2025Activity reportProject-Team​​ACUMES

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 122, or​ to the dynamics of​‌ turbulent flows for which​​ large scale / small​​​‌ scale vortical structures interfere​ 149.

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 160​​​‌.

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 123​​. Further improvements require​​​‌ more advanced models, keeping​ into better account interactions​‌ at the microscopic scale,​​ and adapted control techniques,​​​‌ see 75 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 88, 113​, 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 128.
• 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 124​ 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​ 82, 81),​‌ 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 95‌, 112 or via‌​‌ the Hamilton-Jacobi approach 82​​, 81 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 138.​‌ 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 120​, 130, we​‌ have demonstrated that Nash​​ game approaches are efficient​​​‌ to tackle ill-posedness. We​ intend to extend the​‌ results obtained for Laplace​​ equations in 120,​​​‌ 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.‌​‌ 146, 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 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.

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 optimization is​‌ expected to strikingly improve​​ industrial properties, such as​​​‌ higher energy densities, longer​ lifespans, cost-effectiveness, low maintenance,​‌ and enhanced safety. These​​ objectives are under investigation​​​‌ within the PhD thesis​ of M. Bouchaara (UM6P,​‌ Morocco).

6.1 Awards​​

Paola Goatin : Chevalier​​​‌ de l'Ordre National du​ Mérite, French Republic​‌ President award (2025).

7.1 Latest​ software developments

8.1 Macroscopic​​​‌ traffic flow models on​ networks

Participants: Eric Andoni​‌, Paola Goatin,​​ Rinaldo Colombo [Univ. Brescia,​​​‌ Italy], Chiara Daini​ [KOPERNIC Project-Team, Inria Paris]​‌, Maria Laura Delle​​ Monache [UC Berkeley, USA]​​​‌, Giovanni De Nunzio​ [IFPEN], Antonella Ferrara​‌ [Univ. Pavia, Italy],​​ Agatha Joumaa, Carmen​​​‌ Mezquita Nieto, Benedetto​ Piccoli [Rutgers U, USA]​‌, Elena Rossi [Univ.​​ Modena - Reggio Emilia,​​​‌ Italy].

Traffic control​ by Connected and Automated​‌ Vehicles (CAVs)

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, numerical​ tests show that few,​‌ optimally chosen CAVs are​​ sufficient to significantly improve​​​‌ the selected performance indexes,​ even using a decentralized​‌ control policy. The studies​​ 30 supports 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).

In the aim​​ of modeling the formation​​​‌ of stop-and-go waves (to​ be controlled employing CAVs),​‌ we also consider 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.​​ We then study the​​​‌ boundary stabilization of these​ $2\times 2$ nonlinear​‌ hyperbolic systems of balance​​ equations with a relaxation-type​​​‌ source term in Lagrangian​ coordinates. Since the largest​‌ eigenvalue of the system​​ is null, the boundaries​​​‌ are characteristic, and the​ available results on boundary​‌ controllability do not apply.​​ Yet, we are able​​​‌ to prove that weak​ solutions can be steered​‌ to an equilibrium state​​ by prescribing the corresponding​​​‌ equilibrium speed at the​ right boundary. This corresponds​‌ to controlling the speed​​ of one vehicle to​​​‌ stabilize the upstream traffic​ flow 42.

Traffic​‌ flow optimization on road​​ 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.

Agatha Joumaa​ 's PhD thesis 46​‌ focused in particular on​​ the design and testing​​​‌ of class-specific variable speed​ limits to mitigate the​‌ impact of traffic congestion​​ 43. This paper​​​‌ presents a novel approach​ to traffic management in​‌ road networks, consisting in​​ time-varying class-specific variable speed​​​‌ limit (VSL) restricted to​ a fraction of road​‌ users. In particular, we​​ present a macroscopic approach​​ where traffic dynamics is​​​‌ described by a multi-class‌ Lighthill-Whitham-Richards (LWR) model, with‌​‌ two classes of users​​ (controlled and uncontrolled vehicles).​​​‌ The model can be‌ applied to a general‌​‌ road network and our​​ goal is to optimize​​​‌ traffic performance by minimizing‌ both the average travel‌​‌ time in the network​​ and the total time​​​‌ spent in virtual buffers‌ at the network entries‌​‌ to prevent spill-back scenarios.​​ The optimization is performed​​​‌ by acting on different‌ ratios of controlled vehicles,‌​‌ and we compare the​​ performance of the proposed​​​‌ control strategy with a‌ classical inflow control at‌​‌ the entries of the​​ network. The numerical tests​​​‌ show that class-specific speed‌ control outperforms inflow control,‌​‌ and highlight the importance​​ of tailored traffic control​​​‌ strategies for road networks,‌ offering insights into optimizing‌​‌ mobility, safety, and traffic​​ efficiency.

Besides, in her​​​‌ PhD, Carmen Mezquita Nieto‌ applies a discrete adjoint‌​‌ gradient computation method to​​ the multi-class traffic flow​​​‌ model on networks, where‌ vehicle classes are characterized‌​‌ by their speed functions.​​ The resulting hyperbolic system​​​‌ of conservation laws is‌ discretized using a Godunov-type‌​‌ finite volume scheme based​​ on demand and supply​​​‌ functions, which extends to‌ coupling conditions at junctions‌​‌ and boundary conditions. The​​ optimization of the different​​​‌ travel metrics is accomplished‌ through the definition of‌​‌ a minimization problem using​​ the adjoint gradient method.​​​‌ Numerical simulations are also‌ presented to illustrate the‌​‌ efficiency of the method​​ on a test network,​​​‌ see 54.

Hyperbolic-parabolic‌ models for the management‌​‌ of traffic generated pollution​​

In 28, 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 $L^1$.‌ 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 flow models

Participants: Paola‌​‌ Goatin, Ilaria Ciaramaglia​​, Harold Deivi Contreras​​​‌ [Universidad San Sebastian, Chile]‌, Simone Göttlich [Univ.‌​‌ Mannheim, Germany], 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], Fabian​​ Ziegler [Univ. Mannheim, Germany]​​​‌.

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 $L^{\mathrm{1‌}}$-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.

Besides,​​​‌ in 26, we​ present a class of​‌ systems of non-local conservation​​ laws in one space-dimension​​​‌ incorporating time delay, which​ can be used to​‌ investigate the interaction between​​ autonomous and human-driven vehicles,​​​‌ each characterized by a​ different reaction time and​‌ interaction range. We construct​​ approximate solutions using a​​​‌ Hilliges-Weidlich scheme and we​ provide uniform $L^{\mathrm{\infty ‌}}$ and BV estimates which​​ ensure the convergence of​​​‌ the scheme, thus obtaining​ existence of entropy weak​‌ solutions of bounded variation.​​ Uniqueness follows from an​​​‌ $L^1$ stability result​ derived from the entropy​‌ condition. Additionally, we provide​​ numerical simulations to illustrate​​​‌ applications to mixed autonomous​ / human-driven traffic flow​‌ modeling. In particular, we​​ show that the presence​​​‌ of autonomous vehicles improves​ overall traffic flow and​‌ stability.

In 29,​​ 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 48,​‌ we propose and analyze​​ the well-posedness of a​​​‌ new class of macroscopic​ vehicular traffic model described​‌ by a scalar nonlocal​​ conservation law that simultaneously​​​‌ incorporates both upstream and​ downstream effects in the​‌ flow dynamics. Unlike nonlocal​​ models previously described in​​​‌ the literature, which only​ account for downstream density​‌ averages (look-ahead behavior), the​​ proposed model introduces an​​​‌ additional term depending on​ an upstream average (look-behind),​‌ allowing for a more​​ realistic representation of anticipatory​​​‌ driver behavior under high-density​ conditions. The main novelty​‌ of this work lies​​ in establishing the existence​​​‌ and uniqueness theory for​ entropy weak solutions, together​‌ with a rigorous proof​​ of Lipschitz continuous dependence​​​‌ of solutions not only​ on the initial data,​‌ but also on the​​ kernel functions, under reasonable​​​‌ structural assumptions on the​ flux components. The proofs​‌ are achieved through the​​ design of a conservative​​​‌ numerical scheme that preserves​ key structural properties of​‌ the continuous model, such​​ as maximum principle, mass​​ conservation, BV estimates, and​​​‌ $L^1$-stability. Finally,‌ we present numerical experiments‌​‌ that illustrate the behavior​​ of solutions and the​​​‌ qualitative impact of nonlocal‌ terms on traffic dynamics.‌​‌

In 32, 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 31.‌

Finally, in 53,‌​‌ a finite volume approximation​​ scheme is used to​​​‌ solve a non-local macroscopic‌ material flow model in‌​‌ two space dimensions, accounting​​ for the presence of​​​‌ boundaries in the non-local‌ terms. Based on a‌​‌ previous result for the​​ scalar case, we extend​​​‌ the setting to a‌ system of heterogeneous material‌​‌ on bounded domains. We​​ prove the convergence of​​​‌ the approximate solutions constructed‌ using the Roe scheme‌​‌ with dimensional splitting, where​​ the major challenge lies​​​‌ in the treatment of‌ the discontinuity occurring in‌​‌ the flux function. Numerical​​ tests show a good​​​‌ agreement with microscopic simulations.‌

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 $\mathrm{ϵ‌}-$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 33​‌.

• 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​ $0\le t_{\mathrm{0‌}}<T$, 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 $\mathrm{t‌}=0$. 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 25.​‌

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, by using​ a mixed continuous /​‌ discontinuous Galerkin scheme 45​​. The properties of​​​‌ NURBS basis functions are​ leveraged to enable an​‌ exact transfer of the​​ structural displacement to the​​​‌ fluid domain, while using​ different discretizations and refinements​‌ on the two sides​​ of the coupling interface​​​‌ 50. The proposed​ approach is validated by​‌ the simulation of a​​ compressible flow around an​​​‌ elastic wing membrane and​ the classical fluid-structure benchmark​‌ proposed by Turek &​​ Hron, involving the flow​​​‌ around a cylinder equipped​ with a hyperelastic bar,​‌ demonstrating the interest of​​ high-order treatment of the​​​‌ coupling interface.

8.5 Physics-informed​ neural networks

Participants: Mickaël​‌ Binois, Régis Duvigneau​​, Laurent Monasse,​​​‌ Mathilde Pascal, Nathan​ Ricard, Marion Vidal​‌.

We investigate the​​ use of the novel​​​‌ Physics-Informed Neural Networks (PINNs)​ paradigm in the context​‌ of complex PDE/ODE systems,​​ including the following research​​​‌ axes:

• Multi-objective learning for​ physics-informed neural networks

  It​‌ was observed that PINNs​​ suffer from the difficulty​​​‌ to minimize simultaneously the​ loss function reflecting the​‌ physical principles (ODEs or​​ PDEs) and the one​​​‌ fitting the data (boundary​ conditions and observations). To​‌ overcome this difficulty, we​​ investigated the use of​​​‌ multi-objective strategies for PINN​ learning, based on game​‌ theory and multi-criteria descent​​ algorithms 44, 59​​​‌. In particular, we​ aim at accounting for​‌ the conflicts in the​​ gradient directions and amplitudes​​​‌ to define an efficient​ learning strategy. This work​‌ is achieved in the​​ framework of N. Ricard's​​​‌ PhD thesis.

• Physics-informed neural​ networks for shock fitting​‌

  To overcome the limitation​​ of PINNs in describing​​​‌ discontinuous solutions, we investigated​ a shock-fitting formulation, in​‌ which a set of​​ neural networks are used​​ to represent the solution​​​‌ fields in the continuous‌ regions, coupled with another‌​‌ network that defines the​​ discontinuity characteristics. All of​​​‌ them are trained using‌ physical principles, either PDEs‌​‌ or Rankine-Hugoniot conditions. This​​ approach is investigated for​​​‌ Burgers equation 56,‌ revealing a real potential‌​‌ for the design of​​ a parametric shock fitting​​​‌ method for shock waves.‌

• Physics-informed neural networks for‌​‌ gene dynamics identification

  We​​ also experimented the PINN​​​‌ formulation in a biology‌ context. To defend themselves‌​‌ against biotic threats, plants​​ rely on a complex​​​‌ network of signaling pathways‌ that regulate immune responses.‌​‌ Understanding the dynamic modulation​​ of these defense mechanisms​​​‌ is therefore crucial for‌ advancing research in plant‌​‌ immunity and improving crop​​ resilience. In this context,​​​‌ we employed PINNs to‌ model gene expression dynamics‌​‌ in plants exposed to​​ biotic stress, on the​​​‌ basis of first-order linear‌ ordinary differential equations, whose‌​‌ coefficients are calibrated using​​ experiments during the learning​​​‌ phase 57. Results‌ were encouraging and underlined‌​‌ the necessity to work​​ with larger databases and​​​‌ account for gene interactions.‌

8.6 Including additional information‌​‌ in Gaussian process based​​ surrogates

Participants: Mickaël Binois​​​‌, Anna Flowers [Virginia‌ Tech, USA], Christopher‌​‌ Franck [Virginia Tech, USA]​​, Robert Gramacy [Virginia​​​‌ Tech, USA], Paola‌ Goatin, Ross Hammond‌​‌ [Washington University in Saint-Louis]​​, Chiwoo Park [University​​​‌ of Washington], Alexandra‌ Würth [Fraunhofer Institute, Germany]‌​‌.

Modeling output discontinuities​​

Gaussian processes (GPs) furnish​​​‌ accurate nonlinear predictions with‌ well-calibrated uncertainty. However, the‌​‌ typical GP setup has​​ a built-in stationarity assumption,​​​‌ making it ill-suited for‌ modeling data from processes‌​‌ with sudden changes, or​​ "jumps" in the output​​​‌ variable. The "jump GP"‌ (JGP) was developed for‌​‌ modeling data from such​​ processes, combining local GPs​​​‌ and latent "level" variables‌ under a joint inferential‌​‌ framework. But joint modeling​​ can be fraught with​​​‌ difficulty. In 52 we‌ aim to simplify by‌​‌ suggesting a more modular​​ setup, eschewing joint inference​​​‌ but retaining the main‌ JGP themes: (a) learning‌​‌ optimal neighborhood sizes that​​ locally respect manifolds of​​​‌ discontinuity; and (b) a‌ new cluster-based (latent) feature‌​‌ to capture regions of​​ distinct output levels on​​​‌ both sides of the‌ manifold. We show that‌​‌ each of (a) and​​ (b) separately leads to​​​‌ dramatic improvements when modeling‌ processes with jumps. In‌​‌ tandem (but without requiring​​ joint inference) that benefit​​​‌ is compounded, as illustrated‌ on real and synthetic‌​‌ benchmark examples from the​​ recent literature.

Physics-informed modeling​​​‌

In 38, we‌ propose a physics informed‌​‌ statistical framework for traffic​​ travel time prediction. This​​​‌ combined approach has the‌ merit to address the‌​‌ shortcomings of the purely​​ model-driven or data-driven approaches,​​​‌ while leveraging their respective‌ advantages. Indeed, models are‌​‌ based on physical laws,​​ but cannot capture all​​​‌ the complexity of real‌ phenomena. Plus they are‌​‌ rarely used for prediction​​ since this requires future​​​‌ data such as boundary‌ conditions. On the other‌​‌ hand, pure statistical outputs​​ can violate basic characteristic​​​‌ dynamics in their prediction‌ and do not reconstruct‌​‌ traffic conditions. Here, on​​​‌ one side, the discrepancy​ of the considered mathematical​‌ model with real data​​ is represented by a​​​‌ Gaussian process. On the​ other side, the traffic​‌ simulator is fed with​​ boundary data predicted by​​​‌ a Gaussian process, forced​ to satisfy the mathematical​‌ equations at virtual points,​​ resulting in a multi-objective​​​‌ optimization problem. We validate​ our approach on both​‌ synthetic and real world​​ data, showing that it​​​‌ delivers more reliable results​ compared to other methods.​‌

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​‌, Roman Garnett [Washington​​ University in Saint-Louis],​​​‌ Laëtitia Giraldi [Calisto team]​, Ross Hammond [Washington​‌ University in Saint-Louis],​​ Cliff Kerr [Bill and​​​‌ Melinda Gates foundation],​ Daniel Klein [Bill and​‌ Melinda Gates foundation],​​ Stéphane Lanteri [Atlantis team]​​​‌, Jeffrey Larson [Argonne,​ USA], David O'Gara​‌ [Washington University in Saint-Louis]​​, Jonathan Ozik [Argonne,​​​‌ USA], Lucas Palazzolo​ [Calisto team], Abby​‌ Stevens [Argonne, USA].​​

• Handling of noisy simulators​​​‌

  To reduce the number​ of call to epidemiology​‌ simulators, we show in​​ 36 that heteroscedastic Gaussian​​​‌ process modeling can greatly​ help. It shows a​‌ significant reduction of the​​ number of simulations required​​​‌ to calibrate the parameters​ of a Covid-19 simulator.​‌ A Python version of​​ the corresponding Gaussian process​​​‌ regression code is provided​ with hetGPy35.​‌

  Bayesian optimization (BO) is​​ a powerful framework for​​​‌ estimating parameters of computationally​ expensive simulation models, particularly​‌ in settings where the​​ likelihood is intractable and​​​‌ evaluations are costly. In​ stochastic models every simulation​‌ is run with a​​ specific parameter set and​​​‌ an implicit or explicit​ random seed, where each​‌ parameter set and random​​ seed combination generates an​​​‌ individual realization, or trajectory,​ sampled from an underlying​‌ random process. Existing BO​​ approaches typically rely on​​​‌ summary statistics over the​ realizations, such as means,​‌ medians, or quantiles, potentially​​ limiting their effectiveness when​​​‌ trajectory-level information is desired.​ In 51 we propose​‌ a trajectory-oriented Bayesian optimization​​ method that incorporates a​​​‌ Gaussian process (GP) surrogate​ using both input parameters​‌ and random seeds as​​ inputs, enabling direct inference​​​‌ at the trajectory level.​ Using a common random​‌ number (CRN) approach, we​​ define a surrogate-based likelihood​​​‌ over trajectories and introduce​ an adaptive Thompson Sampling​‌ algorithm that refines a​​ fixed-size input grid through​​​‌ likelihood-based filtering and Metropolis-Hastings-based​ densification. This approach concentrates​‌ computation on statistically promising​​ regions of the input​​​‌ space while balancing exploration​ and exploitation. We apply​‌ the method to stochastic​​ epidemic models, a simple​​​‌ compartmental and a more​ computationally demanding agent-based model,​‌ demonstrating improved sampling efficiency​​ and faster identification of​​​‌ data-consistent trajectories relative to​ parameter-only inference.

  Concerning optimization,​‌ in 47 we develop​​ and analyze a method​​​‌ for stochastic simulation optimization​ relying on Gaussian process​‌ models within a trust-region​​ framework. We are interested​​​‌ in the case when​ the variance of the​‌ objective function is large.​​ We propose to rely​​ on replication and local​​​‌ modeling to cope with‌ this high-throughput regime, where‌​‌ the number of evaluations​​ may become large to​​​‌ get accurate results while‌ still keeping good performance.‌​‌ We propose several schemes​​ to encourage replication, from​​​‌ the choice of the‌ acquisition function to setup‌​‌ evaluation costs. Compared with​​ existing methods, our results​​​‌ indicate good scaling, in‌ terms of both accuracy‌​‌ (several orders of magnitude​​ better than existing methods)​​​‌ and speed (taking into‌ account evaluation costs).

• Bayesian‌​‌ optimization of micro-swimmers

  In​​ 37, 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.​​

  Then in 55 we​​​‌ consider optimizing controls for‌ such micro-swimmers. Unlike macroscopic‌​‌ swimmers, microswimmers operate in​​ a low-Reynolds-number regime dominated​​​‌ by viscous forces. This‌ paper investigates the controllability‌​‌ of a magnetic microswimmer​​ composed of a spherical​​​‌ magnetic head and an‌ elastic, non-magnetic flagellum. The‌​‌ swimmer evolves in a​​ Stokes flow and is​​​‌ modeled using the resistive‌ force theory. We prove‌​‌ that, under planar motion,​​ the system is not​​​‌ small-time locally controllable and‌ numerically identify regions that‌​‌ remain inaccessible. Nevertheless, simulations​​ show that trajectory tracking​​​‌ can still be achieved‌ via Bayesian optimization, though‌​‌ it requires large-amplitude transverse​​ deformations. This work was​​​‌ also presented during the‌ SAMO conference (International Conference‌​‌ on Sensitivity Analysis of​​ Model Output) 58.​​​‌

• 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​​​‌ 24.

  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). Results​​ have been presented in​​​‌ 40.

• Multi-objective Bayesian​ optimization with decoupled objectives​‌

  In 41 we look​​ at the experimental design​​​‌ for multi-objective problems, where​ the objectives can be​‌ evaluated independently (decoupled) and​​ thus it may make​​​‌ sense to evaluate different​ solutions for each objective​‌ if the objectives have​​ different evaluation costs and/or​​​‌ different landscape characteristics. We​ propose to iteratively add​‌ design points in a​​ way that minimizes the​​​‌ total integrated mean squared​ prediction error assuming a​‌ Gaussian process response surface​​ model, and show that​​​‌ allowing decoupled evaluations can​ lead to significantly better​‌ Pareto front estimations than​​ a coupled design of​​​‌ experiments if the evaluation​ costs of the objectives​‌ are different. We also​​ find that our approach​​​‌ of minimizing mean squared​ prediction error yields significantly​‌ better results than standard​​ Latin Hypercube designs even​​​‌ if the evaluation costs​ and landscape characteristics of​‌ the objectives are the​​ same.

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 13 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 49​​​‌.

A publication agreement‌ has been established between‌​‌ the Society for Industrial​​ and Applied Mathematics (SIAM)​​​‌ and Jean-Antoine Désidéri for‌ a book entitled ”Multiobjective‌​‌ of Smooth Functionals -​​ Application to Aeronautics” (Series:​​​‌ Advances in Design and‌ Control).

8.9 Inverse Problems‌​‌ solved as Nash games​​

Participants: Abderrahmane Habbal,​​​‌ Marwa Ouni [U. Tunis‌ al Manar, Tunisia].‌​‌

• Nash games for shape​​ and boundary identification of​​​‌ nonlinear PDEs. We‌ investigate nonlinear Cauchy-type problems‌​‌ arising in quasi-Newtonian Stokes​​ flows, where the viscosity​​​‌ exhibits a nonlinear dependence‌ on the deformation tensor,‌​‌ modeled by the Carreau​​ law. To tackle the​​​‌ inherent ill-posedness of the‌ Cauchy-Stokes problem, we propose‌​‌ three iterative methods, each​​ reformulating the original problem​​​‌ into a sequence of‌ well-posed mixed boundary value‌​‌ problems (BVPs). A classical​​ control framework is employed​​​‌ to construct a control-type‌ algorithm for the nonlinear‌​‌ inverse problem. Then, we​​ introduce two novel algorithms​​​‌ based on a Nash‌ game formulation; the second‌​‌ algorithm enables each player​​ to linearize the adverse​​​‌ state equations, enhancing computational‌ efficiency and convergence. We‌​‌ further extend this linearized​​ Nash approach to simultaneously​​​‌ recover missing boundary data‌ and identify the location‌​‌ and shape of unknown​​ inclusions. Finite element simulations​​​‌ validate the robustness and‌ effectiveness of the proposed‌​‌ methods 34.

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].

Optimal‌​‌ Transport for adaptive isogeometric​​ analysis. Optimal transport offers​​​‌ a powerful mathematical framework‌ for redistributing geometric or‌​‌ computational resources in adaptive​​ isogeometric analysis (IGA). By​​​‌ treating mesh refinement as‌ a mass-transport problem, optimal‌​‌ transport maps allow to​​ smoothly reposition control points​​​‌ or redistribute quadrature/parameterization density‌ according to error indicators.‌​‌ This leads to adaptive​​ refinements that preserve geometric​​​‌ fidelity, avoid mesh distortion,‌ and maintain the smoothness‌​‌ inherent to IGA bases.​​ As a result, optimal-transport–driven​​​‌ adaptivity provides a principled,‌ variational way to enhance‌​‌ accuracy while controlling computational​​ cost.

• Image Registration and​​​‌ Segmentation Using IGA with‌ Optimal Transport. We have‌​‌ developed a novel fast​​​‌ and high order method​ for the problem of​‌ Image Registration, using Optimal​​ Transport and the Isogeometric​​​‌ Analysis paradigm. Our method​ is based on the​‌ resolution of the Monge-Ampère​​ equation and ensures the​​​‌ one-to-one property. In addition,​ the use of B-Splines​‌ allows to create a​​ map that can be​​​‌ evaluated everywhere, and reduces​ the number of degrees​‌ of freedom needed to​​ store the constructed (gradient)​​​‌ map, by using e.g.​ high order B-Splines functions​‌ 39 .

8.11 Shock​​ fitting with cut cell​​​‌ methods

Participants: Laurent Monasse​, Alexandre Vieira,​‌ Régis Duvigneau, Mirco​​ Ciallella [Université Paris Cité]​​​‌.

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 with ADER​ time-integration in space-time.

8.12​‌ 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 underlying 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​

• Consortium CIROQUO - Consortium​‌ Industrie Recherche pour l'Optimisation​​ et la QUantification d'incertitude​​​‌ pour les données Onéreuses​ - gathers academical and​‌ technological partners to work​​ on problems related to​​​‌ the exploitation of numerical​ simulators. This Consortium, created​‌ in January 2021, is​​ the continuation of the​​ projects DICE, ReDICE and​​​‌ OQUAIDO which respectively covered‌ the periods 2006-2009, 2011-2015‌​‌ and 2015-2020. CIROQUO continued​​ from 2025 as CIROQUO​​​‌ 2 with new industrial‌ partners such as EDF‌​‌ or Michelin ciroquo.ec-lyon.fr.​​

  Participants: Mickaël Binois,​​​‌ Régis Duvigneau.

• IFPEN‌ (2022-2025): this research contract‌​‌ financed 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 (2024-2026): these​​​‌ research contracts involving AMIES‌ and PUI Med'Innov finance‌​‌ 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,​​​‌ in two and three‌ space dimensions, first with‌​‌ Lagrangian interface tracking, then​​ with a level-set description.​​​‌

  Participants: Laurent Monasse,‌ Alexandre Vieira.

10.1​​ International research visitors

10.2 European initiatives

10.3​​​‌ National initiatives

11.1 Promoting scientific​​​‌ activities

11.2 Teaching - Supervision​​ - Juries - Educational​​​‌ and pedagogical outreach

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 Analysis522​​2015, 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 online​​​‌2010, 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 Equations2342​​2007, 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 Equations25711​​​‌2014, 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.741​​​‌2014, 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, 54-63HAL​​DOI
• 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. Mathématique​‌Tome 350Fascicule 5-6​​March 2012, 313-318​​​‌HALDOI
• 12 report​J.-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 inproceedings​‌J.-A.Jean-Antoine Désidéri,​​ J.Julien Wintz,​​​‌ M.Mickael Binois,​ N.Nathalie Bartoli,​‌ C.Christophe David and​​ S.Sébastien Defoort.​​​‌ Prioritized multi-objective optimization of​ an aircraft flight performance​‌ based on Nash games​​ from preponderant Pareto-optimal points​​​‌.CM3 Transport 2023​ ConferenceJyvaskyla, Finland2023​‌HALback to text​​
• 14 articleR.R.​​​‌ Duvigneau and P.P.​ Chandrashekar. Kriging-based optimization​‌ applied to flow control​​.Int. J. for​​​‌ Numerical Methods in Fluids​69112012,​‌ 1701-1714
• 15 articleA.​​A. Habbal and M.​​​‌M. Kallel. Neumann-Dirichlet​ Nash strategies for the​‌ solution of elliptic Cauchy​​ problems.SIAM J.​​​‌ Control Optim.515​2013, 4066--4083
• 16​‌ 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.3811-12​‌2014, 3038--3053URL:​​ http://dx.doi.org/10.1016/j.apm.2013.11.034DOI
• 17 article​​​‌M.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 Fluids376​​2010
• 18 articleQ.​​​‌Quentin Mercier, F.​Fabrice Poirion and J.-A.​‌Jean-Antoine Desideri. A​​ stochastic multiple gradient descent​​​‌ algorithm.European Journal​ of Operational ResearchMay​‌ 2018, 10HAL​​DOI
• 19 articleS.​​​‌Souvik Roy, A.​A. Borzì and A.​‌Abderrahmane Habbal. Pedestrian​​ motion modelled by Fokker--Planck​​​‌ Nash games.Royal​ Society open science4​‌92017, 170648​​
• 20 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​​
• 21 articleM.M.​​​‌ Twarogowska, P.P.​ Goatin and R.R.​‌ Duvigneau. Macroscopic modeling​​ and simulations of room​​ evacuation.Appl. Math.​​​‌ Model.38242014‌, 5781--5795
• 22 article‌​‌G.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​​
• 23 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 biosciences​​2922017, 36--45​​​‌

12.2 Publications of the‌ year

12.3 Cited publications

• 61​​​‌ 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
• 62​​​‌ 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 Analysis522​​​‌2015, 963-983HAL​back to textback​‌ to text
• 63 article​​G.Giovanni Alessandrini.​​​‌ Examples of instability in​ inverse boundary-value problems.​‌Inverse Problems134​​1997, 887--897URL:​​​‌ http://dx.doi.org/10.1088/0266-5611/13/4/001DOIback to​ text
• 64 articleD.​‌Debora Amadori and W.​​Wen Shen. An​​​‌ integro-differential conservation law arising​ in a model of​‌ granular flow.J.​​ Hyperbolic Differ. Equ.9​​​‌12012, 105--131​back to text
• 65​‌ articleP.P. Amorim​​, R. M.Rinaldo​​​‌ M. Colombo and A.​A. Teixeira. On​‌ the Numerical Integration of​​ Scalar Nonlocal Conservation Laws​​​‌.ESAIM M2AN49​12015, 19--37​‌back to text
• 66​​ articleP.Paulo Amorim​​​‌. On a nonlocal​ hyperbolic conservation law arising​‌ from a gradient constraint​​ problem.Bull. Braz.​​​‌ Math. Soc. (N.S.)43​42012, 599--614​‌back to text
• 67​​ articleM.M. Annunziato​​​‌ and A.A. Borz\`i​. A Fokker-Planck control​‌ framework for multidimensional stochastic​​ processes.Journal of​​​‌ Computational and Applied Mathematics​2372013, 487-507​‌back to text
• 68​​ articleA.A. Belme​​​‌, F.F. Alauzet​ and A.A. Dervieux​‌. Time accurate anisotropic​​ goal-oriented mesh adaptation for​​​‌ unsteady flows.J.​ Comput. Physics23119​‌2012, 6323--6348back​​ to text
• 69 article​​​‌S.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.462​​​‌20702006, 1791--1803‌back to text
• 70‌​‌ unpublishedE.Enrico Bertino​​, R.Régis Duvigneau​​​‌ and P.Paola Goatin‌. Uncertainties in traffic‌​‌ flow and model validation​​ on GPS data.​​​‌2015back to text‌
• 71 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
• 72 article​​S.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
• 73 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
• 74 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 Structures248​​2008, 1240--1251back​​​‌ to text
• 75 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​​
• 76 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
• 77 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
• 78 techreportA.​​Alessandra Cabassi and P.​​​‌Paola Goatin. Validation‌ of traffic flow models‌​‌ on processed GPS data​​.Research Report RR-8382​​​‌2013HALback to‌ text
• 79 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
• 80 unpublished‌​‌C.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
• 81 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
• 82‌​‌ 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 on55​​5May 2010,​​​‌ 1158-1174back to text​back to text
• 83​‌ 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 Sciences22​‌042012, 1150023​​back to text
• 84​​​‌ 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.17​​​‌22011, 353--379​back to text
• 85​‌ articleR. M.Rinaldo​​ M. Colombo and M.​​​‌Magali Lécureux-Mercier. Nonlocal​ crowd dynamics models for​‌ several populations.Acta​​ Math. Sci. Ser. B​​​‌ Engl. Ed.321​2012, 177--196back​‌ to text
• 86 article​​R. M.Rinaldo M.​​​‌ Colombo and F.Francesca​ Marcellini. A mixed​‌ ODE-PDE model for vehicular​​ traffic.Mathematical Methods​​​‌ in the Applied Sciences​3872015,​‌ 1292--1302back to text​​
• 87 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​​
• 88 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. S7​‌32014, 411--433​​back to text
• 89​​​‌ 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
• 90 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
• 91 bookE.​​​‌Emiliano Cristiani, B.​Benedetto Piccoli and A.​‌Andrea Tosin. Multiscale​​ modeling of pedestrian dynamics​​​‌.12MS&A. Modeling,​ Simulation and ApplicationsSpringer,​‌ Cham2014back to​​ text
• 92 incollectionC.​​​‌ M.C. M. Dafermos​. Solutions in ${\mathrm{L‌}}^{}$ for a conservation law​​ with memory.Analyse​​​‌ mathématique et applicationsMontrouge​Gauthier-Villars1988, 117--128​‌back to text
• 93​​ 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.24​‌12014, 93--115​​URL: http://dx.doi.org/10.1007/s00332-013-9185-2DOIback​​​‌ to text
• 94 article​M. 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. S7​‌32014, 435--447​​back to textback​​​‌ to text
• 95 article​M. 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​‌
• 96 inbookJ.-A.J.-A.​​ Désidéri. Multiple-Gradient Descent​​​‌ Algorithm (\em MGDA) for‌ Pareto-Front Identification.34‌​‌Numerical 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 Jubilees‌​‌Springer-Verlag2014, 1​​back to text
• 97​​​‌ articleJ.-A.Jean-Antoine Désidéri‌. Multiple-gradient descent algorithm‌​‌ (MGDA) for multiobjective optimization​​.Comptes Rendus de​​​‌ l'Académie des Sciences Paris‌3502012, 313-318‌​‌URL: http://dx.doi.org/10.1016/j.crma.2012.03.014back to​​ text
• 98 techreportJ.-A.​​​‌Jean-Antoine Désidéri. Révision‌ de l'algorithme de descente‌​‌ à gradients multiples (MGDA)​​ par orthogonalisation hiérarchique.​​​‌8710INRIAApril 2015‌back to text
• 99‌​‌ 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
• 100‌​‌ 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
• 101 article​​​‌R. J.Ronald J.‌ DiPerna. Measure-valued solutions‌​‌ to conservation laws.​​Arch. Rational Mech. Anal.​​​‌8,831985,‌ 223--270back to text‌​‌
• 102 articleC.Christian​​ Dogbé. Modeling crowd​​​‌ dynamics by the mean-field‌ limit approach.Math.‌​‌ Comput. Modelling529-10​​2010, 1506--1520back​​​‌ to text
• 103 techreport‌R.R. Duvigneau.‌​‌ A Sensitivity Equation Method​​ for Unsteady Compressible Flows:​​​‌ Implementation and Verification.‌INRIA Research Report No‌​‌ 8739June 2015back​​ to text
• 104 article​​​‌R.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 Dynamics20​​​‌7August 2006,‌ 497--512back to text‌​‌back to textback​​ to text
• 105 article​​​‌R.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.764​​​‌2014, 799--818URL:‌ http://dx.doi.org/10.1007/s11538-013-9844-3DOIback to‌​‌ text
• 106 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
• 107 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
• 108 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
• 109 article​​M. 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.2015.01.030DOI​​​‌back to text
• 110​ 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
• 111​​​‌ 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​‌
• 112 articleM.Mauro​​ Garavello and B.Benedetto​​​‌ Piccoli. Coupling of​ microscopic and phase transition​‌ models at boundary.​​Netw. Heterog. Media8​​​‌32013, 649--661​back to textback​‌ to text
• 113 book​​M.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
• 114 article​​​‌P.Paola Goatin and​ M.Matthias Mimault.​‌ A mixed system modeling​​ two-directional pedestrian flows.​​​‌Math. Biosci. Eng.12​22015, 375--392​‌back to text
• 115​​ 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
• 116 article​​P.Paola Goatin and​​​‌ S.Sheila Scialanga.​ Well-posedness and finite volume​‌ approximations of the LWR​​ traffic flow model with​​​‌ non-local velocity.Netw.​ Heterog. Media111​‌2016, 107--121back​​ to textback to​​​‌ textback to text​
• 117 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 Modelling​‌38132014,​​ 3295--3313back to text​​​‌
• 118 articleA.A.​ Griewank. Achieving logarithmic​‌ growth of temporal and​​ spatial complexity in reverse​​​‌ automatic differentiation.Optimization​ Methods and Software1​‌1992, 35-54back​​ to text
• 119 article​​​‌M.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.52​42014, 2141--2163​‌back to text
• 120​​ 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 text‌​‌back to text
• 121​​ articleX.X. Han​​​‌, P.P. Sagaut‌ and D.D. Lucor‌​‌. On sensitivity of​​ RANS simulations to uncertain​​​‌ turbulent inflow conditions.‌Computers & Fluids61‌​‌2-52012back to​​ textback to text​​​‌
• 122 articleD.Dirk‌ Helbing, P.Peter‌​‌ Molnar, I. J.​​Illes J Farkas and​​​‌ K.Kai Bolay.‌ Self-organizing pedestrian movement.‌​‌Environment and planning B​​2832001,​​​‌ 361--384back to text‌
• 123 articleD.Dirk‌​‌ Helbing. Traffic and​​ related self-driven many-particle systems​​​‌.Rev. Mod. Phys.‌7342001,‌​‌ 1067--1141back to text​​
• 124 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
• 125​​ 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 Applications‌​‌41602014,​​ 684--694back to text​​​‌
• 126 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
• 127​​ 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 Engineering​​1942005, 4135--4195​​​‌back to text
• 128‌ 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
• 129 article​​​‌S.S. Jeon and‌ H.H. Choi.‌​‌ Suboptimal feedback control of​​ flow over a sphere​​​‌.Int. J. of‌ Heat and Fluid Flow‌​‌312010back to​​ text
• 130 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
• 131‌​‌ articleO.O.M. Knio​​ and O.O. Le​​​‌ Maitre. Uncertainty propagation‌ in CFD using polynomial‌​‌ chaos decomposition.Fluid​​ Dynamics Research389​​​‌September 2006, 616--640‌back to text
• 132‌​‌ 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
• 133 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: Methodological4510​​2011, 1572--1589back​​​‌ to text
• 134 article​J.-M.Jean-Michel Lasry and​‌ P.-L.Pierre-Louis Lions.​​ Mean field games.​​​‌Jpn. J. Math.2​12007, 229--260​‌back to text
• 135​​ articleM. J.Michael​​​‌ J. Lighthill and G.​ B.Gerald B. Whitham​‌. On kinematic waves.​​ II. A theory of​​​‌ traffic flow on long​ crowded roads.Proc.​‌ Roy. Soc. London. Ser.​​ A.2291955,​​​‌ 317--345back to text​
• 136 articleG.G.​‌ Lin, C.-H.C.-H.​​ Su and G.G.E.​​​‌ Karniadakis. Predicting shock​ dynamics in the presence​‌ of uncertainties.Journal​​ of Computational Physics217​​​‌2006, 260-276back​ to text
• 137 article​‌M.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 Fluids376​2010back to text​‌
• 138 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.2198​‌2002back to text​​
• 139 articleS.Siddhartha​​​‌ Mishra, C.Christoph​ Schwab and J.Jonas​‌ Sukys. Multi-level Monte​​ Carlo finite volume methods​​​‌ for uncertainty quantification in​ nonlinear systems of balance​‌ laws.Lecture Notes​​ in Computational Science and​​​‌ Engineering922013,​ 225--294back to text​‌
• 140 articleW.W.L.​​ Oberkampf and F.F.G.​​​‌ Blottner. Issues in​ Computational Fluid Dynamics code​‌ verification and validation.​​AIAA Journal361998​​​‌, 687--695back to​ text
• 141 bookB.​‌Benoît Perthame. Transport​​ equations in biology.​​​‌Frontiers in MathematicsBirkhäuser​ Verlag, Basel2007back​‌ to text
• 142 article​​B.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
• 143​​ techreportF.F. Poirion​​​‌. Stochastic Multi Gradient​ Descent Algorithm.ONERA​‌July 2014back to​​ text
• 144 articleF.​​​‌ ..F .S. Priuli​. First order mean​‌ field games in crowd​​ dynamics.ArXiv e-prints​​​‌February 2014back to​ text
• 145 inproceedingsM.​‌M.M. Putko, P.​​P.A. Newman, A.​​​‌A.C. Taylor and L.​L.L. Green. Approach​‌ for uncertainty propagation and​​ robust design in CFD​​​‌ using sensitivity derivatives.​15th AIAA Computational Fluid​‌ Dynamics ConferenceAIAA Paper​​ 2001-2528Anaheim, CAJune​​​‌ 2001back to text​
• 146 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 Heidelberg​2010, 183-192URL:​‌ http://dx.doi.org/10.1007/978-3-642-12598-0_16DOIback to​​ text
• 147 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.1672​​​‌2015, 733--760back‌ to text
• 148 article‌​‌P. I.Paul I.​​ Richards. Shock waves​​​‌ on the highway.‌Operations Res.41956‌​‌, 42--51back to​​ text
• 149 bookP.​​​‌P. Sagaut. Large‌ Eddy Simulation for Incompressible‌​‌ Flows An Introduction.​​Springer Berlin Heidelberg2006​​​‌back to textback‌ to text
• 150 inproceedings‌​‌J.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
• 151​​ articleV.V. Schleper​​​‌. A hybrid model‌ for traffic flow and‌​‌ crowd dynamics with random​​ individual properties.Math.​​​‌ Biosci. Eng.122‌2015, 393-413back‌​‌ to text
• 152 article​​A.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.663​​2006, 921--944back​​​‌ to text
• 153 article‌P. R.P. R.‌​‌ Spalart. Detached-Eddy Simulation​​.Annual Review of​​​‌ Fluid Mechanics412009‌, 181-202back to‌​‌ text
• 154 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.​​​‌ ECCOMAS2012back to‌ text
• 155 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, NV‌​‌Jan. 2001back to​​ textback to text​​​‌back to text
• 156‌ bookC.Cédric Villani‌​‌. Optimal transport.​​338Grundlehren der Mathematischen​​​‌ Wissenschaften [Fundamental Principles of‌ Mathematical Sciences]Old and‌​‌ newSpringer-Verlag, Berlin2009​​back to text
• 157​​​‌ bookC.Cédric Villani‌. Topics in optimal‌​‌ transportation.58Graduate​​ Studies in MathematicsAmerican​​​‌ Mathematical Society, Providence, RI‌2003back to text‌​‌
• 158 techreportR.R.W.​​ Walter and L.L.​​​‌ Huyse. Uncertainty analysis‌ for fluid mechanics with‌​‌ applications.2002--1ICASE​​February 2002back to​​​‌ text
• 159 articleD.‌D.B. Xiu and G.‌​‌G.E. Karniadakis. Modeling​​ uncertainty in flow simulations​​​‌ via generalized Polynomial Chaos‌.Journal of Computational‌​‌ Physics1872003,​​ 137-167back to text​​​‌
• 160 articleD.D.‌ You and P.P.‌​‌ Moin. Active control​​ of flow separation over​​​‌ an airfoil using synthetic‌ jets.J. of‌​‌ Fluids and Structures24​​2008, 1349-1357back​​​‌ to text
• 161 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, 116-130back​‌ to text