3.1 Research directions

The project develops along the following two axes:

• modeling complex systems through novel (unconventional) PDE systems, accounting for multi-scale phenomena and uncertainty;
• optimization and optimal control algorithms for systems governed by the above PDE systems.

These themes are motivated by the specific problems treated in the applications, and represent important and up-to-date issues in engineering sciences. For example, improving the design of transportation means and civil buildings, and the control of traffic flows, would result not only in better performances of the object of the optimization strategy (vehicles, buildings or road networks level of service), but also in enhanced safety and lower energy consumption, contributing to reduce costs and pollutant emissions.

3.2 PDE models accounting for multi-scale phenomena and uncertainties

Dynamical models consisting of evolutionary PDEs, mainly of hyperbolic type, appear classically in the applications studied by the previous Project-Team Opale (compressible flows, traffic, cell-dynamics, medicine, etc). Yet, the classical purely macroscopic approach is not able to account for some particular phenomena related to specific interactions occurring at smaller scales. These phenomena can be of greater importance when dealing with particular applications, where the "first order" approximation given by the purely macroscopic approach reveals to be inadequate. We refer for example to self-organizing phenomena observed in pedestrian flows  110, or to the dynamics of turbulent flows for which large scale / small scale vortical structures interfere  138.

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  151.

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  59 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  72, 99, which limit their applicability to real world situations. Hamilton-Jacobi equations could be also an interesting direction of research, following the recent results obtained in  115.
• Data assimilation. In traffic flow modeling, the capability of correctly estimating and predicting the state of the system depends on the availability of rich and accurate data on the network. Up to now, the most classical sensors are fixed ones. They are composed of inductive loops (electrical wires) that are installed at different spatial positions of the network and that can measure the traffic flow, the occupancy rate (i.e. the proportion of time during which a vehicle is detected to be over the loop) and the speed (in case of a system of two distant loops). These data are useful / essential to calibrate the phenomenological relationship between flow and density which is known in the traffic literature as the Fundamental Diagram. Nowadays, thanks to the wide development of mobile internet and geolocalization techniques and its increasing adoption by the road users, smartphones have turned into perfect mobile sensors in many domains, including in traffic flow management. They can provide the research community with a large database of individual trajectory sets that are known as Floating Car Data (FCD), see 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  66, 65), with a stress on the (fixed) Eulerian framework. Up to our knowledge, there exist a few studies in the time-Lagrangian as well as space-Lagrangian frameworks, where data coming from mobile sensors could be easily assimilated, due to the fact that the Lagrangian coordinate (say the label of a vehicle) is fixed.
• Control of autonomous vehicles. Traffic flow is usually controlled via traffic lights or variable speed limits, which have fixed space locations. The deployment of autonomous vehicles opens new perspectives in traffic management, as the use of a small fraction of cars to optimize the overall traffic. In this perspective, the possibility to track vehicles trajectories either by coupled micro-macro models  80, 98 or via the Hamilton-Jacobi approach  66, 65 could allow to optimize the flow by controlling some specific vehicles corresponding to internal conditions.

4.3 Virtual Fractional Flow Reserve in coronary stenting

Atherosclerosis is a chronic inflammatory disease that affects the entire arterial network and especially the coronary arteries. It is an accumulation of lipids over the arterial surface due to a dysfunction of this latter. The objective of clinical intervention, in this case, is to establish a revascularization using different angioplasty techniques, among which the implantation of stents is the most widespread. This intervention involves introducing a stent into the damaged portion in order to allow the blood to circulate in a normal way over all the vessels. Revascularization is based on the principle of remedying ischemia, which is a decrease or an interruption of the supply of oxygen to the various organs. This anomaly is attenuated by the presence of several lesions (multivessel disease patients), which can lead to several complications. The key of a good medical intervention is the fact of establishing a good diagnosis, in order to decide which lesion requires to be treated. In the diagnosis phase, the clinician uses several techniques, among which angiography is the most popular. Angiography is an X-ray technique to show the inside (the lumen) of blood vessels, in order to identify vessel narrowing: stenosis. Despite its widespread use, angiography is often imperfect in determining the physiological significance of coronary stenosis. If the problem remains simple for non significant lesions ($\le 40\%$) or very severe ( $\ge 70\%$), a very important category of intermediate lesions must benefit from a functional evaluation which will determine the strategy of treatment   76.

The technique of the Fractional Flow Reserve (FFR) has derived from the initial coronary physical approaches decades ago. Since then, many studies have demonstrated its effectiveness in improving the patients prognosis, by applying the appropriate approach. Its contribution in the reduction of mortality was statistically proved by the FAME (Fractional Flow Reserve Versus Angiography for Multivessel Evaluation) study   153. It is established that the FFR can be easily measured during coronary angiography by calculating the ratio of distal coronary pressure $P_d$ to aortic pressure $P_a$. These pressures are measured simultaneously with a special guide-wire. FFR in a normal coronary artery equals to 1.0. FFR value of 0.80 or less identifies ischemia-causing coronary lesions with an accuracy of more than 90%   153.

Obviously, from an interventional point of view, the FFR is binding since it is invasive. It should also be noted that this technique induces additional costs, which are not covered by insurances in several countries. For these reasons, it is used only in less than 10% of the cases.

In this perspective, a new virtual version of the FFR, entitled VFFR, has emerged as an attractive and non-invasive alternative to standard FFR, see   144, 127. VFFR is based on computational modeling, mainly fluid and fluid-structural dynamics. However, there are key scientific, logistic and commercial challenges that need to be overcome before VFFR can be translated into routine clinical practice.

While most of the studies related to VFFR use Navier-Stokes models, we focus on the non-newtonian case, starting with a generalized fluid flow approach. These models are more relevant for the coronary arteries, and we expect that the computation of the FFR should then be more accurate. We are also leading numerical studies to assess the impact (on the FFR) of the interaction of the physical devices (catheter, optical captors, spheroids) with the blood flow.

4.4 Other application fields

Besides the above mentioned axes, which constitute the project's identity, the methodological tools described in Section have a wider range of application. We currently carry on also the following research actions, in collaboration with external partners.

• Modeling cell dynamics. Migration and proliferation of epithelial cell sheets are the two keystone aspects of the collective cell dynamics in most biological processes such as morphogenesis, embryogenesis, cancer and wound healing. It is then of utmost importance to understand their underlying mechanisms.

  Semilinear reaction-diffusion equations are widely used to give a phenomenological description of the temporal and spatial changes occurring within cell populations that undergo scattering (moving), spreading (expanding cell surface) and proliferation. We have followed the same methodology and contributed to assess the validity of such approaches in different settings (cell sheets  106, dorsal closure  46, actin organization  45). However, epithelial cell-sheet movement is complex enough to undermine most of the mathematical approaches based on locality, that is mainly traveling wavefront-like partial differential equations. In 93 it is shown that Madin-Darby Canine Kidney (MDCK) cells extend cryptic lamellipodia to drive the migration, several rows behind the wound edge. In 132 MDCK monolayers are shown to exhibit similar non-local behavior (long range velocity fields, very active border-localized leader cells).

  Our aim is to start from a mesoscopic description of cell interaction: considering cells as independent anonymous agents, we plan to investigate the use of mathematical techniques adapted from the mean-field game theory. Otherwise, looking at them as interacting particles, we will use a multi-agent approach (at least for the actin dynamics). We intend also to consider approaches stemming from compartment-based simulation in the spirit of those developed in   90, 95, 97.

• Game strategies for thermoelastography. Thermoelastography is an innovative non-invasive control technology, which has numerous advantages over other techniques, notably in medical imaging 125. Indeed, it is well known that most pathological changes are associated with changes in tissue stiffness, while remaining isoechoic, and hence difficult to detect by ultrasound techniques. Based on elastic waves and heat flux reconstruction, thermoelastography shows no destructive or aggressive medical sequel, unlike X-ray and comparables techniques, making it a potentially prominent choice for patients.

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

  The mathematical formulation of the thermoelasticity reconstruction is based on data completion and material identification, making it a harsh ill posed inverse problem. In previous works  107, 117, we have demonstrated that Nash game approaches are efficient to tackle ill-posedness. We intend to extend the results obtained for Laplace equations in 107, and the algorithms developed in Section 3.3.4 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. 135, that have the desirable convergence property in constrained problems. However, in this approach, the constraints are assumed to be known formally, by explicit expressions.

  In collaboration with ONERA-Meudon, we are exploring the possibility of representing constraints, in successive iterations, through local approximations of the constraint surfaces, splitting the design space locally into tangent and normal sub-spaces, and eliminating the normal coordinates through a linearization, or more generally a finite expansion, and applying the BFGS method through dependencies on the coordinates in the tangent subspace only. Preliminary experiments on the difficult Rosenbrock test-case, although in low dimensions, demonstrate the feasibility of this approach. On-going research is on theorizing this method, and testing cases of higher dimensions.

• Multi-objective optimization for nanotechnologies. Our team takes part in a larger collaboration with CEA/LETI (Grenoble), initiated by the Inria Project-Team Nachos, and related to the Maxwell equations. Our component in this activity relates to the optimization of nanophotonic devices, in particular with respect to the control of thermal loads. We have first identified a gradation of representative test-cases of increasing complexity:

  - infrared micro-source;

  - micro-photoacoustic cell;

  - nanophotonic device.

  These cases involve from a few geometric parameters to be optimized to a functional minimization subject to a finite-element solution involving a large number of dof's. CEA disposes of such codes, but considering the computational cost of the objective functions in the complex cases, the first part of our study is focused on the construction and validation of meta-models, typically of RBF-type. Multi-objective optimization will be carried out subsequently by MGDA, and possibly Nash games.

5.1 Impact of research results

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

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

From medical viewpoint, virtual fractional flow reserve vFFR is a promising technique to support clinicians in cardiostenting with cheap social costs compared to the analogic commercial solutions. Acumes has contributed to improve the involved computational apparatus (nonlinear fluid mechanics with ad hoc boundary conditions).

The research activities related to isogeometric analysis aim at facilitating the use of shape optimization methods in engineering, yielding a gain of efficiency, for instance in transportation industry (cars, aircrafts) or energy industry (air conditioning, turbines).

6.1 Awards

• M. Binois: 2020 Youden Award for best expository paper appearing in the 2019 issues of Technometrics (American Quality Association).
• M. Binois: Finalist for the 2020 Gordon Bell Special Prize for High Performance Computing-Based COVID-19 Research (ACM Gordon Bell prize).
• P. Goatin: ANR-3IA Côte d'Azur Senior Chair for the project “Data driven traffic management” (2020-2024).

7.1 New software

8.1 Macroscopic traffic flow models on networks

Participants: Mickaël Binois, Paola Goatin, Alexandra Würth, Antonella Ferrara, Giulia Piacentini.

Traffic control by vehicle platooning.

In 33, a coupled PDE-ODE model describing the interaction between the bulk traffic flow and a platoon of connected vehicles is adopted to develop a control action aiming at reducing the fuel consumption of the overall traffic flow. The platoon is modeled as a capacity restriction acting on the surrounding traffic. The trajectory of the initial and final points of the platoon are optimized by means of a model predictive control strategy, acting on the speeds of the front-end and back-end of the platoon, thus resulting in controlling both the speed and the length of the platoon. The approach is assessed in simulations.

This work is part of G. Piacentini's PhD thesis.

Traffic flow model calibration by statistical approaches.

In the framework of A. Würth's internship, we have employed a Bayesian approach including a bias term to estimate first and second order model parameters, based on two traffic data sets: a set of loop detector data located on the A50 highway between Marseille and Aubagne provided by DirMED, and publicy available data from the Minnesota Department of transportation (http://data.dot.state.mn.us/datatools/). The method proved to be effective and allowed to infer better performances of second order model, in particular regarding speed and density esimation.

8.2 Non-local conservation laws

Participants: Paola Goatin, Raimund Bürger, Daniel Inzunza, Luis Miguel Villada.

In the framework of the Associated Team NOLOCO, we proposed a revised version often the non-local macroscopic pedestrian flow model proposed in [R. M. Colombo, M. Garavello, and M. Lécureux-Mercier. A class of nonlocal models for pedestrian traffic. Math. Models Methods Appl. Sci., 22(4):1150023, 2012] to account for anisotropic interactions and the presence of walls or other obstacles in the walking domain. We proved the well-posedness of this extended model and we applied high-resolution numerical schemes to illustrate the model characteristics. In particular, numerical simulations highlight the role of different model parameters in the observed pattern formation. The results are published in 18.

8.3 Isogeometric Discontinuous Galerkin method for compressible flows

Participants: Régis Duvigneau, Stefano Pezzano, Maxime Stauffert.

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

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

• CAD-consistent adaptive refinement

  Properties of NURBS functions are used to define an adaptive refinement strategy, which refines locally the discretization according to an error indicator, while describing exactly CAD geometries whatever the refinement level. The resulting approach exhibits an optimal convergence rate and capture efficiently local flow features, like shocks or vortices, avoiding refinement due to geometry approximation 24.

• Arbitrary Eulerian-Lagrangian formulation for high-order meshes

  To enable the simulation of flows around moving bodies, an Arbitrary Eulerian-Lagrangian (ALE) formulation is proposed in the context of the isogeometric DG method. It relies on a NURBS-based grid velocity field, integrated along time over moving NURBS elements. The gain of using exact-geometry representations is clearly quantified, in terms of accuracy and computational efficiency 31, 36.

• Geometrically exact sliding interfaces

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

• Isogeometric shape optimization

  We develop an optimization procedure with shape sensitivity analysis, entirely based on NURBS representations 41. The mesh, the shape to optimize, as well as the flow solutions are represented by NURBS, which avoids any geometrical conversion and allows to exploit NURBS properties regarding regularity or hierarchy.

8.4 Sensitivity analysis for compressible flows

Participants: Régis Duvigneau, Maxime Stauffert, Camilla Fiorini, Christophe Chalons.

The adjoint equation method, classically employed in design optimization to compute functional gradients, is not well suited to complex unsteady problems, because of the necessity to solve it backward in time. Therefore, we investigate the use of the sensitivity equation method, which is integrated forward in time, in the context of compressible flows. More specifically, the following research axes are considered:

• Sensitivity analysis in presence of shocks

  While the sensitivity equation method is a common approach for parabolic systems, its use for hyperbolic ones is still tedious, because of the generation of discontinuities in the state solution, yielding Dirac distributions in the sensitivity solution. To overcome this difficulty, we investigate a modified sensitivity equation, that includes an additional source term when the state solution exhibits discontinuities, to avoid the generation of delta-peaks in the sensitivity solution. We consider as typical example the 1D compressible Euler equations. Different approaches are tested to integrate the additional source term: a Roe solver, a Godunov method and a moving cells approach. Applications to uncertainty quantification in presence of shocks are demonstrated and compared to the classical Monte-Carlo method 26. This study is achieved in collaboration with C. Chalons and C. Fiorini from University of Versailles.

• Shape sensitivity analysis

  When shape parameters are considered, the evaluation of flow sensitivities is more difficult, because equations include an additional term, involving flow gradient, due to the fact that the parameter affects the boundary condition location. To overcome this difficulty, we propose to solve sensitivity equations using an isogeometric Discontinuous Galerkin (DG) method, which allows to estimate accurately flow gradients at boundary and consider boundary control points as shape parameters. First results obtained for 2D compressible Euler equations exhibit a sub-optimal convergence rate, as expected, but a better accuracy with respect to a classical DG method 41.

8.5 Advanced Bayesian optimization

Participants: Mickaël Binois, Régis Duvigneau, Abderrahmane Habbal, Mahmoud Elsawy, Frédéric Hauville, Olivier Lemaitre, Stéphane Lanteri, Victor Picheny, Matthieu Sacher.

Multi-fidelity Bayesian optimization

The objective of multi-fidelity optimization strategies is to account for a set of models of different accuracies and costs to accelerate the optimization procedure. In the context of Bayesian optimization, we develop such a multi-fidelity approach based on non-nested evaluations: each time a new evaluation is required, the algorithm selects a new design point associated to a fidelity level to maximize the expected improvement on the finest modeling level. The proposed approach is applied to the fluid-structure optimization of a sailing boat, which is described by five modeling levels. A significant acceleration of the optimization procedure is reported, without loss of accuracy 35.

The Kalai-Smorodinski solution for many-objective Bayesian optimization

Extending the short paper 16 on the use of the game-theoretic Kalai-Smorodinski solution in Bayesian optimization, we have refined the definition of solutions, discussed underlying assumptions, and shown empirically the improved performance of our proposed approach over naive heuristics. A realistic hyperparameter tuning problem with eight objectives as well as an expensive calibration problem with nine objectives have been considered as well.

In parallel, we have substantially improved the efficiency of the implementation, enabled specific treatment of calibration problems as well as handling noise in the GPGame package https://cran.r-project.org/web/packages/GPGame.

Bayesian optimization of nano-photonic devices

In collaboration with Atlantis Project-Team, we consider the optimization of optical meta-surface devices, which are able to alter light properties by operating at nano-scale. In the contexte of Maxwell equations, modified to account for nano-scale phenomena, the geometrical properties of materials are optimized to achieve a desired electromagnetic wave response, such as change of polarization, intensity or direction. This task is especially challenging due to the computational cost related to the 3D time-accurate simulations, the difficulty to handle the different geometrical scales in optimization and the presence of uncertainties.

First studies achieved using Bayesian optimization algorithms, demonstrate the potentiality of the proposed approach 25, 37.

8.6 Gaussian process based sequential design

Participants: Mickaël Binois, Robert Gramacy, Michael Ludkovski, Xiong Lyu, Stefan Wild, Nathan Wycoff.

Besides Bayesian optimization as above, Gaussian processes are useful for a variety of other related tasks. Here we first present a tutorial on the subject of modeling with input dependent noise with an implementation in the hetGP R package. Then the estimation of level-set in for noisy simulators with complex input noise is studied, before treating sequential design for efficient dimension reduction.

Heteroskedastic Gaussian process modeling and sequential design

An increasing number of time-consuming simulators exhibit a complex noise structure that depends on the inputs. For conducting studies with limited budgets of evaluations, new surrogate methods are required in order to simultaneously model the mean and variance fields. To this end, in 55 we present the hetGP package https://cran.r-project.org/web/packages/hetGP, implementing many recent advances in Gaussian process modeling with input-dependent noise. First, we describe a simple, yet efficient, joint modeling framework that relies on replication for both speed and accuracy. Then we tackle the issue of data acquisition leveraging replication and exploration in a sequential manner for various goals, such as for obtaining a globally accurate model, for optimization, or for contour finding. Reproducible illustrations are provided throughout.

Sequential learning of active subspace

Continuing a work started at Argonne National Laboratory, in 149 we consider the combination of Gaussian process regression modeling with the active subspace methods (ASMs), which have become a popular means of performing subspace sensitivity analysis on black-box functions. Naively applied, however, ASMs require gradient evaluations of the target function. In the event of noisy, expensive, or stochastic simulators, evaluating gradients via finite differencing may be infeasible. In such cases, often a surrogate model is employed, on which finite differencing is performed. When the surrogate model is a Gaussian process, we show that the ASM estimator is available in closed form, rendering the finite-difference approximation unnecessary. We use our closed-form solution to develop acquisition functions focused on sequential learning tailored to sensitivity analysis on top of ASMs. We also show that the traditional ASM estimator may be viewed as a method of moments estimator for a certain class of Gaussian processes. We demonstrate how uncertainty on Gaussian process hyperparameters may be propagated to uncertainty on the sensitivity analysis, allowing model-based confidence intervals on the active subspace. Our methodological developments are illustrated on several examples.

Evaluating Gaussian Process Metamodels and Sequential Designs for Noisy Level Set Estimation

We consider the problem of learning the level set for which a noisy black-box function exceeds a given threshold. To efficiently reconstruct the level set, we investigate Gaussian process (GP) metamodels. Our focus in 40 is on strongly stochastic samplers, in particular with heavy-tailed simulation noise and low signal-to-noise ratio. To guard against noise misspecification, we assess the performance of three variants: (i) GPs with Student-t observations; (ii) Student-t processes (TPs); and (iii) classification GPs modeling the sign of the response. In conjunction with these metamodels, we analyze several acquisition functions for guiding the sequential experimental designs, extending existing stepwise uncertainty reduction criteria to the stochastic contour-finding context. This also motivates our development of (approximate) updating formulas to efficiently compute such acquisition functions. Our schemes are benchmarked by using a variety of synthetic experiments in 1–6 dimensions. We also consider an application of level set estimation for determining the optimal exercise policy of Bermudan options in finance.

8.7 Prioritized multi-objective optimization of a sandwich panel

Participants: Jean-Antoine Désidéri, Pierre Leite, Quentin Mercier.

The potential of the prioritized multi-objective optimization approach has been demonstrated by applying it to the sizing of a sandwich panel with respect to mechanical criteria: mass, critical failure forces under bending load (1st and 2nd modes), and blast mitigation measured by the core energy absorption, deflection, or both. Several objective functions are defined by analytical models. Four numerical test-cases were documented.

In each test-case, in a first phase of optimization, the Primary Pareto Front associated with two criteria only, considered preponderant, was first established. Mass was always retained in these primary objective functions. In a second phase, a starting point on the Primary Pareto Front is selected and the corresponding design is improved with respect to secondary criteria by the construction of a continuum of Nash equilibria tangent to the front in function space at the starting point, thus only marginally degrading the primary criteria.

The second phase of optimization can be viewed as a form of adaptation of the optimization process. Here it permits to account for four criteria, at a more economical and simpler strategy than it would be to evaluate and analyze a complete Pareto front in a four-dimensional function space. 39

(All the numerical experiments were realized using the software platform http://mgda.inria.fr.)

8.8 Inverse Cauchy-Stokes problems solved as Nash games

Participants: Abderrahmane Habbal, Marwa Ouni, Moez Kallel.

We extend in two directions our results published in 108 to tackle ill posed Cauchy-Stokes inverse problems as Nash games. First, we consider the problem of detecting unknown pointwise sources in a stationary viscous fluid, using partial boundary measurements. The considered fluid obeys a steady Stokes regime, the boundary measurements are a single compatible pair of Dirichlet and Neumann data, available only on a partial accessible part of the whole boundary. This inverse source identification for the Cauchy-Stokes problem is ill-posed for both the sources and missing data reconstructions, and designing stable and efficient algorithms is challenging. We reformulate the problem as a three-player Nash game. Thanks to a source identifiability result derived for the Cauchy-Stokes problem, it is enough to set up two Stokes BVP, then use them as state equations. The Nash game is then set between 3 players, the two first targeting the data completion while the third one targets the detection of the number, location and magnitude of the unknown sources. We provided the third player with the location and magnitude parameters as strategy, with a cost functional of Kohn-Vogelius type. In particular, the location is obtained through the computation of the topological sensitivity of the latter function. We propose an original algorithm, which we implemented using Freefem++. We present 2D numerical experiments for many different test-cases.The obtained results corroborate the efficiency of our 3-player Nash game approach to solve parameter or shape identification for Cauchy-Stokes problems.

The second direction is dedicated to the solution of the data completion problem for non-linear flows. We consider two kinds of non linearities leading to either a non newtonian Stokes flow or to Navier-Stokes equations. Our recent numerical results show that it is possible to perform a one-shot approach using Nash games : players exchange their respective state information and solve linear systems. At convergence to a Nash equilibrium, the states converge to the solution of the non linear systems. To the best of our knowledge, this is the first time such an approach is applied to solve Inverse problems for nonlinear systems.

8.9 Virtual FFR quantified with a generalized flow model using Windkessel boundary conditions ; Application to a patient-specific coronary tree

Participants: Abderrahmane Habbal, Keltoum Chahour, Rajae Aboulaich, Nejib Zemzemi, Mickaël Binois.

Fractional flow reserve (FFR) has proved its efficiency in improving patients diagnosis. From both economical and clinical viewpoints, a realistic simulation of vascular blood flow inside the coronary arteries could be a better alternative to the invasive FFR. In this view, we consider a 2D reconstructed left coronary tree with two artificial lesions of different degrees. We use a generalized fluid model with a Carreau law and use a coupled multidomain method to implement Windkessel boundary conditions at the outlets. We introduce our methodology to quantify the virtual FFR, and lead several numerical experiments. We compare FFR results from Navier Stokes versus generalized flow model, and for Windkessel versus traction free outlets boundary conditions or mixed outlets boundary conditions. We also investigate some sources of uncertainty that the FFR index might encounter during the invasive procedure, in particular the arbitrary position of the distal sensor. The computational FFR results show that the degree of stenosis is not enough to classify a lesion, while there is a good agreement between Navier Stokes and the non Newtonian flow model adopted in classifying coronary lesions. Furthermore, we highlight that the lack of standardization while making FFR measurement might be misleading regarding the significance of stenosis 19.

9.1 Bilateral contracts with industry

• Etic Data (2019-2020): Acumes has set up a 12 months research and development contract with the company Etic Data on "Predictive modeling and proactive driving of customers behaviour in massive data BtoC context".
• Mycophyto (2020-...): this research contract involving Université Côte d'Azur is financing the post-doctoral contract of Khadija Musayeva.

10.1 International initiatives

10.2 European initiatives

10.3 National initiatives

11.1 Promoting scientific activities

11.2 Teaching - Supervision - Juries

11.3 Popularization

12.1 Major publications

• 1 articleAektaA. Aggarwal, Rinaldo M.R. Colombo and PaolaP. Goatin. Nonlocal systems of conservation laws in several space dimensionsSIAM Journal on Numerical Analysis5222015, 963-983
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• 2 articleBorisB. Andreianov, PaolaP. Goatin and NicolasN. Seguin. Finite volume schemes for locally constrained conservation lawsNumer. Math.1154With supplementary material available online2010, 609--645
• 3 article SebastienS. Blandin and PaolaP. Goatin. Well-posedness of a conservation law with non-local flux arising in traffic flow modeling Numerische Mathematik 2015
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• 4 articleRinaldo M.R. Colombo and PaolaP. Goatin. A well posed conservation law with a variable unilateral constraintJ. Differential Equations23422007, 654--675
• 5 articleM. L.M. Delle Monache and PaolaP. Goatin. Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence resultJ. Differential Equations257112014, 4015--4029
• 6 articleM. L.M. Delle Monache, J. Reilly, S. Samaranayake, W. Krichene, P. Goatin and AlexandreA. Bayen. A PDE-ODE model for a junction with ramp bufferSIAM J. Appl. Math.7412014, 22--39
• 7 articleR. Duvigneau and P. Chandrashekar. Kriging-based optimization applied to flow controlInt. J. for Numerical Methods in Fluids69112012, 1701-1714
• 8 articleA. Habbal and M. Kallel. Neumann-Dirichlet Nash strategies for the solution of elliptic Cauchy problemsSIAM J. Control Optim.5152013, 4066--4083
• 9 articleMoezM. Kallel, RajaeR. Aboulaich, AbderrahmaneA. Habbal and MaherM. Moakher. A Nash-game approach to joint image restoration and segmentationAppl. Math. Model.3811-122014, 3038--3053URL: http://dx.doi.org/10.1016/j.apm.2013.11.034
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• 10 article M. Martinelli and R. Duvigneau. On the use of second-order derivative and metamodel-based Monte-Carlo for uncertainty estimation in aerodynamics Computers and Fluids 37 6 2010
• 11 articleSouvikS. Roy, A. Borzì and AbderrahmaneA. Habbal. Pedestrian motion modelled by Fokker--Planck Nash gamesRoyal Society open science492017, 170648
• 12 articleM. Twarogowska, P. Goatin and R. Duvigneau. Macroscopic modeling and simulations of room evacuationAppl. Math. Model.38242014, 5781--5795
• 13 article G. Xu, B. Mourrain, A. Galligo and R. Duvigneau. Constructing analysis-suitable parameterization of computational domain from CAD boundary by variational harmonic method J. Comput. Physics 252 November 2013
• 14 articleBoutheinaB. Yahyaoui, MekkiM. Ayadi and AbderrahmaneA. Habbal. Fisher-KPP with time dependent diffusion is able to model cell-sheet activated and inhibited wound closureMathematical biosciences2922017, 36--45

12.2 Publications of the year

12.3 Cited publications

• 42 article R. Abgrall and P. M.P. Congedo. A semi-intrusive deterministic approach to uncertainty quantification in non-linear fluid flow problems J. Comput. Physics 2012
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• 43 articleAektaA. Aggarwal, Rinaldo M.R. Colombo and PaolaP. Goatin. Nonlocal systems of conservation laws in several space dimensionsSIAM Journal on Numerical Analysis5222015, 963-983
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• 44 articleGiovanniG. Alessandrini. Examples of instability in inverse boundary-value problemsInverse Problems1341997, 887--897URL: http://dx.doi.org/10.1088/0266-5611/13/4/001
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• 45 articleLu\'\isL. Almeida, PatriziaP. Bagnerini and AbderrahmaneA. Habbal. Modeling actin cable contractionComput. Math. Appl.6432012, 310--321URL: http://dx.doi.org/10.1016/j.camwa.2012.02.041
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• 46 articleLu\'\isL. Almeida, PatriziaP. Bagnerini, AbderrahmaneA. Habbal, StéphaneS. Noselli and FannyF. Serman. A Mathematical Model for Dorsal ClosureJournal of Theoretical Biology2681January 2011, 105-119URL: http://hal.inria.fr/inria-00544350/en
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• 47 articleDeboraD. Amadori and WenW. Shen. An integro-differential conservation law arising in a model of granular flowJ. Hyperbolic Differ. Equ.912012, 105--131
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• 48 articleP. Amorim, Rinaldo M.R. Colombo and A. Teixeira. On the Numerical Integration of Scalar Nonlocal Conservation LawsESAIM M2AN4912015, 19--37
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• 49 articlePauloP. Amorim. On a nonlocal hyperbolic conservation law arising from a gradient constraint problemBull. Braz. Math. Soc. (N.S.)4342012, 599--614
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• 50 articleM. Annunziato and A. Borz\`i. A Fokker-Planck control framework for multidimensional stochastic processesJournal of Computational and Applied Mathematics2372013, 487-507
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• 51 articleA. Belme, F. Alauzet and A. Dervieux. Time accurate anisotropic goal-oriented mesh adaptation for unsteady flowsJ. Comput. Physics231192012, 6323--6348
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• 52 articleSylvieS. Benzoni-Gavage, Rinaldo M.R. Colombo and PiotrP. Gwiazda. Measure valued solutions to conservation laws motivated by traffic modellingProc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.46220702006, 1791--1803
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• 53 unpublished EnricoE. Bertino, RégisR. Duvigneau and PaolaP. Goatin. Uncertainties in traffic flow and model validation on GPS data 2015
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• 54 articleF. Betancourt, R. Bürger, K. H.K. Karlsen and E. M.E. Tory. On nonlocal conservation laws modelling sedimentationNonlinearity2432011, 855--885
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• 55 unpublishedMickaelM. Binois and Robert B.R. Gramacy. hetGP: Heteroskedastic Gaussian Process Modeling and Sequential Design in RDecember 2019, working paper or preprint
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• 56 articleSebastienS. Blandin and PaolaP. Goatin. Well-posedness of a conservation law with non-local flux arising in traffic flow modelingNumer. Math.13222016, 217--241URL: https://doi.org/10.1007/s00211-015-0717-6
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• 57 articleJeffJ. Borggaard and JohnJ. Burns. A Sensitivity Equation Method for Optimal Aerodynamic Design'Journal of Computational Physics13621997, 366--384URL: http://www.sciencedirect.com/science/article/pii/S0021999197957430
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• 58 articleR. Bourguet, M. Brazza, G. Harran and R. El Akoury. Anisotropic Organised Eddy Simulation for the prediction of non-equilibrium turbulent flows around bodiesJ. of Fluids and Structures2482008, 1240--1251
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• 59 articleAlbertoA. Bressan, SunċicaS. Ċanić, MauroM. Garavello, MichaelM. Herty and BenedettoB. Piccoli. Flows on networks: recent results and perspectivesEMS Surv. Math. Sci.112014, 47--111
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• 60 articleMartinM. Burger, MarcoM. Di Francesco, Peter A.P. Markowich and Marie-ThereseM.-T. Wolfram. Mean field games with nonlinear mobilities in pedestrian dynamicsDiscrete Contin. Dyn. Syst. Ser. B1952014, 1311--1333
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• 61 articleM. Burger, J. Haskovec and M.-T. Wolfram. Individual based and mean-field modelling of direct aggregationPhysica D2602013, 145--158
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• 62 techreport AlessandraA. Cabassi and PaolaP. Goatin. Validation of traffic flow models on processed GPS data Research Report RR-8382 2013
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• 63 unpublishedJ.~A.J. Carrillo, S. Martin and M.-T. Wolfram. A local version of the Hughes model for pedestrian flow2015, Preprint
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• 64 unpublishedChristopheC. Chalons, Maria LauraM. Delle Monache and PaolaP. Goatin. A conservative scheme for non-classical solutions to a strongly coupled PDE-ODE problem2015, Preprint
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• 65 articleChristian G.C. Claudel and Alexandre M.A. Bayen. Convex formulations of data assimilation problems for a class of Hamilton-Jacobi equationsSIAM J. Control Optim.4922011, 383--402
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• 66 articleC.G.C. Claudel and Alexandre M.A. Bayen. Lax-Hopf Based Incorporation of Internal Boundary Conditions Into Hamilton-Jacobi Equation. Part II: Computational MethodsAutomatic Control, IEEE Transactions on555May 2010, 1158-1174
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• 67 articleRinaldo M.R. Colombo, MauroM. Garavello and MagaliM. Lécureux-Mercier. A Class Of Nonloval Models For Pedestrian TrafficMathematical Models and Methods in Applied Sciences22042012, 1150023
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• 68 articleRinaldo M.R. Colombo, MichaelM. Herty and MagaliM. Mercier. Control of the continuity equation with a non local flowESAIM Control Optim. Calc. Var.1722011, 353--379
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• 69 articleRinaldo M.R. Colombo and MagaliM. Lécureux-Mercier. Nonlocal crowd dynamics models for several populationsActa Math. Sci. Ser. B Engl. Ed.3212012, 177--196
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• 70 articleRinaldo M.R. Colombo and FrancescaF. Marcellini. A mixed ODE-PDE model for vehicular trafficMathematical Methods in the Applied Sciences3872015, 1292--1302
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• 71 articleRinaldo M.R. Colombo and E. Rossi. On the micro-macro limit in traffic flowRend. Semin. Mat. Univ. Padova1312014, 217--235
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• 72 articleGuillaumeG. Costeseque and Jean-PatrickJ.-P. Lebacque. Discussion about traffic junction modelling: conservation laws vs Hamilton-Jacobi equationsDiscrete Contin. Dyn. Syst. Ser. S732014, 411--433
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• 73 articleGianlucaG. Crippa and MagaliM. Lécureux-Mercier. Existence and uniqueness of measure solutions for a system of continuity equations with non-local flowNonlinear Differential Equations and Applications NoDEA2012, 1-15
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• 74 inproceedingsE. Cristiani, B. Piccoli and A. Tosin. How can macroscopic models reveal self-organization in traffic flow?Decision and Control (CDC), 2012 IEEE 51st Annual Conference onDec 2012, 6989-6994
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• 75 book EmilianoE. Cristiani, BenedettoB. Piccoli and AndreaA. Tosin. Multiscale modeling of pedestrian dynamics 12 MS&A. Modeling, Simulation and Applications Springer, Cham 2014
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• 76 article T. Cuisset, J. QuiliCi and G. Cayla.. Qu'est-ce que la FFR? Comment l'utiliser? Réalités Cardiologiques Janvier/Février 2013
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• 77 incollectionC. M.C. Dafermos. Solutions in $L^{}$ for a conservation law with memory'Analyse mathématique et applicationsMontrougeGauthier-Villars1988, 117--128
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• 78 articlePierreP. Degond, Jian-GuoJ.-G. Liu and ChristianC. Ringhofer. Large-scale dynamics of mean-field games driven by local Nash equilibriaJ. Nonlinear Sci.2412014, 93--115URL: http://dx.doi.org/10.1007/s00332-013-9185-2
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• 79 articleMaria LauraM. Delle Monache and PaolaP. Goatin. A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flowDiscrete Contin. Dyn. Syst. Ser. S732014, 435--447
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• 80 articleM. L.M. Delle Monache and P. Goatin. Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence resultJ. Differential Equations257112014, 4015--4029
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• 81 inbookJ.-A. Désidéri. Multiple-Gradient Descent Algorithm (\em MGDA) for Pareto-Front Identification34Numerical Methods for Differential Equations, Optimization, and Technological ProblemsModeling, Simulation and Optimization for Science and Technology, Fitzgibbon, W.; Kuznetsov, Y.A.; Neittaanmäki, P.; Pironneau, O. Eds.J. Périaux and R. Glowinski JubileesSpringer-Verlag2014, 1
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• 82 articleJean-AntoineJ.-A. Désidéri. Multiple-gradient descent algorithm (MGDA) for multiobjective optimizationComptes Rendus de l'Académie des Sciences Paris3502012, 313-318URL: http://dx.doi.org/10.1016/j.crma.2012.03.014
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• 83 techreport Jean-AntoineJ.-A. Désidéri. Révision de l'algorithme de descente à gradients multiples (MGDA) par orthogonalisation hiérarchique 8710 INRIA April 2015
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• 84 incollectionBrunoB. Després, GaëlG. Poëtte and DidierD. Lucor. Robust uncertainty propagation in systems of conservation laws with the entropy closure methodUncertainty quantification in computational fluid dynamics92Lect. Notes Comput. Sci. Eng.Springer, Heidelberg2013, 105--149
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• 85 article M. Di Francesco and M.~D.M. Rosini. Rigorous Derivation of Nonlinear Scalar Conservation Laws from Follow-the-Leader Type Models via Many Particle Limit Archive for Rational Mechanics and Analysis 2015
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• 86 articleRonald J.R. DiPerna. Measure-valued solutions to conservation lawsArch. Rational Mech. Anal.8831985, 223--270
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• 87 articleChristianC. Dogbé. Modeling crowd dynamics by the mean-field limit approachMath. Comput. Modelling529-102010, 1506--1520
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• 88 techreport R. Duvigneau. A Sensitivity Equation Method for Unsteady Compressible Flows: Implementation and Verification INRIA Research Report No 8739 June 2015
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• 89 articleR. Duvigneau and D. Pelletier. A sensitivity equation method for fast evaluation of nearby flows and uncertainty analysis for shape parametersInt. J. of Computational Fluid Dynamics207August 2006, 497--512
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• 90 articleRadekR. Erban, Mark B.M. Flegg and Garegin A.G. Papoian. Multiscale stochastic reaction-diffusion modeling: application to actin dynamics in filopodiaBull. Math. Biol.7642014, 799--818URL: http://dx.doi.org/10.1007/s11538-013-9844-3
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• 91 articleR. Etikyala, S. Göttlich, A. Klar and S. Tiwari. Particle methods for pedestrian flow models: from microscopic to nonlocal continuum modelsMath. Models Methods Appl. Sci.24122014, 2503--2523
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• 92 incollectionRobertR. Eymard, ThierryT. Gallouët and RaphaèleR. Herbin. Finite volume methodsHandbook of numerical analysis, Vol. VIIHandb. Numer. Anal., VIINorth-Holland, Amsterdam2000, 713--1020
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• 93 articleRizwanR. Farooqui and GabrielG. Fenteany. Multiple rows of cells behind an epithelial wound edge extend cryptic lamellipodia to collectively drive cell-sheet movement Journal of Cell Science118Pt 12005, 51-63
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• 94 techreport UlrikU. Fjordholm, RogerR. Kappeli, SiddharthaS. Mishra and EitanE. Tadmor. Construction of approximate entropy measure valued solutions for systems of conservation laws 2014-33 Seminar for Applied Mathematics, ETH Zürich 2014
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• 95 articleMark B.M. Flegg, StefanS. Hellander and RadekR. Erban. Convergence of methods for coupling of microscopic and mesoscopic reaction-diffusion simulationsJ. Comput. Phys.2892015, 1--17URL: http://dx.doi.org/10.1016/j.jcp.2015.01.030
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• 96 inproceedings F. Fleuret and 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) 2 1999
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• 97 articleBenjaminB. Franz, Mark B.M. Flegg, S. JonathanS. Chapman and RadekR. Erban. Multiscale reaction-diffusion algorithms: PDE-assisted Brownian dynamicsSIAM J. Appl. Math.7332013, 1224--1247
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• 98 articleMauroM. Garavello and BenedettoB. Piccoli. Coupling of microscopic and phase transition models at boundaryNetw. Heterog. Media832013, 649--661
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• 99 book MauroM. Garavello and BenedettoB. Piccoli. Traffic flow on networks 1 AIMS Series on Applied Mathematics Conservation laws models American Institute of Mathematical Sciences (AIMS), Springfield, MO 2006
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• 100 articlePaolaP. Goatin and MatthiasM. Mimault. A mixed system modeling two-directional pedestrian flowsMath. Biosci. Eng.1222015, 375--392
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• 101 unpublishedPaolaP. Goatin and FrancescoF. Rossi. A traffic flow model with non-smooth metric interaction: well-posedness and micro-macro limit2015, PreprintURL: http://arxiv.org/abs/1510.04461
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• 102 articlePaolaP. Goatin and SheilaS. Scialanga. Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocityNetw. Heterog. Media1112016, 107--121
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• 103 articleSimoneS. Göttlich, SimonS. Hoher, PatrickP. Schindler, VeronikaV. Schleper and AlexanderA. Verl. Modeling, simulation and validation of material flow on conveyor beltsApplied Mathematical Modelling38132014, 3295--3313
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• 104 articleA. Griewank. Achieving logarithmic growth of temporal and spatial complexity in reverse automatic differentiationOptimization Methods and Software11992, 35-54
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• 105 articleMichaelM. Gröschel, AlexanderA. Keimer, GünterG. Leugering and ZhiqiangZ. Wang. Regularity theory and adjoint-based optimality conditions for a nonlinear transport equation with nonlocal velocitySIAM J. Control Optim.5242014, 2141--2163
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• 106 articleAbderrahmaneA. Habbal, HélèneH. Barelli and GrégoireG. Malandain. Assessing the ability of the 2D Fisher-KPP equation to model cell-sheet wound closureMath. Biosci.2522014, 45--59URL: http://dx.doi.org/10.1016/j.mbs.2014.03.009
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• 107 articleA. Habbal and MoezM. Kallel. Neumann-Dirichlet Nash strategies for the solution of elliptic Cauchy problemsSIAM J. Control Optim.5152013, 4066--4083
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• 108 articleAbderrahmaneA. Habbal, MoezM. Kallel and MarwaM. Ouni. Nash strategies for the inverse inclusion Cauchy-Stokes problemInverse Problems and Imaging 1342019, 36
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• 109 article X. Han, P. Sagaut and D. Lucor. On sensitivity of RANS simulations to uncertain turbulent inflow conditions Computers & Fluids 61 2-5 2012
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• 110 articleDirkD. Helbing, PeterP. Molnar, Illes JI. Farkas and KaiK. Bolay. Self-organizing pedestrian movementEnvironment and planning B2832001, 361--384
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• 111 articleDirkD. Helbing. Traffic and related self-driven many-particle systemsRev. Mod. Phys.7342001, 1067--1141
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• 112 articleJuan CJ. Herrera, Daniel BD. Work, RyanR. Herring, Xuegang JeffX. Ban, QuinnQ. Jacobson and Alexandre M.A. Bayen. Evaluation of traffic data obtained via GPS-enabled mobile phones: The Mobile Century field experimentTransportation Research Part C: Emerging Technologies1842010, 568--583
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• 113 articleSerge P.S. Hoogendoorn, Femke L.M.F. van Wageningen-Kessels, WinnieW. Daamen and Dorine C.D. Duives. Continuum modelling of pedestrian flows: From microscopic principles to self-organised macroscopic phenomenaPhysica A: Statistical Mechanics and its Applications41602014, 684--694
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• 114 articleH. Hristova, S. Etienne, D. Pelletier and J. Borggaard. A continuous sensitivity equation method for time-dependent incompressible laminar flowsInt. J. for Numerical Methods in Fluids502004, 817-844
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• 115 article CyrilC. Imbert and RégisR. Monneau. Flux-limited solutions for quasi-convex Hamilton--Jacobi equations on networks arXiv preprint arXiv:1306.2428 October 2014
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• 116 article S. Jeon and H. Choi. Suboptimal feedback control of flow over a sphere Int. J. of Heat and Fluid Flow 31 2010
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• 117 articleMoezM. Kallel, RajaeR. Aboulaich, AbderrahmaneA. Habbal and MaherM. Moakher. A Nash-game approach to joint image restoration and segmentationAppl. Math. Model.3811-122014, 3038--3053URL: http://dx.doi.org/10.1016/j.apm.2013.11.034
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• 118 articleO.M.O. Knio and O. Le Maitre. Uncertainty propagation in CFD using polynomial chaos decompositionFluid Dynamics Research389September 2006, 616--640
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• 119 articleAlexanderA. Kurganov and AnthonyA. Polizzi. Non-Oscillatory Central Schemes for a Traffic Flow Model with Arrehenius Look-Ahead DynamicsNetw. Heterog. Media432009, 431-451
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• 120 articleAimeA. Lachapelle and Marie-ThereseM.-T. Wolfram. On a mean field game approach modeling congestion and aversion in pedestrian crowdsTransportation Research Part B: Methodological45102011, 1572--1589
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• 121 articleJean-MichelJ.-M. Lasry and Pierre-LouisP.-L. Lions. Mean field gamesJpn. J. Math.212007, 229--260
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• 122 articleMichael J.M. Lighthill and Gerald B.G. Whitham. On kinematic waves. II. A theory of traffic flow on long crowded roadsProc. Roy. Soc. London. Ser. A.2291955, 317--345
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• 123 articleG. Lin, C.-H. Su and G.E.G. Karniadakis. Predicting shock dynamics in the presence of uncertaintiesJournal of Computational Physics2172006, 260-276
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• 124 article M. Martinelli and R. Duvigneau. On the use of second-order derivative and metamodel-based Monte-Carlo for uncertainty estimation in aerodynamics Computers and Fluids 37 6 2010
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• 125 article C.R.C. Merritt, F. Forsberg, J. Liu and F. Kallel. In-vivo elastography in animal models: Feasibility studies, (abstract) J. Ultrasound Med. 21 98 2002
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• 126 articleSiddharthaS. Mishra, ChristophC. Schwab and JonasJ. Sukys. Multi-level Monte Carlo finite volume methods for uncertainty quantification in nonlinear systems of balance lawsLecture Notes in Computational Science and Engineering922013, 225--294
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• 127 articlePaul DP. Morris, Frans NF. van de Vosse, Patricia VP. Lawford, D RodneyD. Hose and Julian PJ. Gunn. “Virtual”(computed) fractional flow reserve: current challenges and limitationsJACC: Cardiovascular Interventions882015, 1009--1017
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• 128 articleW.L.W. Oberkampf and F.G.F. Blottner. Issues in Computational Fluid Dynamics code verification and validationAIAA Journal361998, 687--695
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• 129 book Benô\itB. Perthame. Transport equations in biology Frontiers in Mathematics Birkhäuser Verlag, Basel 2007
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• 130 articleBenedettoB. Piccoli and FrancescoF. Rossi. Transport equation with nonlocal velocity in Wasserstein spaces: convergence of numerical schemesActa Appl. Math.1242013, 73--105
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• 131 techreport F. Poirion. Stochastic Multi Gradient Descent Algorithm ONERA July 2014
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• 132 articleM. Poujade, E. Grasland-Mongrain, A. Hertzog, J. Jouanneau, P. Chavrier, B. Ladoux, A. Buguin and P. Silberzan. Collective migration of an epithelial monolayer in response to a model woundProceedings of the National Academy of Sciences104412007, 15988-15993
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• 133 article F.~S.F. Priuli. First order mean field games in crowd dynamics ArXiv e-prints February 2014
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• 134 inproceedings M.M.M. Putko, P.A.P. Newman, A.C.A. Taylor and L.L.L. Green. Approach for uncertainty propagation and robust design in CFD using sensitivity derivatives 15th AIAA Computational Fluid Dynamics Conference AIAA Paper 2001-2528 Anaheim, CA June 2001
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• 135 incollectionChunhongC. Qi, KyleA.K. Gallivan and P.-A. Absil. Riemannian BFGS Algorithm with ApplicationsRecent Advances in Optimization and its Applications in EngineeringSpringer Berlin Heidelberg2010, 183-192URL: http://dx.doi.org/10.1007/978-3-642-12598-0_16
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• 136 articleJ. Reilly, W. Krichene, M. L.M. Delle Monache, S. Samaranayake, P. Goatin and Alexandre M.A. Bayen. Adjoint-based optimization on a network of discretized scalar conservation law PDEs with applications to coordinated ramp meteringJ. Optim. Theory Appl.16722015, 733--760
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• 137 articlePaul I.P. Richards. Shock waves on the highwayOperations Res.41956, 42--51
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• 138 book P. Sagaut. Large Eddy Simulation for Incompressible Flows An Introduction Springer Berlin Heidelberg 2006
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• 139 inproceedings J. Schaefer, T. West, S. Hosder, C. Rumsey, J.-R. Carlson and 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
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• 140 articleV. Schleper. A hybrid model for traffic flow and crowd dynamics with random individual propertiesMath. Biosci. Eng.1222015, 393-413
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• 141 articleAlexandrosA. Sopasakis and Markos A.M. Katsoulakis. Stochastic modeling and simulation of traffic flow: asymmetric single exclusion process with Arrhenius look-ahead dynamicsSIAM J. Appl. Math.6632006, 921--944
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• 142 articleP. R.P. Spalart. Detached-Eddy SimulationAnnual Review of Fluid Mechanics412009, 181-202
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• 143 inproceedings SvetlanaS. Tokareva, SiddharthaS. Mishra and ChristophC. Schwab. High Order Stochastic Finite Volume Method for the Uncertainty Quantification in Hyperbolic Conservtion Laws with Random Initial Data and Flux Coefficients Proc. ECCOMAS Proc. ECCOMAS 2012
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• 144 articleShengxianS. Tu, EmanueleE. Barbato, ZsoltZ. Köszegi, JunqingJ. Yang, ZhonghuaZ. Sun, NielsN. Holm, BalázsB. Tar, YingguangY. Li, DanD. Rusinaru and WilliamW. Wijns. Fractional flow reserve calculation from 3-dimensional quantitative coronary angiography and TIMI frame count: a fast computer model to quantify the functional significance of moderately obstructed coronary arteriesJACC: Cardiovascular Interventions772014, 768--777
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• 145 inproceedings É. Turgeon, D. Pelletier and J. Borggaard. Sensitivity and Uncertainty Analysis for Variable Property Flows 39th AIAA Aerospace Sciences Meeting and Exhibit AIAA Paper 2001-0139 Reno, NV Jan. 2001
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• 146 book CédricC. Villani. Optimal transport 338 Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Old and new Springer-Verlag, Berlin 2009
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• 147 book CédricC. Villani. Topics in optimal transportation 58 Graduate Studies in Mathematics American Mathematical Society, Providence, RI 2003
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• 148 techreport R.W.R. Walter and L. Huyse. Uncertainty analysis for fluid mechanics with applications 2002--1 ICASE February 2002
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• 149 unpublishedNathanN. Wycoff, MickaelM. Binois and Stefan M.S. Wild. Sequential Learning of Active SubspacesNovember 2019, https://arxiv.org/abs/1907.11572 - working paper or preprint
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• 150 articleD.B.D. Xiu and G.E.G. Karniadakis. Modeling uncertainty in flow simulations via generalized Polynomial ChaosJournal of Computational Physics1872003, 137-167
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• 151 articleD. You and P. Moin. Active control of flow separation over an airfoil using synthetic jetsJ. of Fluids and Structures242008, 1349-1357
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• 152 articleA. Zerbinati, A. Minelli, I. Ghazlane and J.-A. Désidéri. Meta-Model-Assisted MGDA for Multi-Objective Functional OptimizationComputers and Fluids102http://www.sciencedirect.com/science/article/pii/S0045793014002576#2014, 116-130
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• 153 articleLokienL. van Nunen, FrederikF. Zimmermann, PimP. Tonino, EmanueleE. Barbato, AndreasA. Baumbach, ThomasT. Engstr\om, VolkerV. Klauss, PhilipP. MacCarthy, GaneshG. Manoharan and KeithK. Oldroyd. Fractional flow reserve versus angiography for guidance of PCI in patients with multivessel coronary artery disease (FAME): 5-year follow-up of a randomised controlled trialThe Lancet386100062015, 1853--1860
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