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In this paper, we discuss a particular class of Nash games, where the participants of the game (the players) are divided into two groups (leaders and followers) according to their position or influence on the other players. Moreover, we consider the case, when the leaders’ and/or the followers’ game can be described as a potential game. This is a subclass of Nash games that has been introduced by Monderer and Shapley in 1996 and has beneficial properties to reformulate the bilevel Nash game. We develope necessary and sufficient conditions for Nash equilibria and present existence and uniqueness results. Furthermore, we discuss some Examples to illustrate our results. In this paper, we discussed analytical properties for multi-leader follower potential games, that form a subclass of hierarchical Nash games. The application of these theoretical results to various fields of applications are a future research topic. Moreover, they are meant to serve as a starting point for the developement of efficient numerical solution methods for multi-leader-follower games.

Numerical simulation codes are very common tools to study complex phenomena, but they are often time-consuming and considered as black boxes. For some statistical studies (e.g. asset management, sensitivity analysis) or optimization problems (e.g. tuning of a molecular model), a high number of runs of such codes is needed. Therefore it is more convenient to build a fast-running approximation – or metamodel – of this code based on a design of experiments. The topic of this paper is the definition of metamodels for stochastic codes. Contrary to deterministic codes, stochastic codes can give different results when they are called several times with the same input. In this paper, two approaches are proposed to build a metamodel of the probability density function of a stochastic code output. The first one is based on kernel regression and the second one consists in decomposing the output density on a basis of well-chosen probability density functions, with a metamodel linking the coefficients and the input parameters. For the second approach, two types of decomposition are proposed, but no metamodel has been designed for the coefficients yet. This is a topic of future research. These methods are applied to two analytical models and three industrial cases.

This paper presents a new method for global optimization. We use exact quadratic regularization for the transformation of the multimodal problems to a problem of a maximum norm vector on a convex set. Quadratic regularization often allows you to convert a multimodal problem into a unimodal problem. For this, we use the shift of the feasible region along the bisector of the positive orthant. We use only local search (primal-dual interior point method) and a dichotomy method for search of a global extremum in the multimodal problems. The comparative numerical experiments have shown that this method is very efficient and promising.

In this paper, we present the optimization procedure for computing the discrete boxconstrained minimax classifier introduced in [1, 2]. Our approach processes discrete or beforehand discretized features. A box-constrained region defines some bounds for each class proportion independently. The box-constrained minimax classifier is obtained from the computation of the least favorable prior which maximizes the minimum empirical risk of error over the box-constrained region. After studying the discrete empirical Bayes risk over the probabilistic simplex, we consider a projected subgradient algorithm which computes the prior maximizing this concave multivariate piecewise affine function over a polyhedral domain. The convergence of our algorithm is established.

We provide an overview of a class of iterative convex approximation methods for nonlinear optimization problems with convex-over-nonlinear substructure. These problems are characterized by outer convexities on the one hand, and nonlinear, generally nonconvex, but differentiable functions on the other hand. All methods from this class use only first order derivatives of the nonlinear functions and sequentially solve convex optimization problems. All of them are different generalizations of the classical Gauss-Newton (GN) method. We focus on the smooth constrained case and on three methods to address it: Sequential Convex Programming (SCP), Sequential Convex Quadratic Programming (SCQP), and Sequential Quadratically Constrained Quadratic Programming (SQCQP). While the first two methods were previously known, the last is newly proposed and investigated in this paper. We show under mild assumptions that SCP, SCQP and SQCQP have exactly the same local linear convergence – or divergence – rate. We then discuss the special case in which the solution is fully determined by the active constraints, and show that for this case the KKT conditions are sufficient for local optimality and that SCP, SCQP and SQCQP even converge quadratically. In the context of parameter estimation with symmetric convex loss functions, the possible divergence of the methods can in fact be an advantage that helps them to avoid some undesirable local minima: generalizing existing results, we show that the presented methods converge to a local minimum if and only if this local minimum is stable against a mirroring operation applied to the measurement data of the estimation problem. All results are illustrated by numerical experiments on a tutorial example.

The Sea Surface Temperature (SST) plays a significant role in analyzing and assessing the dynamics of weather and also biological systems. It has various applications such as weather forecasting or planning of coastal activities. On the one hand, standard physical methods for forecasting SST use coupled ocean- atmosphere prediction systems, based on the Navier-Stokes equations. These models rely on multiple physical hypotheses and do not optimally exploit the information available in the data. On the other hand, despite the availability of large amounts of data, direct applications of machine learning methods do not always lead to competitive state of the art results. Another approach is to combine these two methods: this is data-model coupling. The aim of this paper is to use a model in another domain. This model is based on a data-model coupling approach to simulate and predict SST. We first introduce the original model. Then, the modified model is described, to finish with some numerical results.

This paper is concerned by statistical inference problems from a data set whose elements may be modeled as random probability measures such as multiple histograms or point clouds. We propose to review recent contributions in statistics on the use of Wasserstein distances and tools from optimal transport to analyse such data. In particular, we highlight the benefits of using the notions of barycenter and geodesic PCA in the Wasserstein space for the purpose of learning the principal modes of geometric variation in a dataset. In this setting, we discuss existing works and we present some research perspectives related to the emerging field of statistical optimal transport.

This paper presents a FEM-BEM coupling method suitable for the numerical simulation of the electromagnetic scattering of objects composed of dielectric materials and perfect electric conduc- tors. The originality of the approach lies in part in the use of the newly proposed Sparse Cardinal Sine Decomposition SCSD) method for the BEM part of the computation and the fact that the simulation software is almost entirely written in MATLAB. The performance of the method is illustrated by the computation of the electromagnetic scattering by an UAV-like object with two RAM regions proposed in the workshop ISAE EM 2016.

When the martingale representation property holds, we call any local martingale which realizes the representation a representation process . There are two properties of the representation process which can greatly facilitate the computations under the martingale representation property . On the one hand, the representation process is not unique and there always exists a representation process which is locally bounded and has pathwise orthogonal components outside of a predictable thin set. On the other hand, the jump measure of a representation process satisfies the finite predictable constraint , which implies the martingale projection property . In this paper, we give a detailed account of these properties. As application, we will prove that, under the martingale representation property , the full viability of an expansion of market information flow implies the drift multiplier assumption .

Breiman's (2001) random forests are a very popular class of learning algorithms often able to produce good predictions even in high-dimensional frameworks, with no need to accurately tune its inner parameters. Unfortunately, there are no theoretical findings to support the default values used for these parameters in Breiman's algorithm. The aim of this paper is therefore to present recent theoretical results providing some insights on the role and the tuning of these parameters.

Automatic differentiation has been involved for long in applied mathematics as an alternative to finite difference to improve the accuracy of numerical computation of derivatives. Each time a numerical minimization is involved, automatic differentiation can be used. In between formal derivation and standard numerical schemes, this approach is based on software solutions applying mechanically the chain rule formula to obtain an exact value for the desired derivative. It has a cost in memory and cpu consumption. For participants of financial markets (banks, insurances, financial intermediaries, etc), computing derivatives is needed to obtain the sensitivity of their exposure to well-defined potential market moves. It is a way to understand variations of their balance sheets in specific cases. Since the 2008 crisis, regulation demands to compute this kind of exposure to many different cases, to be sure that market participants are aware and ready to face a wide spectrum of market configurations. This paper shows how automatic differentiation provides a partial answer to this recent explosion of computations to be performed. One part of the answer is a straightforward application of Adjoint Algorithmic Differentiation (AAD), but it is not enough. Since financial sensitivities involve specific functions and mix differentiation with Monte-Carlo simulations, dedicated tools and associated theoretical results are needed. We give here short introductions to typical cases arising when one uses AAD on financial markets.

This paper studies the dynamics and stability of a stage-structured (immature/mature) population. The emphasis is on how resource available to the immature stage indirectly regulates the dynamics of the mature stage, where the resource is defined by a Holling-II function. We present the results from numerical experiments, where inference about stability and predictability of the system dynamics is based on the theory of Lyapunov exponents.

International audience; We are interested in this paper in the modelling and numerical simulation of some phenomena that are observed in the context of car dynamics, in particular the appearance of persistent jams upstream critical points with no real cause of flux limitation. We shall consider the case of a stable jam on a freeway upstream an accident that took place on the opposite lane. This situation is not properly handled by most models, either micro-or macroscopic ones, since it corresponds to a phenomenon that does not have a counterpart in gas dynamics, for which only entropy solution are usually considered as physically feasible. The approach we propose consists in accounting for the very behaviour of agents in the neighbourhood of the discontinuity, and makes it possible to numerically recover in a robust way steady traffic jams.; Nous nous intéressons dans cet article à la modélisation et à la simulation numérique de phénomènes particuliers observés dans le contexte du trafic routier, plus particulièrement le phénomène d’apparition et de persistance de bouchons en amont de points critiques sans cause objective de ralentissement. Nous considérerons notamment le cas d’un bouchon persistant sur une autoroute en amont d’un accident qui s’est produit sur la voie d’en face. Cette situation n’est pas reproduite par la plupart des modèles, qu’ils soient microscopiques ou macroscopiques, car elle correspond à un phénomène qui n’a pas d’équivalent en dynamique des gaz, pour lesquels seules les solutions dites entropiques sont en général considérées comme correspondant à un comportement observable dans la réalité. L’approche que nous proposons, aux niveaux microscopique et macroscopique, consiste à prendre en compte le comportement particulier des agents humains au voisinage de la discontinuité, et permet de retrouver numériquement de façon robuste des bouchons stables.

Gyrokinetic modeling is appropriate for describing Tokamak plasma turbulence, and the gyroaverage operator is a cornerstone of this approach. In a gyrokinetic code, the gyroaveraging scheme needs to be accurate enough to avoid spoiling the data but also requires a low computation cost because it is applied often on the main unknown, the 5D guiding-center distribution function, and on the 3D electric potentials. In the present paper, we improve a gyroaverage scheme based on Hermite interpolation used in the Gysela code. This initial implementation represents a too large fraction of the total execution time. The gyroaverage operator has been reformulated and is now expressed as a matrix-vector product and a cache-friendly algorithm has been setup. Different techniques have been investigated to quicken the computations by more than a factor two. Description of the algorithms is given, together with an analysis of the achieved performance.

International audience; This paper introduces a Semi-Lagrangian solver for the Vlasov-Poisson equations on a uniform hexagonal mesh. The latter is composed of equilateral triangles, thus it doesn't contain any singularities, unlike polar meshes. We focus on the guiding-center model, for which we need to develop a Poisson solver for the hexagonal mesh in addition to the Vlasov solver. For the interpolation step of the Semi-Lagrangian scheme, a comparison is made between the use of Box-splines and of Hermite finite elements. The code will be adapted to more complex models and geometries in the future.; Dans cet article nous présentons un solveur semi-Lagrangien pour les équations de Vlasov-Poisson sur un maillage hexagonal uniforme. Ce dernier est composé de triangles équilatéraux, ainsi il ne présente aucune singularité, contrairement au maillage polaire. Nous nous concentrons ici sur le modèle centre-guide. À cette fin nous avons développé en plus du solveur pour Vlasov, un solveur de l'équation de Poisson pour maillage hexagonal. Nous comparons les résultats obtenus avec une interpolation par éléments finis d'Hermite et par des Box-splines. Dans l'avenir, ce code sera adapté à des géométries et modèles plus complexes.

This paper collects the four contributions which were presented during the session devoted to Stochastic Geometry at the journées MAS 2014. It is focused in particular on several questions related to the transmission of information in a general sense in different random media. The underlying models include the Boolean model, simplicial complexes or geometric random graphs induced by a point process.

The goal of this paper is to present a series of recent contributions on some various problems of numerical probability. Beginning with the Richardson-Romberg Multilevel Monte-Carlo method which, among other fields of applications, is a very efficient method for the approximation of diffusion processes, we focus on some adaptive multilevel splitting algorithms for rare event simulation. Then, the third part is devoted to the simulation of McKean-Vlasov forward and decoupled forward-backward stochastic differential equations by some cubature algorithms. Finally, we tackle the problem of the weak error estimation in total variation norm for a general Markov semi-group.

In this paper, we consider the problem of optimal regional controllability of a distributed bilinear system evolving on Ω. The question is to obtain a control with minimum energy that drives such a system from an initial state to a final state close to a desired one in finite time, only on a subregion ω of Ω. Our purpose is to prove that a regional optimal control exists and characterized in both bounded and unbounded cases. The obtained results are successfully illustrated by simulations.

This paper deals with the control design for a class of nonlinear distributed parameter systems, i.e. convection-diffusion-reaction systems, encountered under the form of non-isothermal tubular reactors in chemical engineering applications. More specifically the design concentrates on the boundary control of the temperature profile in an exothermic chemical reactor under input constraints. Our objective is to analyze the global stability of the closed-loop system by considering a simple control structure, i.e. an adaptive λ-tracking controller. It is shown that for all initial temperature and under simple feasibility assumptions, the tracking error tends asymptotically to a ball of arbitrary prescribed radius λ> 0, centered at the given temperature profile.

2D Free-surface hydraulic modeling tools are commonly used to assess flood hazard for production of maximal water depth (hmax) maps, as support for flood risk assessment. High Resolution (HR) topographic data are big data getting commonly available and used by hydraulic modeling community. Topographical information and its strategy of inclusion in models, are inputs of great importance for overland flow hmax calculation. To strengthen the assessment of confidence level in these deterministic hydraulic models outputs, uncertainty analysis (UA) and global sensitivity analysis (SA) can provide useful information that is required by practitioners and decision makers. UA and SA approaches allow to identify effective strategies to reduce the uncertainty of a model output. In this paper, developed approach consists in parameterizing three factors which introduce uncertainty related to HR topographic data use with hydraulic models: the measurement error (var. E), the level of details of above ground element representation in DEM (buildings, sidewalks, etc.) (var. S), and the spatial discretization resolution (grid cell size of a regular mesh) (var. R). Parameter var. E follows a probability density function whereas parameter var. S and var. R are discrete operator choices. The coupling of an environment for parametric computation (Prométhée) and a code relying on 2D shallow water equation (FullSWOF2D), Promethée-FullSWOF2D (P-FS) tool has been set up. P-FS tool allows launching directly numerous set of computation using R software. 1200 simulations of a river flood event scenario were performed on the regular computational mesh, spatially discretizing a 17.5 km2 urban area (Nice, France). The aim is to produce UA over points of interests and SA through Sobol index maps production.

In many engineering applications, the investigation of the vibro-acoustic response of structures is of great interest. Hence, great effort has been dedicated to improve methods in this field in the last twenty years. Classical techniques have the main drawback that they become unaffordable when high frequency impact waves are considered. In that sense, the Energy Finite Element Analysis (EFEA) is a good alternative to those methods. Based on an approximate model, EFEA gives time and locally space-averaged energy densities and has been proven to yield accurate results. However, when dealing with structural-acoustic design, it is necessary to obtain the energy density varying a large number of parameters. It is computationally unaffordable and too expensive to compute such solutions for each set of parameters. To prevent this drawback, we introduce a reduced order model which allows to drastically decrease those computational costs, while yielding a reliable and accurate approximation. In this paper, we present an approximation of the EFEA solution considering the Reduced Basis (RB) method. The RB method has already been applied successfully to many different problems. A complete development of this procedure in the context of EFEA is introduced here. Numerical tests and examples are provided for both geometrical and physical parameters.

We consider compartmental models in epidemiology. For the study of the divergence of the stochastic model from its corresponding deterministic limit (i.e., the solution of an ODE) for long time horizon, a large deviations principle suggests a thorough numerical analysis of the two models. The aim of this paper is to present three such motivated numerical works. We first compute the solution of the ODE model by means of a non-standard finite difference scheme. Next we solve a constraint optimization problem via discrete-time dynamic programming: this enables us to compute the leading term in the large deviations principle of the time of extinction of a given disease. Finally, we apply the {\tau}-leaping algorithm to the stochastic model in order to simulate its solution efficiently. We illustrate these numerical methods by applying them to two examples., Comment: Conference Proceedings

This paper proposes different levels of modeling around the notion of resistance: from the single parameter that is commonly measured in medical practice, to more sophisticated settings that account for the geometrical characteristics of the respiratory tract.

Generation of chaos is of the highest interest for many kind of applications as secure transmissions, image processing or telecommunications. In order to obtain chaotic signals that can be used in applications, simple circuits including switches have been considered those last years. Their interest is of two types: first, they belong to the class of hybrid systems that have recently attracted a great interest, secondly, they permit to obtain chaos in a very easy way. Such circuits are very easy to implement and robust chaos can be obtained, depending upon parameter values. For this aim, it is necessary to study and understand the bifurcation structures of the circuit model. In this paper, we propose two kinds of chaos generators obtained from simple RC circuits including switches managed using a clock (impulse waveform) and the charging/discharging of the capacitors. Our models are given via one-dimensional (1D) or two-dimensional (2D) nonlinear maps. We present the bifurcation studies permitting to understand the evolution of the behaviour of our systems and the appearance of chaos.

In this paper, we introduce the notion of dimensioned numbers and extended dimensioned numbers. The notion of dimensioned numbers originates in geometric measurements, and it can be used to describe iterated exponential functions of a single variable. We apply the theory of dimensioned numbers to solving some functional equations of a single variable.

In this paper, the author addresses the issue of designing a theoretically well-motivated and computationally efficient method ensuring topology preservation on image-registration-related deformation fields. The model is motivated by a mathematical characterization of topology preservation for a deformation field mapping two subsets of Z2, namely, positivity of the four approximations to the Jacobian determinant of the deformation on a square patch. The first step of the proposed algorithm thus consists in correcting the gradient vector field of the deformation at the discrete level in order to fulfill this positivity condition. Once this step is achieved, it thus remains to reconstruct the deformation field, given its full set of discrete gradient vectors. The author propose to decompose the reconstruction problem into independent problems of smaller dimensions, yielding a natural parallelization of the computations and enabling us to reduce drastically the computational time (up to 80 in some applications). For each subdomain, a functional minimization problem under Lagrange interpolation constraints is introduced and its well-posedness is studied: existence/uniqueness of the solution, characterization of the solution, convergence of the method when the number of data increases to infinity, discretization with the Finite Element Method and discussion on the properties of the matrix involved in the linear system. Numerical simulations based on OpenMP parallelization and MKL multi-threading demonstrating the ability of the model to handle large deformations (contrary to classical methods) and the interest of having decomposed the problem into smaller ones are provided.

In this paper, we are interested in scalable Domain Decomposition Methods (DDM). To this end, we introduce and study a new Coarse Space Correction algorithm for Optimized Schwarz Methods(OSM): the DCS-DGLC algorithm. The main idea is to use a Discontinuous Galerkin like formulation to compute a discontinuous coarse space correction. While the local spaces remain continuous, the coarse space should be discontinuous to compensate the discontinuities introduced by the OSM at the interface between neighboring subdomains. The discontinuous coarse correction algorithm can be used not only with OSM but also with any one-level DDM that produce discontinuous iterates. While ideas from Discontinuous Galerkin(DG) are used in the computation of the coarse correction, the final aim of the DCS-DGLC algorithm is to compute in parallel the discrete solution to the classical non-DG finite element problem.

This paper describes a method for analyzing the structure and documenting a fairly large chunk of source code for image analysis with a literate programming tool called Lepton. We propose a step-by-step approach to deconstructing the source code and indicate where Lepton’s specific features are most useful.

International audience; In this paper we present three-dimensional numerical simulations of electromagnetic waves. The Maxwell equations are solved by the Discontinuous Galerkin (DG) method. For achieving high performance, we exploit two levels of parallelism. The coarse grain parallelism is managed through MPI and a classical domain decomposition. The fine grain parallelism is managed with OpenCL in order to optimize the local computations on multicore processors or GPU's. We present several numerical experiments and performance comparisons.; Dans cet article, nous présentons des simulations numériques tridimensionnelles d’ondes électromagnétiques. Les équations de Maxwell sont résolues par la méthode de Galerkin Discontinue (GD). Pour accélérer les calculs, nous exploitons deux niveaux de parallélisme. Le large grain est basé sur MPI. Le parallélisme à grain fin repose sur OpenCL afin d’exploiter les processeurs massivement multicoeur (GPU ou CPU) récents. Nous présentons plusieurs expériences numériques et des tests de performance.

This paper is a short account on phase-field approximations of the Willmore functional and the associated L2-flows.

The modeling of swimming at the micro scale, either for biological understanding of microorganisms’ motility, e.g. bacteria, or for the conceptual building of small mechanisms capable of underwater displacement, involves low Reynolds number flows. This paper is a short review of recent links unveiled by the author and his collaborators between this problem and control theory.

International audience; Numerical simulations of the magnetohydrodynamics (MHD) equations have played a significant role in plasma research over the years. The need of obtaining physical and stable solutions to these equations has led to the development of several schemes, all requiring to satisfy and preserve the divergence constraint of the magnetic field numerically. In this paper, we aim to show the importance of maintaining this constraint numerically. We investigate in particular the hyperbolic divergence cleaning technique applied to the ideal MHD equations on a collocated grid and compare it to the constrained transport technique that uses a staggered grid to maintain the property. The methods are implemented in the software HERACLES and several numerical tests are presented, where the robustness and accuracy of the different schemes can be directly compared.; Au fil des ans, la simulation numérique des équations de la magnétohydrodynamique (MHD) a joué un rôle important dans la recherche en physique des plasmas. La nécessité de trouver des solutions physiques et stables à ces équations a conduit à l'élaboration de plusieurs schémas numériques, tous devant satisfaire et préserver la contrainte de divergence nulle pour le champ magnétique numérique. Dans cet article, nous tenterons de montrer l'importance de maintenir cette contrainte numériquement. En particulier, nous étudions la technique de nettoyage hyperbolique de la divergence appliquée aux équations de la MHD idéale discrétisées sur une grille colocalisée et nous la comparons à la technique du transport constraint qui utilise une grille décalée pour maintenir cette propriété. Les méthodes sont implémentées dans le code HERACLES et des tests numériques sont présentés. Il est ainsi possible de comparer directement la robustesse et la précision des méthodes.

Gyrokinetic simulations lead to huge computational needs. Up to now, the semi- Lagrangian code Gysela performed large simulations using a few thousands cores (8k cores typically). Simulation with finer resolutions and with kinetic electrons are expected to increase those needs by a huge factor, providing a good example of applications requiring Exascale machines. This paper presents our work to improve Gysela in order to target an architecture that presents one possible way towards Exascale: the Blue Gene/Q. After analyzing the limitations of the code on this architecture, we have implemented three kinds of improvement: computational performance improvements, memory consumption improvements and disk i/o improvements. As a result, we show that the code now scales beyond 32k cores with much improved performances. This will make it possible to target the most powerful machines available and thus handle much larger physical cases.

This paper deals with the representation of the trace of iterative Schwarz solutions at the interfaces of domain decomposition to approximate adaptively the interface error operator. This allows to build a cost-effectively accelerating of the convergence of the iterative method by extending to the vectorial case the Aitken’s accelerating convergence technique. The first representation is based on the building of a nonuniform discrete Fourier transform defined on a non-regular grid. We show how to construct a Fourier basis of dimension N+1 on this grid by building numerically a sesquilinear form, its exact accuracy to represent trigonometric polynomials of degree N / 2, and its spectral approximation property that depends on the continuity of the function to approximate. The decay of Fourier-like modes of the approximation of the trace of the iterative solution at the interfaces provides an estimate to adaptively select the modes involved in the acceleration. The drawback of this approach is to be dependent on the continuity of the trace of the iterated solution at the interfaces. The second representation, purely algebraic, uses a singular value decomposition of the trace of the iterative solution at the interfaces to provide a set of orthogonal singular vectors of which the associated singular values provide an estimate to adapt the acceleration. The resulting Aitken-Schwarz methodology is then applied to large scale computing on 3D linear Darcy flow where the permeability follows a log normal random distribution. Cet acte traite de la représentation des solutions itérées aux interfaces de la méthode de décomposition de domaine de type Schwarz afin d’approximer de manière adaptative son opérateur d’erreur aux interfaces des sous domaines. Ceci permet de construire de manière économique l’accélération de la convergence de la méthode itérative en étendant la technique d’accélération de la convergence de Aitken au cas vectoriel. La première représentation est fondée sur la construction d’une transformée de Fourier discrète non uniforme définie sur un maillage non régulier. Nous montrons comment construire une base de Fourier de dimension N+1 sur ce maillage à partir de la construction numérique d’une forme sesquilinéaire, son exactitude pour les polynômes trigonométriques de degré N/2, et numériquement sa capacité d’approximation spectrale dépendante de la continuité de la fonction à approximer. La décroissance des modes de Fourier de la trace de la solution itérée aux interfaces nous fournis une estimation pour sélectionner de manière adaptive les modes intervenant dans l’accélération. Le défaut de cette approche est d’être dépendant de la continuité de la solution itérée aux interfaces. La deuxième représentation, purement algébrique, utilise une décomposition en valeurs singulières des solutions itérées aux interfaces pour fournir un ensemble de vecteurs singuliers orthogonaux dont les valeurs singulières associées fournissent une estimation pour adapter l’accélération. La méthode Aitken-Schwarz résultante est ensuite appliquée au calcul à grande échelle pour résoudre un écoulement 3D en milieu poreux modélisé par l’équation de Darcy linéaire où la perméabilité suit une distribution aléatoire log-normale.

Models of hydrogeology must deal with both heterogeneity and lack of data. We consider in this paper a flow and transport model for an inert solute. The conductivity is a random field following a stationary log normal distribution with an exponential or Gaussian covariance function, with a very small correlation length. The quantities of interest studied here are the expectation of the spatial mean velocity, the equivalent permeability and the macro spreading. In particular, the asymptotic behavior of the plume is characterized, leading to large simulation times, consequently to large physical domains. Uncertainty is dealt with a classical Monte Carlo method, which turns out to be very efficient, thanks to the ergodicity of the conductivity field and to the very large domain. These large scale simulations are achieved by means of high performance computing algorithms and tools.

In this paper, we consider a more general model describing the dynamics of Hematopoietic Stem Cells (HSC) model with one delay. Its dynamics are studied in terms of local stability and Hopf bifurcation. We prove the existence of the possible steady state and their stability with respect to the time delay and without delay. We show that a sequence of Hopf bifurcations occur at the positive steady state as the delay crosses some critical values. We illustrate our results by some numerical simulations.

CEMRACS'11: Multiscale Coupling of Complex Models in Scientific Computing; International audience; This paper presents an overview of a unified framework for finite element and spectral element methods in 1D, 2D and 3D in \cpp called \Feel. The article is divided in two parts. The first part provides a digression through the design of the library as well as the main abstractions handled by it, namely, meshes, function spaces, operators, linear and bilinear forms and an embedded variational language. In every case, the closeness between the language developed in \Feel and the equivalent mathematical objects is highlighted. In the second part, examples using the mortar, Schwartz (non)overlapping, three fields and two ficticious domain-like methods (the Fat Boundary Method and the Penalty Method) are presented and numerically solved in the scope of the library.

We propose to formally derive a low Mach number model adapted to the modeling of a water nuclear core (e.g. of PWR- or BWR-type) in the forced convection regime or in the natural convection regime by filtering out the acoustic waves in the compressible Navier-Stokes system. Then, we propose a monodimensional stationary analytical solution with regular and singular charge loss when the equation of state is a stiffened gas equation. Moreover, we show that this solution may not be admissible from a physical or a mathematical point of view for a particular choice of the mass flux and we study the consistency between this solution and the solution obtained from a Boussinesq approximation. Let us underline that the modeling of the nuclear core is simplified in this paper. For example, the flow is a single-phase flow and we do not model neither the porosity nor the turbulence. Nevertheless, it will be possible to enrich the modeling in the future. On se propose de formellement dériver un modèle bas Mach adapté à la modélisation d’un cœur de réacteur nucléaire à eau (par exemple de type REP ou REB) en régime de convection forcée ou en régime de convection naturelle en filtrant les ondes acoustiques dans un modèle de type Navier-Stokes compressible. On construit ensuite une solution analytique stationnaire monodimensionnelle avec perte de charge régulière et singulière dans le cas où l’équation d’état est de type gaz raidi. Puis, on montre que cette solution peut ne pas être physiquement ou mathématiquement admissible pour un choix particulier du flux de masse et on étudie la cohérence entre cette solution et la solution obtenue à partir d’une approximation de Boussinesq. Soulignons que la modélisation proposée du cœur nucléaire est ici simplifiée. Par exemple, l’écoulement est monophasique et on ne modélise ni la porosité, ni la turbulence. Il sera par contre tout à fait possible d’enrichir la modélisation par la suite.

The aim of this paper is to present and solve a mathematical model of a two-aircraft optimal control problem reducing the noise on the ground during the approach. The mathematical modelization of this problem is a non-convex optimal control governed by ordinary non-linear dif- ferential equations. To solve this problem, A direct method and a Runge-Kutta RK4 discretization schema are used. This discretization schema is chosed because it is a sufficiently high order and it does-not require computation of the partial derivatives of the aircraft dynamic. The Nonlinear Interior point Trust Region Optimization solver KNITRO is applied. A large set of numerical experiments is presented. The obtained results give feasible trajectories with a significant noise reduction.

Forewords The European Conference on Iteration Theory, ECIT 2010, has been held on September 2010, 12-17, in Midi-Pyrenees, France. It was organized at “Le Roc nantais”, in Nant, Aveyron, by the “Institut National des Sciences Appliquees de Toulouse” (INSA) with the help of the Conseil Regional of Midi-Pyrenees and the Laboratory for Analysis and Architecture of Systems (LAAS). ECIT 2010 followed the sequence of meetings started in Toulouse 1973 and continued in Graz, Austria, 1977, Marburg, Germany, 1980, Toulouse, France, 1982, Lochau, Austria, 1984, Caldes de Malavella, Spain, 1987, Batschuns, Austria, 1989, Lisboa, Portugal, 1991, Batschuns, Austria, 1992, Opava, Czech Republic, 1994, Urbino, Italy, 1996, Muszyna, Poland, 1998, La Manga del Mar Menor, Spain, 2000, Evora, Portugal, 2002, Batschuns, Austria, 2004, Gargnano, Italy, 2006 and Yalta, Ukraine, 2008. The main topic of ECIT is iteration theory. This theory is a branch of mathematics and includes topics related to discrete dynamical systems and functional equations. The aim of these theoretical studies is to model, analyze and understand iterative process, for example, systems with discrete time. Such systems can be obtained from many different fields of application: engineering (control theory, electronics,...), physics, mechanics, economics, biology, ecology, ... All these questions have given rise to the introduction and the development of several branches of mathematics, related with the iteration theory, as, for instance:  Theory of singularities  Bifurcation theory  Dynamical systems with small perturbations  Topological dynamics  Ergodic theory  Symbolic dynamics  Embedding problems  Iteration theory, in particular also iteration groups and semi-groups  Iterative and linear functional equations In this ECIT 2010 conference, there were 73 participants, issued from many different countries including Algeria, Austria, China, Czech Republic, Germany, India, Israel, Italy, Poland, Portugal, Russia, Spain, Tunisia and Ukraine. 57 talks have been given and there were many very fruitful exchanges and discussions during the special sessions “Problems and remarks” and during the whole conference. The present issue of ESAIM Proceedings gathers 18 contributions by participants to the ECIT 2010 conference. These papers are related to the different thematics, which have been proposed during the conference, among them, symbolic dynamics, dynamical systems, functional equations, bifurcation theory,... We thank all authors for their contribution to this issue of ESAIM Proceedings.