estimates under suitable assumptions on the diffusion coefficient and the initial state $t=0$. The proofs are based on Carleman’s estimates with a simple weight function. We first prove a Carleman estimate for the Hamilton-Jacobi-Bellman (HJB) equation. A second Carleman estimate will be derived for the Fokker-Planck (FP) equation. Then, by combining the two estimates, we obtain a Carleman estimate for the mean-field game system, leading to the stability of the backward problems 26.

8.4 Fluid-structure interaction using isogeometric analysis

Participants: Régis Duvigneau.

The isogeometric analysis framework is used to develop an accurate numerical scheme for fluid-structure interaction problems. The approach relies on a Discontinuous Galerkin method for the flow (compressible Navier-Stokes equations) whereas a Continuous Galerkin method is employed for the structure (non-linear membrane or linear elasticity), both of them based on NURBS bases allowing an exact matching interface with a different resolution for each discipline. The approach is applied to the study of membrane wings.

8.5 Simulation of windblown sand

Participants: Régis Duvigneau, Adrien Bousseau [GraphDeco Project-Team], Guillaume Cordonnier [GraphDeco Project-Team], Nicolas Rosset [GraphDeco Project-Team].

In collaboration with GraphDeco team, numerical methods for fast simulations of sand erosion and deposition patterns has been developed for computer graphics applications in the context of N. Rosset's PhD thesis. The proposed approach based on geosciences principles has been validated against experimental results as well as real world observations in 34.

8.6 Learning strategies for PDEs

Participants: Guillaume Coulaud, Régis Duvigneau, Nathan Ricard, Maxime Le [SED Centre Inria d'Université Côte d'Azur].

We investigate the use of novel machine learning paradigms in the context of complex PDE systems, including the following research axes:

• Multiphysic coupling using physics-informed neural networks

  Physics-Informed Neural Networks (PINNs) have emerged as a promising paradigm to handle diverse scenarios to simulate multiphysic systems, by embedding the different PDE models and coupling conditions in a single learning task. This is investigated in 35 in the context of conjugate heat transfer, during G. Coulaud's Master thesis.

• Turbulence inference 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. PINNs 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 in 35 to simulate turbulent flows by infering turbulence characteristics from mean flow data in replacement to classical turbulence closures.

• Non-classical training approaches for physics-informed neural networks

  Above-mentioned studies put in light some specific issues for the training of PINNs, related to the presence of unbalanced terms in the loss functions, yielding the failure of first-order gradient algorithms classically used in machine learning. Thus, we investigate non-classical training strategies based on Nash games or hierarchical methods. This work is part of N. Ricard's PhD thesis.

The three latter activities have benefited from SED support for the development of pinnacle software (7.1.2) devoted to PINNs.

8.7 Advanced Bayesian optimization

Participants: Ayoub Bellouch [Atlantis team], Luca Berti [IRMA, Institut de Recherche Mathématique Avancée], Mickaël Binois, Nicholson Collier [Argonne, USA], Régis Duvigneau, Arindam Fadikar, Laëtitia Giraldi [Calisto team], Stéphane Lanteri [Atlantis team], Jonathan Ozik [Argonne, USA], Lucas Palazzolo [Calisto team], Victor Picheny [SecondMind, GB], Abby Stevens [Argonne, USA].

• Handling of noisy simulators

  In 38, we present specific aspects of Gaussian process modeling in the presence of complex noise. Starting from the standard homoscedastic model, various generalizations from the literature are presented: input varying noise variance, non-Gaussian noise, or quantile modeling. These approaches are compared in terms of goal, data availability and inference procedure. A distinction is made between methods depending on their handling of repeated observations at the same location, also called replication. The chapter concludes with the corresponding adaptations of the sequential design procedures. These are illustrated in an example from epidemiology.

  A more realistic example is described by 46, showing a significant reduction of the number of simulations required to calibrate the parameters of a Covid-19 simulator.

• High-dimensional Gaussian process modeling

  In an effort to improve the scalability of Bayesian optimization based on Gaussian processes, we propose a new approach in 39. Gaussian processes are a widely embraced technique for regression and classification due to their good prediction accuracy, analytical tractability and built-in capabilities for uncertainty quantification. However, they suffer from the curse of dimensionality whenever the number of variables increases. This challenge is generally addressed by assuming additional structure in the problem, the preferred options being either additivity or low intrinsic dimensionality. Our contribution for high-dimensional Gaussian process modeling is to combine them with a multi-fidelity strategy, showcasing the advantages through experiments on synthetic functions and datasets.

• Bayesian optimization of micro-swimmers

  In 47, we are interested in understanding and optimizing the design of helical micro-swimmers. This is crucial for advancing their application in various fields. This study presents an innovative approach combining Free-Form Deformation with Bayesian Optimization to enhance the shape of these swimmers. Our method facilitates the computation of generic swimmer shapes that achieve optimal average speed and efficiency. Applied to both monoflagellated and biflagellated swimmers, our optimization framework has led to the identification of new optimal shapes. These shapes are compared with biological counterparts, highlighting a diverse range of swimmers, including both pushers and pullers.

  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 25.

  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.

• 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 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. In the context of design optimization, we especially investigate the use of a multi-task entropy search approach, with applications to aerodynamics and nano-photonics (in collaboration with the Atlantis team).

8.8 Pareto optimality and Nash games

Participants: Jean-Antoine Désidéri, Mickaël Binois, Nathalie Bartoli [ONERA/DTIS, Université de Toulouse], Christophe David [ONERA/DTIS, Université de Toulouse], Sébastien Defoort [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 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 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) 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 in 43 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.

In the present prioritized approach for multiobjective optimization, after a first phase of optimization has produced a Primary Pareto Front relative to the sole primary cost functions (under functional constraints), considered to be preponderant, a second phase of optimization is initiated to yield a continuum of Nash equilibria of quasi-Pareto-optimal solutions with respect to the whole set of objective functions (primary and secondary). This second phase relies on an orthogonal decomposition of the working space, a subset of ${𝐑}^N$, referred to as “territory splitting”.

We have generalized the original method 7 by relaxing the convergence condition on the construction of the territory splitting to isolate the affine subspace locally tangent to the constraints, proposed several alternatives, and tested their efficacy on a testcase of optimal sizing of an aluminum sandwich panel 42.

8.9 Modeling, Simulation and Optimization of Solid State Electrolyte batteries

Participants: Abderrahmane Habbal, Mustapha Bouchaara [U Mohamed VI Polytech, Morocco, UM6P], Ahmed Ratnani [UM6P].

The demand for advanced energy storage drives an urgency to accelerate material discovery in solid-state electrolytes. In pursuit of this aim, this study presents an innovative methodology that integrates materials science insights with machine learning techniques to improve the ionic conductivity prediction in garnet-based solid electrolytes. Utilizing an expanded dataset comprising 362 data points, and exploiting easily obtainable pre-synthesis inputs, our approach incorporates rigorous data preprocessing inspired by materials science and machine learning methodologies. Through systematic feature selection and hyperparameter tuning, the model achieved an improved R-squared value of 0.85. This study highlights the efficacy of the proposed approach and underscores the potential of machine learning in streamlining materials discovery and design for next-generation solid-state batteries 33.

8.10 Optimal transport and isogeometric analysis

Participants: Abderrahmane Habbal, Mustapha Bahari [U Mohamed VI Polytech, Morocco,UM6P], Ahmed Ratnani [UM6P], Eric Sonnendrücker [Max Planck Institute].

• Adaptive mesh generation using fast diagonalization 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 24.
• Mesh adaptation for time dependent problems We introduce a new algorithm designed to create a dynamic r-adaptive mesh within the framework of isogeometric analysis. The approach is based on the simultaneous computation of adaptive meshes using a nonlinear parabolic Monge–Ampere equation with a resolution of partial differential equations in multidimensional spaces. The technique ensures the absence of geometric boundary errors and is simple to implement, requiring the solution of only one Laplace scalar equation at each time step. It utilizes a fast diagonalization method that can be adapted to any dimension. Various numerical experiments were conducted to validate an original parabolic Monge–Ampere solver. The solver was respectively applied to Burgers, Allen–Cahn, and Cahn–Hilliard problems to demonstrate the efficiency of the new approach 23.

8.11 Cut-cells for Saint-Venant equations with moving obstacles

Participants: Laurent Monasse, Amal Chakib, Konstantin Brenner [Université Côte d'Azur].

During Amal Chakib's M2 internship, we investigated the extension of the cut-cell methodology developed in code CELIA2D for Euler equations to Saint-Venant equations for free-surface flows with moving obstacles. The ultimate goal is to simulate flooding phenomena where vehicles float and block the fluid flow, increasing the damage to structures and casualties. The investigations allowed to extend the numerical scheme for well-balanced shallow water on real topography data with bottom friction. Validation and benchmarking against state-of-the-art methods are in development.

8.12 Shock fitting with cut cell methods

Participants: Laurent Monasse, Alexandre Vieira, Régis Duvigneau.

Compressible fluids develop shocks in finite time and transport initial material discontinuities. Accurately tracking these discontinuities in the fluid state is numerically challenging and can lead to numerical smearing of discontinuities and loss of accuracy. This is especially difficult for discontinuities in material density and behavior laws. Instead of discontinuity capturing methods, we have proposed to use discontinuity tracking methods by following material discontinuities in a Lagrangian way. In order to enable wave interaction and topology changes, we combine discontinuity tracking with cut cell methods. We have extended the classical Finite Volume framework to Discontinuous Galerkin methods in space-time.

8.13 Fungal growth modeling and simulation

Participants: Laurent Monasse, Yves D'Angelo [Université Côte d'Azur], Rémi Catellier [Université Côte d'Azur], Claire Guerrier [Université Côte d'Azur], Nicolas Fricker [Université Côte d'Azur].

Fungi develop growing networks in the form of mycellium which explore space by branching at the tips (apex branching) and on the existing network (lateral branching). We have studied fungal growth at two scales: a large scale (of the order of the Petri dish) where the behavior of the fungus is homogenized, and a small scale (of the order of the fungus cell) in order to understand the undelying biochemical phenomena at hand in growth and branching. On the large scale, we have proposed a new partial differential equation (PDE) model for the dynamics of growth. We have computed front propagation velocities and proposed numerical schemes to approximate the solution in space and time of the PDE. The work is in its final process to submission. On the other hand, on the small scale, we have proposed a biologically informed model of filament growth. We have developed a fast and accurate numerical scheme accounting for the absorption of nutrients, their conversion to vesicles and the transport of vesicles in the cell, inducing the growth of the filament. The model is in the process of fitting unknown parameters to experimental biological data.

9.1 Bilateral contracts with industry

• Mycophyto (2020-2024): 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.

• IFPEN (2022-2025): this research contract is financing the PhD thesis of Agatha Joumaa on “A multi-mode macroscopic traffic model for the improvement of mobility and air quality in our cities via optimal modal share and routing”'.

  Participants: Paola Goatin, Agatha Joumaa.

• Altair (2024-2025): this research contract involving AMIES support finances the post-doctoral contract of Alexandre Vieira. The goal is to develop compressible multimaterial flow simulation using cut-cell methods in the open-source code OpenRadioss.

  Participants: Laurent Monasse, Alexandre Vieira.

10.1 European initiatives

10.2 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, 963-983HAL
• 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 textback 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

12.2 Publications of the year

12.3 Cited publications

• 48 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
• 49 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
• 50 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
• 51 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
• 52 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
• 53 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
• 54 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
• 55 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
• 56 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
• 57 unpublishedE.Enrico Bertino, R.Régis Duvigneau and P.Paola Goatin. Uncertainties in traffic flow and model validation on GPS data.2015back to text
• 58 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
• 59 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
• 60 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
• 61 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
• 62 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
• 63 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
• 64 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
• 65 techreportA.Alessandra Cabassi and P.Paola Goatin. Validation of traffic flow models on processed GPS data.Research Report RR-83822013HALback to text
• 66 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
• 67 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
• 68 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
• 69 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
• 70 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
• 71 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
• 72 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
• 73 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
• 74 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
• 75 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
• 76 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
• 77 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
• 78 bookE.Emiliano Cristiani, B.Benedetto Piccoli and A.Andrea Tosin. Multiscale modeling of pedestrian dynamics.12MS&A. Modeling, Simulation and ApplicationsSpringer, Cham2014back to text
• 79 incollectionC. M.C. M. Dafermos. Solutions in $L^{}$ for a conservation law with memory.Analyse mathématique et applicationsMontrougeGauthier-Villars1988, 117--128back to text
• 80 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
• 81 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
• 82 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
• 83 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
• 84 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
• 85 techreportJ.-A.Jean-Antoine Désidéri. Révision de l'algorithme de descente à gradients multiples (MGDA) par orthogonalisation hiérarchique.8710INRIAApril 2015back to text
• 86 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
• 87 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
• 88 articleR. J.Ronald J. DiPerna. Measure-valued solutions to conservation laws.Arch. Rational Mech. Anal.8,831985, 223--270back to text
• 89 articleC.Christian Dogbé. Modeling crowd dynamics by the mean-field limit approach.Math. Comput. Modelling529-102010, 1506--1520back to text
• 90 techreportR.R. Duvigneau. A Sensitivity Equation Method for Unsteady Compressible Flows: Implementation and Verification.INRIA Research Report No 8739June 2015back to text
• 91 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
• 92 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
• 93 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
• 94 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
• 95 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
• 96 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
• 97 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
• 98 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
• 99 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
• 100 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
• 101 articleP.Paola Goatin and M.Matthias Mimault. A mixed system modeling two-directional pedestrian flows.Math. Biosci. Eng.1222015, 375--392back to text
• 102 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
• 103 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
• 104 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
• 105 articleA.A. Griewank. Achieving logarithmic growth of temporal and spatial complexity in reverse automatic differentiation.Optimization Methods and Software11992, 35-54back to text
• 106 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
• 107 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
• 108 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
• 109 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
• 110 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
• 111 articleD.Dirk Helbing. Traffic and related self-driven many-particle systems.Rev. Mod. Phys.7342001, 1067--1141back to text
• 112 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
• 113 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
• 114 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
• 115 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
• 116 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
• 117 articleS.S. Jeon and H.H. Choi. Suboptimal feedback control of flow over a sphere.Int. J. of Heat and Fluid Flow312010back to text
• 118 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
• 119 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
• 120 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
• 121 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
• 122 articleJ.-M.Jean-Michel Lasry and P.-L.Pierre-Louis Lions. Mean field games.Jpn. J. Math.212007, 229--260back to text
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