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We consider the Poisson problem in a domain with small holes, as a template for developing efficient and accurate numerical approximation schemes for partial differential equations defined on domains with low-dimensional inclusions, such as embedded fibers. We propose a reduced model based on the projection of Dirichlet boundary constraints on a finite dimensional approximation space, obtaining in this way a Poisson problem with defective interface conditions. We analyze the existence of the solution of the reduced problem and for arbitrarily small holes we prove its convergence towards the original problem, the rate of which depends on the size of the inclusion and on the number of modes of the finite dimensional space. The numerical discretization of the reduced problem is addressed by the finite element method, using a computational mesh that does not fit to the holes in the framework of a fictitious domain approach. We propose a stable, optimally convergent and robust formulation with respect to the hole size that exploits an augmented Galerkin formulation based on the addition of suitable non-polynomial functions to the finite element approximation space. The properties of the discretization method are supported by numerical experiments.

We develop a fictitious domain method to approximate a Dirichlet problem on a domain with small circular holes (simply called a perforated domain). To address the case of many small inclusions or exclusions, we propose a reduced model based on the projection of the homogeneous Dirichlet boundary constraint on a finite dimensional approximation space. We analyze the existence of the solution of this reduced problem and prove its convergence towards the limit problem without holes. We next obtain an estimate of the gap between the solution of the reduced model and the solution of the full initial model with small holes, the convergence rate depending on the size of the inclusion and on the number of modes of the finite dimensional space. The numerical discretization of the reduced problem is addressed by the finite element method, using a computational mesh that does not fit to the holes. The approximation properties of the finite element method are analyzed by a-priori estimates and confirmed by numerical experiments. elliptic differential equations, small inclusions, asymptotic analysis, approximated numerical method.

National audience; We study the problem of inferring the conditional independence structure between the entries of a Gaussian random vector. Our focus is on finding groups of independent variables. This can be translated into the estimation of a precision matrix (inverse of the covariance matrix) with a block-diagonal structure. We borrow ideas from spectral graph theory and spectral clustering and propose a novel prior called Fiedler prior showing shrinkage properties towards block-diagonal precision matrices. We compare the shrinkage induced by our prior and the popular Graphical Lasso prior, and compare their performance on a simulated dataset.; Nousétudions le problème de l'inférence de la structure d'indépendance conditionnelle entre les entrées d'un vecteur aléatoire gaussien, principalement dans le but d'obtenir des groupes de variables indépendantes. Cela peut se traduire par l'estimation d'une matrice de précision (inverse de la matrice de covariance) avec une structure blocdiagonale. Cette approche se base sur des techniques de théorie spectrale des graphes et de clustering spectral. Nous proposons une nouvelle loi a priori, le prior de Fiedler, qui satisfait une propriété de shrinkage vers les matrices de précisionà structure blocdiagonale. Nous comparons le shrinkage induit par ce prior de Fiedler et par le Graphical Lasso, et comparons leurs performances sur un ensemble de données simulées.

International audience; In Computer Vision, images of dynamic or segmented scenes are modeled as linear projections from P k to P 2. The reconstruction problem consists in recovering the position of the projected objects and the projections themselves from their images, after identifying many enough correspondences between the images. A critical locus for the reconstruction problem is a variety in P k containing the objects for which the reconstruction fails. In this paper, we deal with projections both of points from P 4 → P 2 and of lines from P 3 → P 2. In both cases, we consider 3 projections, minimal number for a uniquely determined reconstruction. In the case of projections of points, we declinate the Grassmann tensors introduced in [21] in our context, and we use them to compute the equations of the critical locus. Then, given the ideal that defines this locus, we prove that, in the general case, it defines a Bordiga surface, or a scheme in the same irreducible component of the associated Hilbert scheme. Furthermore, we prove that every Bordiga surface is actually the critical locus for the reconstruction for suitable projections. In the case of projections of lines, we compute the defining ideal of the critical locus, that is the union of 3 α–planes and a line congruence of bi–degree (3, 6) and sectional genus 5 in the Grassmannian G(1, 3) ⊂ P 5. This last surface is biregular to a Bordiga surface [40]. We use this fact to link the two reconstruction problems by showing how to compute the projections of one of the two settings, given the projections of the other one. The link is effective, in the sense that we describe an algorithm to compute the projection matrices.

Modeling species distributions over space and time is one of the major research topics in both ecology and conservation biology. Joint Species Distribution models (JSDMs) have recently been introduced as a tool to better model community data, by inferring a residual covariance matrix between species, after accounting for species' response to the environment. However, these models are computationally demanding, even when latent factors, a common tool for dimension reduction, are used. To address this issue, Taylor-Rodriguez et al. (2017) proposed to use a Dirichlet process, a Bayesian nonparametric prior, to further reduce model dimension by clustering species in the residual covariance matrix. Here, we built on this approach to include a prior knowledge on the potential number of clusters, and instead used a Pitman–Yor process to address some critical limitations of the Dirichlet process. We therefore propose a framework that includes prior knowledge in the residual covariance matrix, providing a tool to analyze clusters of species that share the same residual associations with respect to other species. We applied our methodology to a case study of plant communities in a protected area of the French Alps (the Bauges Regional Park), and demonstrated that our extensions improve dimension reduction and reveal additional information from the residual covariance matrix, notably showing how the estimated clusters are compatible with plant traits, endorsing their importance in shaping communities.

Most climate change mitigation scenarios rely on increased use of bioenergy to decarbonize the energy system. Here we use results from the 33rd Energy Modeling Forum study (EMF-33) to investigate projected international bioenergy trade for different integrated assessment models across several climate change mitigation scenarios. Results show that in scenarios with no climate policy, international bioenergy trade is likely to increase over time, and becomes even more important when climate targets are set. More stringent climate targets, however, do not necessarily imply greater bioenergy trade compared to weaker targets, as final energy demand may be reduced. However, the scaling up of bioenergy trade happens sooner and at a faster rate with increasing climate target stringency. Across models, for a scenario likely to achieve a 2 °C target, 10–45 EJ/year out of a total global bioenergy consumption of 72–214 EJ/year are expected to be traded across nine world regions by 2050. While this projection is greater than the present trade volumes of coal or natural gas, it remains below the present trade of crude oil. This growth in bioenergy trade largely replaces the trade in fossil fuels (especially oil) which is projected to decrease significantly over the twenty-first century. As climate change mitigation scenarios often show diversified energy systems, in which numerous world regions can act as bioenergy suppliers, the projections do not necessarily lead to energy security concerns. Nonetheless, rapid growth in the trade of bioenergy is projected in strict climate mitigation scenarios, raising questions about infrastructure, logistics, financing options, and global standards for bioenergy production and trade.

International audience; Flow in fractured porous media occurs in the earth's subsurface, in biological tissues, and in man-made materials. Fractures have a dominating influence on flow processes, and the last decade has seen an extensive development of models and numerical methods that explicitly account for their presence. To support these developments, we present a portfolio of four benchmark cases for single-phase flow in three-dimensional fractured porous media. The cases are specifically designed to test the methods' capabilities in handling various complexities common to the geometrical structures of fracture networks. Based on an open call for participation, results obtained with 17 numerical methods were collected. This paper presents the underlying mathematical model, an overview of the features of the participating numerical methods, and their performance in solving the benchmark cases.

In this work, we introduce a novel abstract framework for the stability and convergence analysis of fully coupled discretisations of the poroelasticity problem and apply it to the analysis of Hybrid High-Order (HHO) schemes. A relevant feature of the proposed framework is that it rests on mild time regularity assumptions that can be derived from an appropriate weak formulation of the continuous problem. To the best of our knowledge, these regularity results for the Biot problem are new. A novel family of HHO discretisation schemes is also proposed and analysed, and their performance numerically evaluated.

International audience; We develop a novel Hybrid High-Order method for the simulation of Darcy flows in fractured porous media. The discretization hinges on a mixed formulation in the bulk region and on a primal formulation inside the fracture. Salient features of the method include a seamless treatment of nonconforming discretizations of the fracture, as well as the support of arbitrary approximation orders on fairly general meshes. For the version of the method corresponding to a polynomial degree $k\ge 0$, we prove convergence in $h^{k+1}$ of the discretization error measured in an energy-like norm. In the error estimate, we explicitly track the dependence of the constants on the problem data, showing that the method is fully robust with respect to the heterogeneity of the permeability coefficients, and it exhibits only a mild dependence on the square root of the local anisotropy of the bulk permeability. The numerical validation on a comprehensive set of test cases confirms the theoretical results.

International audience; We use the formalism of the Rényi entropies to establish the symmetry range of extremal functions in a family of subcriti-cal Caffarelli-Kohn-Nirenberg inequalities. By extremal functions we mean functions which realize the equality case in the inequalities, written with optimal constants. The method extends recent results on critical Caffarelli-Kohn-Nirenberg inequalities. Using heuristics given by a nonlinear diffusion equation, we give a variational proof of a symmetry result, by establishing a rigidity theorem: in the symmetry region, all positive critical points have radial symmetry and are therefore equal to the unique positive, radial critical point, up to scalings and multiplications. This result is sharp. The condition on the parameters is indeed complementary of the condition which determines the region in which symmetry breaking holds as a consequence of the linear instability of radial optimal functions. Compared to the critical case, the subcritical range requires new tools. The Fisher information has to be replaced by Rényi entropy powers, and since some invariances are lost, the estimates based on the Emden-Fowler transformation have to be modified.

In this work we construct a low-order nonconforming approximation method for linear elasticity problems supporting general meshes and valid in two and three space dimensions. The method is obtained by hacking the Hybrid High-Order method, that requires the use of polynomials of degree $k\ge1$ for stability. Specifically, we show that coercivity can be recovered for $k=0$ by introducing a novel term that penalises the jumps of the displacement reconstruction across mesh faces. This term plays a key role in the fulfillment of a discrete Korn inequality on broken polynomial spaces, for which a novel proof valid for general polyhedral meshes is provided. Locking-free error estimates are derived for both the energy- and the $L^2$-norms of the error, that are shown to convergence, for smooth solutions, as $h$ and $h^2$, respectively (here, $h$ denotes the meshsize). A thorough numerical validation on a complete panel of two- and three-dimensional test cases is provided., Comment: 26 pages, 6 tables, and 4 Figures

34 pages; International audience; In this work, we propose a model for the passive transport of a solute in a fractured porous medium, for which we develop a Hybrid High-Order (HHO) space discretization. We consider, for the sake of simplicity, the case where the flow problem is fully decoupled from the transport problem. The novel transmission conditions in our model mimic at the discrete level the property that the advection terms do not contribute to the energy balance. This choice enables us to handle the case where the concentration of the solute jumps across the fracture. The HHO discretization hinges on a mixed formulation in the bulk region and on a primal formulation inside the fracture for the flow problem, and on a primal formulation both in the bulk region and inside the fracture for the transport problem. Relevant features of the method include the treatment of nonconforming discretizations of the fracture, as well as the support of arbitrary approximation orders on fairly general meshes.

Cet article est distribué selon les termes de la licence internationale Creative Commons Attribution 4.0 (http://creativecommons.org/licenses/by/4.0/); International audience; We study the problem of identifying relevant genes in a co-expression network using a (cooperative) game theoretic approach. The Shapley value of a cooperative game is used to asses the relevance of each gene in interaction with the others, and to stress the role of nodes in the periphery of a co-expression network for the regulation of complex biological pathways of interest. An application of the method to the analysis of gene expression data from microarrays is presented, as well as a comparison with classical centrality indices. Finally, making further assumptions about the a priori importance of genes, we combine the game theoretic model with other techniques from cluster analysis.

A transferable utility (TU) game with n players specifies a vector of \(2^n-1\) real numbers, i.e. a number for each non-empty coalition, and this can be difficult to handle for large n. Therefore, several models from the literature focus on interaction situations which are characterized by a compact representation of a TU-game, and such that the worth of each coalition can be easily computed. Sometimes, the worth of each coalition is computed from the values of single players by means of a mechanism describing how the individual abilities interact within groups of players. In this paper we introduce the class of Generalized additive games (GAGs), where the worth of a coalition \(S { \subseteq } N\) is evaluated by means of an interaction filter, that is a map \(\mathcal {M}\) which returns the valuable players involved in the cooperation among players in S. Moreover, we investigate the subclass of basic GAGs, where the filter \(\mathcal {M}\) selects, for each coalition S, those players that have friends but not enemies in S. We show that well-known classes of TU-games can be represented in terms of such basic GAGs, and we investigate the problem of computing the core and the semivalues for specific families of GAGs.

Studies in Computational Intelligence book series (SCI, volume 689); Centrality measures are used in network analysis to identify the relevant elements in a network. Recently, several centrality measures based on coalitional game theory have been successfully applied to different kinds of biological networks, such as brain networks, gene networks, and metabolic networks. We propose an approach, using coalitional games, to the problem of identifying relevant genes in a biological network. Our model generalizes the notion of degree centrality, whose correlation with the relevance of genes for different biological functions is supported by several practical evidences in the literature. The new relevance index we propose is characterized by a set of axioms defined on gene networks and a formula for its computation is provided. Furthermore, an application to the analysis of a large co-expression network is shortly presented.

International audience; This paper is the second part of the study. In Part~I, self-similar solutions of a weighted fast diffusion equation (WFD) were related to optimal functions in a family of subcritical Caffarelli-Kohn-Nirenberg inequalities (CKN) applied to radially symmetric functions. For these inequalities, the linear instability (symmetry breaking) of the optimal radial solutions relies on the spectral properties of the linearized evolution operator. Symmetry breaking in (CKN) was also related to large-time asymptotics of (WFD), at formal level. A first purpose of Part~II is to give a rigorous justification of this point, that is, to determine the asymptotic rates of convergence of the solutions to (WFD) in the symmetry range of (CKN) as well as in the symmetry breaking range, and even in regimes beyond the supercritical exponent in (CKN). Global rates of convergence with respect to a free energy (or entropy) functional are also investigated, as well as uniform convergence to self-similar solutions in the strong sense of the relative error. Differences with large-time asymptotics of fast diffusion equations without weights will be emphasized.

In this paper we consider a proper generalized decomposition method to solve the steady incompressible Navier--Stokes equations with random Reynolds number and forcing term. The aim of such a technique is to compute a low-cost reduced basis approximation of the full stochastic Galerkin solution of the problem at hand. A particular algorithm, inspired by the Arnoldi method for solving eigenproblems, is proposed for an efficient greedy construction of a deterministic reduced basis approximation. This algorithm decouples the computation of the deterministic and stochastic components of the solution, thus allowing reuse of preexisting deterministic Navier--Stokes solvers. It has the remarkable property of only requiring the solution of $m$ uncoupled deterministic problems for the construction of an $m$-dimensional reduced basis rather than $M$ coupled problems of the full stochastic Galerkin approximation space, with $m \ll M$ (up to one order of magnitude for the problem at hand in this work).

International audience; We study optimal functions in a family of Caffarelli-Kohn-Nirenberg inequalities with a power-law weight, in a regime for which standard symmetrization techniques fail. We establish the existence of optimal functions, study their properties and prove that they are radial when the power in the weight is small enough. Radial symmetry up to translations is true for the limiting case where the weight vanishes, a case which corresponds to a well-known subfamily of Gagliardo-Nirenberg inequalities. Our approach is based on a concentration-compactness analysis and on a perturbation method which uses a spectral gap inequality. As a consequence, we prove that optimal functions are explicit and given by Barenblatt-type profiles in the perturbative regime.

International audience; The purpose of this work is to use a variational method to identify some of the parameters of one-dimensional models for blood flow in arteries. These parameters can be fit to approach as much as possible some data coming from experimental measurements or from numerical simulations performed using more complex models. A nonlinear least squares approach to parameter estimation was taken, based on the optimization of a cost function. The resolution of such an optimization problem generally requires the efficient and accurate computation of the gradient of the cost function with respect to the parameters. This gradient is computed analytically when the one-dimensional hyperbolic model is discretized with a second order Taylor-Galerkin scheme. An adjoint approach was used. Some preliminary numerical tests are shown. In these simulations, we mainly focused on determining a parameter that is linked to the mechanical properties of the arterial walls, the compliance. The synthetic data we used to estimate the parameter were obtained from a numerical computation performed with a more accurate model: a three-dimensional fluid-structure interaction model. The first results seem to be promising. In particular, it is worth noticing that the estimated compliance which gives the best fit is quite different from the values that are commonly used in practice.; Le but de ce travail est d'identifier certains des paramètres existant dans des modèles 1-d d'écoulement sanguin dans des artères. Ces paramètres peuvent permettre d'approcher autant que possible des configurations géométriques réalistes ou des données expérimentales. Une approche de l'estimation de paramètres par moindres carrés non-linéaires a été adoptée, basée sur l'optimisation d'une certaine fonction coût. La résolution d'un tel problème de minimisation requiert le calcul efficace et précis du gradient de la fonction coût par rapport aux paramètres. Le gradient est discrétisé analytiquement dans le cas d'une discrétisation du modèle hyperbolique 1-d par le schema de Taylor-Galerkin. Une approche par l'état adjoint a été employée. Des premiers résultats numériques sont fournis. Pour ces simulations, nous nous sommes concentrés sur la détermination d'un paramètre lié aux propriétés mécaniques de la paroi artérielle. Les données synthétiques utilisées pour l'estimation de ce paramètre ont été obtenues à partir d'un modèle beaucoup plus raffiné : un modèle 3-d d'interaction fluide-structure. Les résultats semblent intéressants car le paramètre estimé est assez différent de ce à quoi on s'attendrait a priori.

International audience; This paper presents the consistency and stability analyses of the Generalized-α methods applied to non-linear dynamical systems. The second-order accuracy of this class of algorithms is proved also in the non-linear regime, independently of the quadrature rule for non-linear internal forces. Conversely, the G-stability notion which is suitable for linear multistep schemes devoted to non-linear dynamic problems cannot be applied, as the non-linear structural dynamics equations are not contractive. Nonetheless, it is proved that the Generalized-α methods are endowed with stability in an energy sense and guarantee energy decay in the high-frequency range as well as asymptotic annihilation. However, overshoot and heavy energy oscillations in the intermediate-frequency range are exhibited. The results of representative numerical simulations performed on relatively simple single- and multiple-degrees-of-freedom non-linear systems are presented in order to confirm the analytical estimates.

Morphogenetic theories investigate the mechanisms of creation and regulation of definite biological forms in living organisms. The incredible diversity of shapes and sizes is generated through a barely unknown coordination of biochemical processes occurring at molecular levels. Such a crosstalk not only defines the rules of a robust scheme of matter differentiation, but it also has the capacity to adapt with respect to some variations of the environmental conditions. In this work, we propose a continuum model of growth and mass transport for biological materials during morphogenetic processes. Using the theory of configurational forces, we define the thermomechanical bases for understanding how both the mechanical and the biochemical states can orchestrate growth. The model is successfully applied to describe the morphogen-driven growth control in the imaginal wing disc of Drosophila melanogaster.

Automata, Logic and Semantics, We characterize the relations which are first-order definable in the model of the group of integers with the constant 1. This allows us to show that given a relation defined by a first-order formula in this model enriched with the usual ordering, it is recursively decidable whether or not it is first-order definable without the ordering.

International audience; In this chapter we derive the equations governing the mechanics of the fluid (the blood) and the structure (the vessel wall). The derivation will be made in a rigorous way, yet trying to provide the reader a physical understanding. Reason of space obliged us to focus on the most important models for haemodynamic computations and to omit several details and the proofs of the propositions. For instance, all issues related to energy conservation principles have been ignored and we have eventually considered only incompressible fluids.

In this paper we establish a link between the sigma transformation approach and the arbitrary Lagrangian–Eulerian (ALE) approach. For that purpose we introduce the ALE-sigma (ALES) approach, which consists in an ALE interpretation of the sigma transformation. Taking advantage of this new approach, we propose a general ALES transformation, allowing for a great adaptability of the vertical discretization and therefore overcoming some drawbacks of the classical sigma transformation. Numerical results are presented, showing the advantages of this general coordinate system, as, for example, a better representation of horizontal stratifications. Copyright © 2008 John Wiley & Sons, Ltd.

28 Janvier 2005; International audience; We establish the existence and uniqueness of weak solutions to a suitable class of non-degenerate FBSDEs with a one-dimensional backward component. The classical Lipschitz framework is weakened: the diffusion matrix and the final condition are space Holder continuous whereas the drift and the backward driver may be discontinuous in x. The growth of the backward driver is also allowed to be at most quadratic with respect to the gradient term.

Cotutelle de thèse franco italienne entre Univ. Lille 1 et Politecnico de Milan Directeurs de Thèse : Paolo Dai Pra et Sylvie Roelly Président du jury et rapporteur : G.-B. Giacomin Rapporteurs : R. Fernandez, F. Martinelli, E. Olivieri Examinateurs : M. Fradon, G. Posta; Used in many scientific areas, the discrete time random dynamics called Probabilistic Cellular Automata (P.C.A.) are Markov stochastic processes with values in an infinite space S^G where S is a finite set and G an infinite graph. Here we assume G=Z^d. The main feature of these dynamics is the parallel, or synchronous, evolution of all the coordinates or interacting elementary components. We are first interested in the existence and uniqueness of stationary measures for non degenerated PCA dynamics, i.e. whose local behaviour is never deterministic. Based on results of Dai Pra, Kozlov, Künsch, Lebowitz, Vasilyev et al., we give for the class of reversible PCA dynamics precise relations between the sets of stationary measures, reversible measures and some Gibbs measures. For a typical parametrized family of reversible PCA dynamics, we prove the existence of a phase transition phenomenon and establish the behaviour of the different Gibbs measures under the dynamics' action. In particular, we exhibit non-stationary Gibbs measures. Secondly, we study the convergence to equilibrium for PCA dynamics which are attractive (so called ergodicity). One of our tools is a special coupling of these dynamics, preserving the stochastic order. When there is no phase transition, inspired by the ideas of Martinelli and Olivieri on Glauber dynamics, we prove the ergodicity of PCA dynamics, and more precisely, the convergence exponentially fast to the unique equilibrium state. This result truly improves previous ones existing in the litterature. Finally, we present some numerical simulations of reversible PCA dynamics, and, in particular, a parallel algorithm which converges to extremal Gibbs measures associated to the Ising model.; Utilisés dans de nombreux domaines scientifiques, les Automates Cellulaires Probabilistes, usuellement abrégés en PCA, de l'anglais "Probabilistic Cellular Automata", constituent, au sein des dynamiques aléatoires à temps discret, une classe de systèmes infinis de particules, c'est à dire de processus stochastiques markoviens à valeurs dans un espace infini S^G où S désigne un ensemble fini et G est un graphe infini. On considère ici toujours le cas où G=Z^d. La particularité de ces dynamiques est l'évolution en parallèle, ou synchrone, de chacune des coordonnées ou composants élémentaires en interaction. Nous nous intéressons dans un premier temps à l'existence et à l'unicité des mesures stationnaires pour les dynamiques PCA non dégénérées i.e. dont le comportement local n'est jamais déterministe, ainsi qu'à la caractérisation de ces états d'équilibre en tant que mesures gibbsiennes. Nous fondant sur les résultats de Dai Pra, Kozlov, Künsch, Lebowitz, Vasilyev et al., nous précisons, pour la classe des dynamiques PCA réversibles, les relations existant entre les mesures stationnaires, les mesures réversibles et les mesures de Gibbs associées à un potentiel dont le lien avec la dynamique est explicité. Pour une famille paramétrée de dynamiques PCA réversibles, nous démontrons l'existence d'un phénomène de transition de phase et explicitons dans ce cas le comportement de différentes mesures de Gibbs sous l'action de ces dynamiques. En particulier, nous exhibons des mesures de Gibbs non-stationnaires. Dans un second temps, nous étudions l'ergodicité, i.e. la convergence vers l'équilibre des dynamiques PCA qui sont de plus attractives. Nous construisons à cet effet un couplage de ces dynamiques préservant l'ordre stochastique. En nous référant aux travaux de Martinelli et Olivieri pour les dynamiques de Glauber, nous établissons qu'en l'absence de transition de phase, dès que l'unique mesure de Gibbs vérifie une condition de faible mélange, il y a ergodicité et convergence à vitesse exponentielle vers cet unique état d'équilibre, améliorant en cela grandement les critères d'ergodicité pour les PCA existant dans la littérature. Enfin, nous illustrons ces résultats par la réalisation de simulations numériques de certaines des dynamiques réversibles précédemment étudiées, et présentons un algorithme parallèle convergeant vers les mesures de Gibbs extrémales du modèle d'Ising.