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  122, or to the dynamics of turbulent flows for which large scale / small scale vortical structures interfere  152.

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  165.

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  72 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  85, 111, 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  79, 78), 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  93, 110 or via the Hamilton-Jacobi approach  79, 78 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   89.

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   141. 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%   141.

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   158, 140. 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.

• 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  119, 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 119, 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. 149, that have the desirable convergence property in constrained problems. However, in this approach, the constraints are assumed to be known formally, by explicit expressions.

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

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

  - infrared micro-source;

  - micro-photoacoustic cell;

  - nanophotonic device.

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

5.1 Impact of research results

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

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

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 New software

7.1 Macroscopic traffic flow models on networks

Participants: Mickaël Binois [U Pavia, Italy], Paola Goatin [U Pavia, Italy], Alexandra Würth [U Pavia, Italy], Chiara Daini [U Pavia, Italy], Antonella Ferrara [U Pavia, Italy], Simone Göttlich [U Mannheim, Germany].

Traffic control by Connected and Automated Vehicles.

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

This work was partly achieved during of C. Daini's internship, see 49.

Traffic flow model calibration by statistical approaches.

In the framework of A. Würth's PhD thesis, we employ a Bayesian approach including a bias term to estimate first and second order model parameters, based on two traffic data sets: a set of loop detector data located on the A50 highway between Marseille and Aubagne provided by DirMED, and publicly available data from the Minnesota Department of transportation (MnDOT). In 56, we propose a Bayesian approach for parameter uncertainty quantification in macroscopic traffic flow models from cross-sectional data. A bias term is introduced and modeled as a Gaussian process to account for the traffic flow models limitations. We validate the results comparing the error metrics of both first and second order models, showing that second order models globally perform better in reconstructing traffic quantities of interest.

We also account for real data information to design improved models to better account for observations 25, 54.

7.2 Isogeometric Discontinuous Galerkin method for compressible flows

Participants: Régis Duvigneau [CEA Saclay], Stefano Pezzano [CEA Saclay], Maxime Stauffert [CEA Saclay].

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 40. The following research axes are considered in this context:

• Arbitrary Eulerian-Lagrangian formulation for high-order meshes

  To enable the simulation of flows around moving or deforming bodies, an Arbitrary Eulerian-Lagrangian (ALE) formulation is proposed in the context of the isogeometric DG method 44. 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 32. The approach has been applied to the simulation of morphing airfoils 41.

• Geometrically exact sliding interfaces

  In the context of rotating machines (compressors, turbines, etc), computations are achieved using a rotating inner grid interfaced to an outer fixed grid. This coupling is cumbersome using classical piecewise-linear grids due to a lack of common geometrical interface. Thus, we have developed a method based on a geometrically exact sliding interface using NURBS elements, ensuring a fully conservative scheme 53.

• Isogeometric shape optimization

  We develop an optimization procedure with shape sensitivity analysis, entirely based on NURBS representations 34. The mesh, the shape to be optimized, as well as the flow solutions are represented by NURBS, which avoid any geometrical conversion and allows to exploit NURBS properties regarding regularity or hierarchy. The approach has also been employed in the framework of Bayesian optimization for airfoil design 42.

7.3 Sensitivity analysis for compressible flows

Participants: Régis Duvigneau [CEA Saclay], Maxime Stauffert [CEA Saclay].

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.

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 34.

7.4 Advanced Bayesian optimization

Participants: Mickaël Binois [Université de Strasbourg], Régis Duvigneau [Université de Strasbourg], Abderrahmane Habbal [Université de Strasbourg], Luca Berti [Université de Strasbourg], Nicholson Collier [Argonne, USA], Mahmoud Elsawy [Atlantis team], Laetitia Giraldi [Calisto team], Frédéric Hauville [Ecole Navale Brest], Olivier Lemaitre [CNRS-LIMSI], Stéphane Lanteri [Atlantis team], Charles Macal [Argonne, USA], Jonathan Ozik [Argonne, USA], Victor Picheny [Secondmind, UK], Matthieu Sacher [ENSTA Bretagne], Justin Wozniak [Argonne, USA].

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 33.

Bayesian optimization of nano-photonic devices

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

First studies achieved using Bayesian optimization algorithms, demonstrate the potentiality of the proposed approach 38. In further studies 27, 28, 37, we tackle robust optimization in the presence of manufacturing uncertainties and a multi-objective approach for improving RGB lenses.

Bayesian optimization of micro-swimmers

In 45 we are interested in optimizing the shape of multi-flagellated helical microswimmers. Mimicking the propagation of helical waves along the flagella, they self-propel by rotating their tails. The swimmer's dynamics is computed using the Boundary Element Method, implemented in the open source Matlab library Gypsilab. We exploit a Bayesian optimization algorithm to maximize the swimmer's speeds through their shape optimization. Our results show that the optimal tail shapes are helices with large wavelength, such that the shape periodicity is disregarded. Moreover, the best propulsion speed is achieved for elongated heads when the swimmer has one or two flagella. Surprisingly, a round head is obtained when more flagella are considered. Our results indicate that the position and number of flagella modify the propulsion pattern and play a significant role in the optimal design of the head. It appears that Bayesian optimization is a promising method for performance improvement in microswimming.

Massively parallel Bayesian optimization

Motivated by a large scale multi-objective optimization problem for which thousands of evaluations can be conducted in parallel 31, we develop an efficient approach to tackle this issue in 46.

CityCOVID is a detailed agent-based model that represents the behaviors and social interactions of 2.7 million residents of Chicago as they move between and colocate in 1.2 million distinct places, including households, schools, workplaces, and hospitals, as determined by individual hourly activity schedules and dynamic behaviors such as isolating because of symptom onset. Disease progression dynamics incorporated within each agent track transitions between possible COVID-19 disease states, based on heterogeneous agent attributes, exposure through colocation, and effects of protective behaviors of individuals on viral transmissibility. Throughout the COVID-19 epidemic, CityCOVID model outputs have been provided to city, county, and state stakeholders in response to evolving decision-making priorities, while incorporating emerging information on SARS-CoV-2 epidemiology. Here we demonstrate our efforts in integrating our high-performance epidemiological simulation model with large-scale machine learning to develop a generalizable, flexible, and performant analytical platform for planning and crisis response.

One way to reduce the time of conducting optimization studies is to evaluate designs in parallel rather than just one-at-a-time. For expensive-to-evaluate black-boxes, batch versions of Bayesian optimization have been proposed. They work by building a surrogate model of the black-box that can be used to select the designs to evaluate efficiently via an infill criterion. Still, with higher levels of parallelization becoming available, the strategies that work for a few tens of parallel evaluations become limiting, in particular due to the complexity of selecting more evaluations. It is even more crucial when the black-box is noisy, necessitating more evaluations as well as repeating experiments. Here we propose a scalable strategy that can keep up with massive batching natively, focused on the exploration/exploitation trade-off and a portfolio allocation. We compare the approach with related methods on deterministic and noisy functions, for mono and multiobjective optimization tasks. These experiments show similar or better performance than existing methods, while being orders of magnitude faster.

A game theoretic perspective on Bayesian multi-objective optimization

In 47, a forthcoming book chapter, we address the question of how to efficiently solve many-objective optimization problems in a computationally demanding black-box simulation context. We motivate the question by applications in machine learning and engineering, and discuss specific harsh challenges in using classical Pareto approaches when the number of objectives is four or more. Then, we review solutions combining approaches from Bayesian optimization, e.g., with Gaussian processes, and concepts from game theory like Nash equilibria, Kalai-Smorodinsky solutions and detail extensions like Nash-Kalai-Smorodinsky solutions. We finally introduce the corresponding algorithms and provide some illustrating results.

7.5 Gaussian process based sequential design

Participants: Mickaël Binois [Virginia Tech, USA], Robert Gramacy [Virginia Tech, USA], Michael Ludkovski [UCSB, USA], Xiong Lyu [UCSB, USA], Stefan Wild [Argonne National Laboratory, USA], Nathan Wycoff [Virginia Tech, USA].

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

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 23 we present the hetGP package , 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.

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 29 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.

Sequential learning of active subspace

Continuing a work started at Argonne National Laboratory, in 35 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.

Sensitivity prewarping for local surrogate modeling

In the continual effort to improve product quality and decrease operations costs, computational modeling is increasingly being deployed to determine feasibility of product designs or configurations. Surrogate modeling of these computer experiments via local models, which induce sparsity by only considering short range interactions, can tackle huge analyses of complicated input-output relationships. However, narrowing focus to local scale means that global trends must be re-learned over and over again. In 57, we propose a framework for incorporating information from a global sensitivity analysis into the surrogate model as an input rotation and rescaling preprocessing step. We discuss the relationship between several sensitivity analysis methods based on kernel regression before describing how they give rise to a transformation of the input variables. Specifically, we perform an input warping such that the "warped simulator" is equally sensitive to all input directions, freeing local models to focus on local dynamics. Numerical experiments on observational data and benchmark test functions, including a high-dimensional computer simulator from the automotive industry, provide empirical validation.

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

In 48 we propose a review of high-dimensional GP modeling. Extending the efficiency of Bayesian optimization (BO) to larger number of parameters has received a lot of attention over the years. Even more so has Gaussian process regression modeling in such contexts, on which most BO methods are based. A variety of structural assumptions have been tested to tame high dimension, ranging from variable selection and additive decomposition to low dimensional embeddings and beyond. Most of these approaches in turn require modifications of the acquisition function optimization strategy as well. Here we review the defining assumptions, and discuss the benefits and drawbacks of these approaches in practice.

7.6 Policy-based optimization

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

This work concerns the development of black-box optimization methods based on single-step deep reinforcement learning (DRL) and their conceptual similarity to evolution strategy (ES) techniques 55. The connection of policy-based optimization (PBO) to evolutionary strategies (especially covariance matrix adaptation evolutionary strategy) is discussed. Relevance is assessed by benchmarking PBO against classical ES techniques on analytic functions minimization problems, and by optimizing various parametric control laws intended for the Lorenz attractor. This contribution definitely establishes PBO as a valid, versatile black-box optimization technique, and opens the way to multiple future improvements building on the inherent flexibility of the neural networks approach.

7.7 Prioritized Multi-Objective/Multi-Disciplinary Optimization

Participants: Jean-Antoine Désidéri [Essilor Créteil], Régis Duvigneau [Essilor Créteil], Pierre Leite [Essilor Créteil], Quentin Mercier [CNAM CNRS, EHESS Paris], Michel Ravachol [Dassault Aviation], Marc Vésin [Inria SED].

Our long-term aim is to contribute to Multidisciplinary Optimization (MDO), although in this area, we have not yet been able to address problems governed by one or more PDE systems. In the perspective of this ambitious target, we observe that calculating a Pareto front associated with more than two cost functions is a complex simulation enterprise, seldomly accomplished in size engineering problems 143. Analyzing the result in three or more dimensions is not a simple task either. Additionally, in many physical situations, the computational challenge of directly accounting for three or more criteria may be superfluous from the start: the performance of a complex system can often be evaluated first by a reduced set of criteria (say two or three), and other criteria be introduced in a second step only, as an adaptive refinement. Our method addresses precisely this problematics.

A numerical method has been developed to conduct multi-objective optimization in two phases. In the first phase, the primary cost functions, considered of preponderant importance, are minimized under constraints by some effective optimizer of appropriate type (gradient-based, genetic, or bayesian). From a selected Pareto-optimal point, a path parametrized by a new variable, $\epsilon$, is constructed as a continuum of Nash equilibria. The formulation is defined for each given $\epsilon$, by a “split of territories” consisting of a decomposition the admissible set into two supplementary (and not simply complementary) sub-spaces, taken to be the strategies of two competing virtual players, one in charge of the primary cost functions, and the other in charge of one or several secondary cost functions for which adaptation is considered necessary.

The formulation is “compatible” with the first phase of optimization, in the sense that the selected initial point is indeed the Nash equilibrium point achieved by the formulation for $\epsilon =0$. We have established theoretically that the Nash equilibrium point exists for all sufficiently small $\epsilon$ (existence), and that as $\epsilon$ increases: (i) the secondary cost functions diminish linearly with $\epsilon$ at a calculated rate; (ii) the Pareto-optimality condition of the primary cost functions is degraded by a term $O\left({\epsilon }^2\right)$ only. Hence the secondary criteria have been improved at the cost of a smaller degradation of the primary ones in orders of magnitude 7.

A special chapter of the software platform MGDA has been developed with the assistance of the Inria Service for Software Development and Experimentation to facilitate the application of this strategy by external users. (See Section Software).

The method was successfully applied in two problems of technical relevance:

• Optimization of the flight performance of a supersonic business jet (SSBJ), evaluated via 15 sizing variables by a software provided by Dassault Aviation within a former ANR project on MDO by application of the classical Breguet's laws. In the first phase of optimization, the Pareto front between mass and range subject to a bound constraint on take-off distance was computed by the Pareto-Archived Evolutionary Strategy. In the second phase, the solution was “adapted” to diminish the approach speed 10.
• Structural optimization of an aluminum sandwich element. The element was sized via 3 thicknesses and 1 ratio. We demonstrated the method through several test-cases, in which the element was subject to bending loads, and/or blast. The cost functions were: mass, critical failure forces under bending loads (first 2 modes considered), 2 measures of blast mitigation (absorbed energy, deflection) 7.

This promising method is currently being applied to another aircraft performance optimization in cooperation with Onera Toulouse (N. Bartoli, Ch. David, S. Defoort). In this case study, we are using the open-source Fast-OAD software developed by Onera to evaluate the performance (two masses at take-off, and the ascent time) and our platform to accomplish the prioritized optimization, aiming at documenting a reproducible case study, and vitalizing a technical cooperation with Onera.

7.8 Inverse Cauchy-Stokes problems solved as Nash games

Participants: Abderrahmane Habbal [PhD, LAMSIN, Univ. Tunis Al Manar], Marwa Ouni [PhD, LAMSIN, Univ. Tunis Al Manar], Moez Kallel [LAMSIN, Univ. Tunis Al Manar].

We extend in two directions our results published in 120 to tackle ill posed Cauchy-Stokes inverse problems as Nash games. First, we consider the problem of detecting unknown pointwise sources in a stationary viscous fluid, using partial boundary measurements. The considered fluid obeys a steady Stokes regime, the boundary measurements are a single compatible pair of Dirichlet and Neumann data, available only on a partial accessible part of the whole boundary. This inverse source identification for the Cauchy-Stokes problem is ill-posed for both the sources and missing data reconstructions, and designing stable and efficient algorithms is challenging. We reformulate the problem as a three-player Nash game. Thanks to a source identifiability result derived for the Cauchy-Stokes problem, it is enough to set up two Stokes BVP, then use them as state equations. The Nash game is then set between 3 players, the first two targeting the data completion while the third one targets the detection of the number, location and magnitude of the unknown sources. We provided the third player with the location and magnitude parameters as strategy, with a cost functional of Kohn-Vogelius type. In particular, the location is obtained through the computation of the topological sensitivity of the latter function. We propose an original algorithm, which we implemented using Freefem++. We present 2D numerical experiments for many different test-cases.The obtained results corroborate the efficiency of our 3-player Nash game approach to solve parameter or shape identification for Cauchy-Stokes problems 43.

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 50, 39.

7.9 Classical PDEs and non classical hybrid ABM/PDEs models for cell dynamics

Participants: Abderrahmane Habbal [PhD, ACUMES and EMI, Univ. Mohammed V], Salma Chabbar [PhD, ACUMES and EMI, Univ. Mohammed V], Rajae Aboulaich [EMI, Univ. Mohammed V], Mekki Ayadi [Sousse University], Talha Achouri [Shaqra University], Boutheina ahyaoui [Taibah University].

We have introduced and analyzed 22 a non-linear Cranck-Nicolson Finite Difference scheme, dedicated to the numerical solution of the Fisher and KPP equation, a non-linear parabolic reaction-diffusion equation we have formerly used to model wound closure in the absence and presence of activators or inhibitors 118, 164. For the present numerical analysis, we take into consideration mixed boundary conditions. We first have established that the non-linear discretized system is well posed, and proved both consistency and, using a Energy functional, its stability. We also proved its second order convergence in the ad hoc Sobolev norm. For each time step, the non-linear -scalar- problem was solved by means of an exact Newton method.

Numerical investigations corroborate the theoretical error estimates, and convergence order. A challenging perspective is to analyse the numerical schemes dedicated to non constant diffusion-proliferation parameters.

Moving from the above well established PDE equations used to model cell dynamics, we develop an hybrid model coupling agent-based modeling to PDEs : an ABM-PDEs multi-scale tumor growth model is developed in 24, where micro and macro scales communicate through a hybrid formulation: cells as microscopic agents, with ABM handling complex cell-cell interactions, and nutrient concentration as a macroscopic field, which evolution is governed by reaction-diffusion PDEs.

8.1 Bilateral contracts with industry

• E-Genius2 (2019, extended to 2021): Acumes has set up a 12 months research and development contract with the company E-Genius2, Data-ia group, Montpellier, (ex Etic Data) on "Predictive modeling and proactive driving of customers behaviour in massive data BtoC context" (22 keuro).

  Participants: Mickaël Binois, Abderrahmane Habbal.

• Mycophyto (2020-...): this research contract involving Université Côte d'Azur is financing the post-doctoral contract of Khadija Musayeva.

  Participants: Mickaël Binois, Khadija Musayeva.

9.1 International initiatives

9.2 International research visitors

9.3 European initiatives

9.4 National initiatives

10.1 Promoting scientific activities

10.2 Teaching - Supervision - Juries

11.1 Major publications

• 1 articleA.Aekta Aggarwal, R. M.Rinaldo M. Colombo and P.Paola Goatin. Nonlocal systems of conservation laws in several space dimensions.SIAM Journal on Numerical Analysis5222015, 963-983
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• 2 articleB.Boris Andreianov, P.Paola Goatin and N.Nicolas Seguin. Finite volume schemes for locally constrained conservation laws.Numer. Math.1154With supplementary material available online2010, 609--645
• 3 articleS.Sebastien Blandin and P.Paola Goatin. Well-posedness of a conservation law with non-local flux arising in traffic flow modeling.Numerische Mathematik2015
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• 4 articleR. M.Rinaldo M. Colombo and P.Paola Goatin. A well posed conservation law with a variable unilateral constraint.J. Differential Equations23422007, 654--675
• 5 articleM. L.M. L. Delle Monache and P.Paola Goatin. Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result.J. Differential Equations257112014, 4015--4029
• 6 articleM. L.M. L. Delle Monache, J.J. Reilly, S.S. Samaranayake, W.W. Krichene, P.P. Goatin and A.Alexandre Bayen. A PDE-ODE model for a junction with ramp buffer.SIAM J. Appl. Math.7412014, 22--39
• 7 inproceedingsJ.-A.Jean-Antoine Désidéri. Adaptation by Nash games in gradient-based multi-objective/multi-disciplinary optimization.JANO13 - Mathematical Control and Numerical Applications372Springer Proceedings in Mathematics & Statistics SeriesKhouribga, MoroccoFebruary 2021
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• 8 articleJ.-A.Jean-Antoine Désidéri. COOPERATION AND COMPETITION IN MULTIDISCIPLINARY OPTIMIZATION Application to the aero-structural aircraft wing shape optimization.Computational Optimization and Applications5212012, 29-68
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• 9 inbookJ.-A.Jean-Antoine Desideri and R.Régis Duvigneau. Parametric optimization of pulsating jets in unsteady flow by Multiple-Gradient Descent Algorithm (MGDA).Numerical Methods for Differential Equations, Optimization, and Technological Problems, Modeling, Simulation and Optimization for Science and TechnologyJanuary 2017
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• 10 articleJ.-A.Jean-Antoine Désidéri and R.Régis Duvigneau. Prioritized optimization by Nash games : towards an adaptive multi-objective strategy.ESAIM: Proceedings and Surveys71August 2021, 54-63
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• 11 articleJ.-A.Jean-Antoine Désidéri. Multiple-gradient descent algorithm (MGDA) for multiobjective optimization / Algorithme de descente à gradients multiples pour l'optimisation multiobjectif.Comptes Rendus. MathématiqueTome 350Fascicule 5-6March 2012, 313-318
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• 12 articleR.R. Duvigneau and P.P. Chandrashekar. Kriging-based optimization applied to flow control.Int. J. for Numerical Methods in Fluids69112012, 1701-1714
• 13 articleA.A. Habbal and M.M. Kallel. Neumann-Dirichlet Nash strategies for the solution of elliptic Cauchy problems.SIAM J. Control Optim.5152013, 4066--4083
• 14 articleM.Moez Kallel, R.Rajae Aboulaich, A.Abderrahmane Habbal and M.Maher Moakher. A Nash-game approach to joint image restoration and segmentation.Appl. Math. Model.3811-122014, 3038--3053URL: http://dx.doi.org/10.1016/j.apm.2013.11.034
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• 15 articleM.M. Martinelli and R.R. Duvigneau. On the use of second-order derivative and metamodel-based Monte-Carlo for uncertainty estimation in aerodynamics.Computers and Fluids3762010
• 16 articleQ.Quentin Mercier, F.Fabrice Poirion and J.-A.Jean-Antoine Desideri. A stochastic multiple gradient descent algorithm.European Journal of Operational ResearchMay 2018, 10
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• 17 articleS.Souvik Roy, A.A. Borzì and A.Abderrahmane Habbal. Pedestrian motion modelled by Fokker--Planck Nash games.Royal Society open science492017, 170648
• 18 articleG.Giovanni Todarello, F.Floris Vonck, S.Sébastien Bourasseau, J.Jacques Peter and J.-A.Jean-Antoine Desideri. Finite-volume goal-oriented mesh adaptation for aerodynamics using functional derivative with respect to nodal coordinates.Journal of Computational Physics313May 2016, 21
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• 19 articleM.M. Twarogowska, P.P. Goatin and R.R. Duvigneau. Macroscopic modeling and simulations of room evacuation.Appl. Math. Model.38242014, 5781--5795
• 20 articleG.G. Xu, B.B. Mourrain, A.A. Galligo and R.R. Duvigneau. Constructing analysis-suitable parameterization of computational domain from CAD boundary by variational harmonic method.J. Comput. Physics252November 2013
• 21 articleB.Boutheina Yahyaoui, M.Mekki Ayadi and A.Abderrahmane Habbal. Fisher-KPP with time dependent diffusion is able to model cell-sheet activated and inhibited wound closure.Mathematical biosciences2922017, 36--45

11.2 Publications of the year

11.3 Cited publications

• 58 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
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• 59 articleA.Aekta Aggarwal, R. M.Rinaldo M. Colombo and P.Paola Goatin. Nonlocal systems of conservation laws in several space dimensions.SIAM Journal on Numerical Analysis5222015, 963-983
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• 60 articleG.Giovanni Alessandrini. Examples of instability in inverse boundary-value problems.Inverse Problems1341997, 887--897URL: http://dx.doi.org/10.1088/0266-5611/13/4/001
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• 61 articleD.Debora Amadori and W.Wen Shen. An integro-differential conservation law arising in a model of granular flow.J. Hyperbolic Differ. Equ.912012, 105--131
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• 62 articleP.P. Amorim, R. M.Rinaldo M. Colombo and A.A. Teixeira. On the Numerical Integration of Scalar Nonlocal Conservation Laws.ESAIM M2AN4912015, 19--37
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• 63 articleP.Paulo Amorim. On a nonlocal hyperbolic conservation law arising from a gradient constraint problem.Bull. Braz. Math. Soc. (N.S.)4342012, 599--614
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• 64 articleM.M. Annunziato and A.A. Borz\`i. A Fokker-Planck control framework for multidimensional stochastic processes.Journal of Computational and Applied Mathematics2372013, 487-507
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• 65 articleA.A. Belme, F.F. Alauzet and A.A. Dervieux. Time accurate anisotropic goal-oriented mesh adaptation for unsteady flows.J. Comput. Physics231192012, 6323--6348
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• 66 articleS.Sylvie Benzoni-Gavage, R. M.Rinaldo M. Colombo and P.Piotr Gwiazda. Measure valued solutions to conservation laws motivated by traffic modelling.Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.46220702006, 1791--1803
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• 67 unpublishedE.Enrico Bertino, R.Régis Duvigneau and P.Paola Goatin. Uncertainties in traffic flow and model validation on GPS data.2015
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• 68 articleF.F. Betancourt, R.R. Bürger, K. H.K. H. Karlsen and E. M.E. M. Tory. On nonlocal conservation laws modelling sedimentation.Nonlinearity2432011, 855--885
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• 69 articleS.Sebastien Blandin and P.Paola Goatin. Well-posedness of a conservation law with non-local flux arising in traffic flow modeling.Numer. Math.13222016, 217--241URL: https://doi.org/10.1007/s00211-015-0717-6
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• 70 articleJ.Jeff Borggaard and J.John 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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• 71 articleR.R. Bourguet, M.M. Brazza, G.G. Harran and R.R. El Akoury. Anisotropic Organised Eddy Simulation for the prediction of non-equilibrium turbulent flows around bodies.J. of Fluids and Structures2482008, 1240--1251
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• 72 articleA.Alberto Bressan, S.Sunċica Ċanić, M.Mauro Garavello, M.Michael Herty and B.Benedetto Piccoli. Flows on networks: recent results and perspectives.EMS Surv. Math. Sci.112014, 47--111
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• 73 articleM.Martin Burger, M.Marco Di Francesco, P. A.Peter A. Markowich and M.-T.Marie-Therese Wolfram. Mean field games with nonlinear mobilities in pedestrian dynamics.Discrete Contin. Dyn. Syst. Ser. B1952014, 1311--1333
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• 74 articleM.M. Burger, J.J. Haskovec and M.-T.M.-T. Wolfram. Individual based and mean-field modelling of direct aggregation.Physica D2602013, 145--158
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• 75 techreportA.Alessandra Cabassi and P.Paola Goatin. Validation of traffic flow models on processed GPS data.Research Report RR-83822013
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• 76 unpublishedJ.J.~A. Carrillo, S.S. Martin and M.-T.M.-T. Wolfram. A local version of the Hughes model for pedestrian flow.2015, Preprint
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• 77 unpublishedC.Christophe Chalons, M. L.Maria Laura Delle Monache and P.Paola Goatin. A conservative scheme for non-classical solutions to a strongly coupled PDE-ODE problem.2015, Preprint
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• 78 articleC. G.Christian G. Claudel and A. M.Alexandre M. Bayen. Convex formulations of data assimilation problems for a class of Hamilton-Jacobi equations.SIAM J. Control Optim.4922011, 383--402
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• 79 articleC.C.G. Claudel and A. M.Alexandre M. Bayen. Lax-Hopf Based Incorporation of Internal Boundary Conditions Into Hamilton-Jacobi Equation. Part II: Computational Methods.Automatic Control, IEEE Transactions on555May 2010, 1158-1174
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• 80 articleR. M.Rinaldo M. Colombo, M.Mauro Garavello and M.Magali Lécureux-Mercier. A Class Of Nonloval Models For Pedestrian Traffic.Mathematical Models and Methods in Applied Sciences22042012, 1150023
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• 81 articleR. M.Rinaldo M. Colombo, M.Michael Herty and M.Magali Mercier. Control of the continuity equation with a non local flow.ESAIM Control Optim. Calc. Var.1722011, 353--379
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• 82 articleR. M.Rinaldo M. Colombo and M.Magali Lécureux-Mercier. Nonlocal crowd dynamics models for several populations.Acta Math. Sci. Ser. B Engl. Ed.3212012, 177--196
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• 83 articleR. M.Rinaldo M. Colombo and F.Francesca Marcellini. A mixed ODE-PDE model for vehicular traffic.Mathematical Methods in the Applied Sciences3872015, 1292--1302
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• 84 articleR. M.Rinaldo M. Colombo and E.E. Rossi. On the micro-macro limit in traffic flow.Rend. Semin. Mat. Univ. Padova1312014, 217--235
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• 85 articleG.Guillaume Costeseque and J.-P.Jean-Patrick Lebacque. Discussion about traffic junction modelling: conservation laws vs Hamilton-Jacobi equations.Discrete Contin. Dyn. Syst. Ser. S732014, 411--433
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• 86 articleG.Gianluca Crippa and M.Magali Lécureux-Mercier. Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow.Nonlinear Differential Equations and Applications NoDEA2012, 1-15
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• 87 inproceedingsE.E. Cristiani, B.B. Piccoli and A.A. Tosin. How can macroscopic models reveal self-organization in traffic flow?Decision and Control (CDC), 2012 IEEE 51st Annual Conference onDec 2012, 6989-6994
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• 88 bookE.Emiliano Cristiani, B.Benedetto Piccoli and A.Andrea Tosin. Multiscale modeling of pedestrian dynamics.12MS&A. Modeling, Simulation and ApplicationsSpringer, Cham2014
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• 89 article T.T. Cuisset, J.J. QuiliCi and G.G. Cayla.. Qu'est-ce que la FFR? Comment l'utiliser? Réalités Cardiologiques Janvier/Février 2013
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• 90 incollectionC. M.C. M. Dafermos. Solutions in $L^{}$ for a conservation law with memory.Analyse mathématique et applicationsMontrougeGauthier-Villars1988, 117--128
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• 91 articleP.Pierre Degond, J.-G.Jian-Guo Liu and C.Christian Ringhofer. Large-scale dynamics of mean-field games driven by local Nash equilibria.J. Nonlinear Sci.2412014, 93--115URL: http://dx.doi.org/10.1007/s00332-013-9185-2
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• 92 articleM. L.Maria Laura Delle Monache and P.Paola Goatin. A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow.Discrete Contin. Dyn. Syst. Ser. S732014, 435--447
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• 93 articleM. L.M. L. Delle Monache and P.P. Goatin. Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result.J. Differential Equations257112014, 4015--4029
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• 94 inbookJ.-A.J.-A. Désidéri. Multiple-Gradient Descent Algorithm (\em MGDA) for Pareto-Front Identification.34Numerical Methods for Differential Equations, Optimization, and Technological ProblemsModeling, Simulation and Optimization for Science and Technology, Fitzgibbon, W.; Kuznetsov, Y.A.; Neittaanmäki, P.; Pironneau, O. Eds.J. Périaux and R. Glowinski JubileesSpringer-Verlag2014, 1
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• 95 articleJ.-A.Jean-Antoine Désidéri. Multiple-gradient descent algorithm (MGDA) for multiobjective optimization.Comptes Rendus de l'Académie des Sciences Paris3502012, 313-318URL: http://dx.doi.org/10.1016/j.crma.2012.03.014
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• 96 techreportJ.-A.Jean-Antoine Désidéri. Révision de l'algorithme de descente à gradients multiples (MGDA) par orthogonalisation hiérarchique.8710INRIAApril 2015
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• 97 incollectionB.Bruno Després, G.Gaël Poëtte and D.Didier Lucor. Robust uncertainty propagation in systems of conservation laws with the entropy closure method.Uncertainty quantification in computational fluid dynamics92Lect. Notes Comput. Sci. Eng.Springer, Heidelberg2013, 105--149
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• 98 articleM.M. Di Francesco and M.M.~D. Rosini. Rigorous Derivation of Nonlinear Scalar Conservation Laws from Follow-the-Leader Type Models via Many Particle Limit.Archive for Rational Mechanics and Analysis2015
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• 99 articleR. J.Ronald J. DiPerna. Measure-valued solutions to conservation laws.Arch. Rational Mech. Anal.8831985, 223--270
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• 100 articleC.Christian Dogbé. Modeling crowd dynamics by the mean-field limit approach.Math. Comput. Modelling529-102010, 1506--1520
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• 101 techreportR.R. Duvigneau. A Sensitivity Equation Method for Unsteady Compressible Flows: Implementation and Verification.INRIA Research Report No 8739June 2015
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• 102 articleR.R. Duvigneau and D.D. Pelletier. A sensitivity equation method for fast evaluation of nearby flows and uncertainty analysis for shape parameters.Int. J. of Computational Fluid Dynamics207August 2006, 497--512
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• 103 articleR.Radek Erban, M. B.Mark B. Flegg and G. A.Garegin A. Papoian. Multiscale stochastic reaction-diffusion modeling: application to actin dynamics in filopodia.Bull. Math. Biol.7642014, 799--818URL: http://dx.doi.org/10.1007/s11538-013-9844-3
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• 104 articleR.R. Etikyala, S.S. Göttlich, A.A. Klar and S.S. Tiwari. Particle methods for pedestrian flow models: from microscopic to nonlocal continuum models.Math. Models Methods Appl. Sci.24122014, 2503--2523
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• 105 incollectionR.Robert Eymard, T.Thierry Gallouët and R.Raphaèle Herbin. Finite volume methods.Handbook of numerical analysis, Vol. VIIHandb. Numer. Anal., VIINorth-Holland, Amsterdam2000, 713--1020
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• 106 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
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• 107 articleM. B.Mark B. Flegg, S.Stefan Hellander and R.Radek Erban. Convergence of methods for coupling of microscopic and mesoscopic reaction-diffusion simulations.J. Comput. Phys.2892015, 1--17URL: http://dx.doi.org/10.1016/j.jcp.2015.01.030
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• 108 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
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• 109 articleB.Benjamin Franz, M. B.Mark B. Flegg, S. J.S. Jonathan Chapman and R.Radek Erban. Multiscale reaction-diffusion algorithms: PDE-assisted Brownian dynamics.SIAM J. Appl. Math.7332013, 1224--1247
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• 110 articleM.Mauro Garavello and B.Benedetto Piccoli. Coupling of microscopic and phase transition models at boundary.Netw. Heterog. Media832013, 649--661
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• 111 bookM.Mauro Garavello and B.Benedetto Piccoli. Traffic flow on networks.1AIMS Series on Applied MathematicsConservation laws modelsAmerican Institute of Mathematical Sciences (AIMS), Springfield, MO2006
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• 112 articleP.Paola Goatin and M.Matthias Mimault. A mixed system modeling two-directional pedestrian flows.Math. Biosci. Eng.1222015, 375--392
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• 113 unpublishedP.Paola Goatin and F.Francesco Rossi. A traffic flow model with non-smooth metric interaction: well-posedness and micro-macro limit.2015, PreprintURL: http://arxiv.org/abs/1510.04461
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• 114 articleP.Paola Goatin and S.Sheila Scialanga. Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity.Netw. Heterog. Media1112016, 107--121
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• 115 articleS.Simone Göttlich, S.Simon Hoher, P.Patrick Schindler, V.Veronika Schleper and A.Alexander Verl. Modeling, simulation and validation of material flow on conveyor belts.Applied Mathematical Modelling38132014, 3295--3313
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• 116 articleA.A. Griewank. Achieving logarithmic growth of temporal and spatial complexity in reverse automatic differentiation.Optimization Methods and Software11992, 35-54
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• 117 articleM.Michael Gröschel, A.Alexander Keimer, G.Günter Leugering and Z.Zhiqiang Wang. Regularity theory and adjoint-based optimality conditions for a nonlinear transport equation with nonlocal velocity.SIAM J. Control Optim.5242014, 2141--2163
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• 118 articleA.Abderrahmane Habbal, H.Hélène Barelli and G.Grégoire Malandain. Assessing the ability of the 2D Fisher-KPP equation to model cell-sheet wound closure.Math. Biosci.2522014, 45--59URL: http://dx.doi.org/10.1016/j.mbs.2014.03.009
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• 119 articleA.A. Habbal and M.Moez Kallel. Neumann-Dirichlet Nash strategies for the solution of elliptic Cauchy problems.SIAM J. Control Optim.5152013, 4066--4083
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• 120 articleA.Abderrahmane Habbal, M.Moez Kallel and M.Marwa Ouni. Nash strategies for the inverse inclusion Cauchy-Stokes problem.Inverse Problems and Imaging 1342019, 36
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• 121 articleX.X. Han, P.P. Sagaut and D.D. Lucor. On sensitivity of RANS simulations to uncertain turbulent inflow conditions.Computers & Fluids612-52012
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• 122 articleD.Dirk Helbing, P.Peter Molnar, I. J.Illes J Farkas and K.Kai Bolay. Self-organizing pedestrian movement.Environment and planning B2832001, 361--384
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• 123 articleD.Dirk Helbing. Traffic and related self-driven many-particle systems.Rev. Mod. Phys.7342001, 1067--1141
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• 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
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• 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 Applications41602014, 684--694
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• 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
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• 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 Engineering1942005, 4135--4195
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