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Paola Antonietti (MOX - Dipartimento di Matematica - Politecnico di Milano)

PDF Non-conforming high order approximations of the elastodynamic equation

Recent developments in computational seismology have been based on high order numerical modelling of wave propagation. High order non-conforming methods employ a spectral finite element discretisation on a non-overlapping subdomain partition of the computational domain, and the continuity of the solution at the skeleton of the decomposition is imposed in a suitable weak sense. The meshes do not have to match between neighbouring subdomains, and different spectral approximation degrees are allowed in different subdomains. In such a way, the spatial discretisation and/or the local polynomial degree can be tailored to the region of interest. These methods are particularly well suited for the simulation of complex wave phenomena, such as soil-structure interaction problems or the seismic response of sedimentary basins, where geometrical and polynomial flexibility is crucial in order to simulate correctly the wave-front field. In this talk work, starting from a displacement-based weak formulation of the elastodynamic equation, we present a family of non-conforming methods for the space discretisation combined with a second order time integration scheme. In particular we prove a priori error bounds for both the semi-discrete and fully-discrete formulations. The theoretical results are validated through numerical experiments. Geophysical applications are also presented.

Lourenco Beirao Da Veiga (Università di Milano)

PDF Mimetic discretizations of arbitrary local order and regularity (Dicrétisations mimétiques d'ordre et de régularité localement arbitraires)

The Mimetic Discretization (MD) method is an approach for the approximation of PDE problems that shares common features with the finite element and the finite difference schemes. The MD method enjoys the same variational background of finite elements, but focuses the attention on the degrees of freedom rather than the underlying basis functions. This construction allows for MD schemes with general polygonal/polyhedral meshes and, in general, it grants a richer flexibility with respect to FE methods. In the present talk, an MD method for scalar elliptic problems with arbitrary and fully local polynomial degree is investigated. Moreover, the possibility of obtaining arbitrarily regular underlying spaces is also studied.

Annalisa Buffa (IMATI Enrico Magenes - CNR Pavia)

PDF On local refinement in isogeometric analysis (Sur le raffinement local en analyse isogéométrique)

Isogeometric methodologies are designed with the aim of improving the connection between numerical simulation of physical phenomena and the Computer Aided Design systems. Indeed, the ultimate goal is to eliminate or drastically reduce the approximation of the computational domain and the re-meshing by the use of the "exact" geometry directly on the coarsest level of discretization. This is achieved by using B-Splines or Non Uniform Rational B-Splines for the geometry description as well as for the representation of the unknown fields. The use of Spline or NURBS functions, together with isoparametric concepts, results in an extremely successfully idea and paves the way to many new numerical schemes enjoying features that would be extremely hard to achieve within a standard finite element framework. One of the big challenges in the context of isogeometric analysis is the possibility to break the tensor product structure of Splines and so design adaptive methods. T-splines, introduced by T. Sederberg et al. in 2004, have been used to this aim in geometric modeling and design, but their use in isogeometric analysis requires a lot of care.

Elisabetta Chiodaroli (Institut für Mathematik der Universität Zürich)

PDF A non-uniqueness result for entropy solutions to the compressible Euler system

The deceivingly simple-looking compressible Euler equations of gas dynamics have a long history of important contributions over more than two centuries. If we allow for discontinuous solutions, uniqueness and stability are lost. In order to restore such properties, further restrictions on weak solutions have been proposed in the form of entropy inequalities. In this talk we will discuss a counterexample to the well-posedness of entropy solutions to the multi-dimensional compressible Euler equations: in our construction the entropy condition is not sufficient as a selection criteria for unique solutions. Our methods are inspired by a new analysis of the incompressible Euler equations recently carried out by De Lellis and Székelyhidi and based on a revisited "h-principle".

Jérémi Dardé (Inverse problems group, Department of Mathematics and Systems Analysis - Aalto University - Helsinki)

PDF A mixed formulation of the quasi-reversibility method to solve elliptic Cauchy problems in 2d and 3d (Une formulation mixte de la méthode de quasi-réversibilité pour résoudre des problèmes de Cauchy elliptiques 2d et 3d)

Introduced by Jacques-Louis Lions and Robert Lattes in the sixties, the quasi-reversibility (QR) is a method to solve elliptic Cauchy problems. The main idea of the method is to approach the original second-order ill-posed problem by a family of variational fourth-order well-posed problems, depending on a regularization parameter. The quasi-reversibility is a non-iterative method, suitable for a discretization by finite elements. However, the standard formulation of the quasi-reversibility is a variational problem posed in H^2. Its discretization requires finite elements adapted to this framework (finite elements of class C^1, or non-conforming finite elements), more complicated than standard C^0 finite elements. This causes many difficulties, especially in the case of three-dimensional problems. We propose a new mixed formulation of the quasi-reversibility, based on a family of variational problems posed in H^1 x H(div), which can therefore be solved using standard finite elements (typically Lagrange and Raviart-Thomas finite elements). We prove the convergence of the QR solution to the solution of the Cauchy problem, and we illustrate the functionality of the method using 2D and 3D numerical examples.

Introduite par Jacques-Louis Lions et Robert Lattès dans les années soixante, la méthode de quasi-réversibilité (QR) est une méthode de résolution des problèmes de Cauchy elliptiques. L'idée principale de la méthode est d'approcher le problème initial, problème mal-posé d'ordre 2, par une famille de problèmes variationnels bien posés d'ordre 4, dépendant d'un paramètre de régularisation. La QR est une méthode non-itérative, adaptée à une discrétisation par éléments finis. Cependant, la formulation standard de la quasi-réversibilité est un problème variationnel posé dans H^2. Sa discrétisation nécessite des éléments finis adaptés à ce cadre (éléments finis de classe C^1, ou encore éléments finis non-conformes), plus compliqués que des éléments finis classiques C^0. Cela entraîne de nombreuses difficultés, notamment dans le cas de problèmes tridimensionnels.
Nous proposons une nouvelle formulation mixte de la quasi-réversibilité, basée sur une famille de problèmes variationnels posés dans H^1 x H(div), et pouvant donc être résolus en utilisant des éléments finis conformes classiques (typiquement éléments finis de Lagrange et de Raviart-Thomas). Nous démontrons la convergence de la solution du problème QR vers le solution du problème de Cauchy, et nous illustrons la fonctionnalité de la méthode à l'aide d'exemples numériques 2d et 3d.

Anne-Claire Egloffe (Inria Rocquencourt et LJLL - Université Pierre et Marie Curie - Paris 6)

PDF Study of inverse problems arising from a multiscale modeling of the lungs (Etude de problèmes inverses issus d'une modélisation multi-échelle de l'écoulement de l'air dans les poumons)

We are interested in inverse problems arising from a multiscale modeling of the mechanical properties of the lungs. The aim of our work is to obtain stability inequalities for some parameters involved in the modeling (resistance to the air flow in the lungs, stiffness constant of the diaphragm) compared to measurements made at mouth. For example, if two volume measurements made at mouth are "close", then the parameters should be "close" too. We consider a simplified version of the air flow in the lungs described by the Stokes equations with mixed Robin/Neumann boundary conditions. We are interested in identifying a Robin coefficient defined on a non accessible part of the boundary from measurements available on another part of the boundary. Even if being simplified compared to the original problem, solving this inverse problem is a relevant question which was an open problem. This is joint work with Muriel Boulakia et Céline Grandmont.

Nous nous intéressons à des problèmes inverses provenant d'une modélisation multi-échelles des propriétés mécaniques de l'appareil respiratoire. Le but de notre travail est d'obtenir des inégalités de stabilité pour certains paramètres intervenant dans la modélisation (résistance à l'écoulement de l'air dans les poumons, constante de raideur du diaphragme) par rapport à des mesures effectuées au niveau de la bouche. Par exemple, si deux mesures de volume faites au niveau de la bouche sont "proches", alors les paramètres doivent également être "proches". Nous considérons une version simplifiée de l'écoulement de l'air dans les poumons décrit par les équations de Stokes avec des conditions aux limites mixtes de type Robin/Neumann. Nous nous intéressons à l'identification d'un coefficient de Robin défini sur une partie non accessible de la frontière à partir des mesures disponibles sur une autre partie de la frontière. Quoiqu'il s'agisse d'une simplification du problème initial, la résolution de ce problème inverse est une question pertinente et était un problème ouvert. Ce travail est en collaboration avec Muriel Boulakia et Céline Grandmont.

Marie Kray (LJLL - Université Pierre et Marie Curie - Paris 6)

PDF Time reversed absorbing condition: recreate the past and application to inverse problems (Retournement temporel avec condition aux limites absorbante : recréer le passé et application aux problèmes inverses)

We introduce time reversed absorbing conditions (TRAC) in time reversal methods. They enable one to "recreate the past" without knowing the source which has emitted the signals that are back-propagated. We present two applications in inverse problems: the reduction of the size of the computational domain and the determination, from boundary measurements, of the location and volume of an unknown inclusion. The method does not rely on any a priori knowledge of the physical properties of the inclusion. Numerical tests with the wave equation illustrate the efficiency of the method. This technique is fairly insensitive with respect to noise in the data. Joint work with Franck Assous and Frédéric Nataf.

Nous présentons une méthode de retournement temporel avec conditions aux limites absorbantes (TRAC). Cette méthode permet de "recréer le passé" sans connaissance de la source qui a émis les signaux rétro-propagés. Nous proposons deux applications aux problèmes inverses : la réduction de la taille du domaine de calcul en redéfinissant une surface de référence virtuelle sur laquelle les récepteurs semblent positionnés, et la détermination de la localisation d~une inclusion inconnue à partir de mesures au bord. La méthode TRAC ne nécessite aucune connaissance a priori des propriétés physiques de l~inclusion. Des tests numériques effectués sur l'équation des ondes illustrent l'efficacité de cette méthode, qui se révèle être très robuste vis-à-vis du bruit sur les données. Ce travail est en collaboration avec Franck Assous et Frédéric Nataf.

Matthieu Léautaud (Université Paris Sud - Orsay)

PDF Controllability of a diffusive interface problem (Contrôlabilité d'un modèle d'interface diffusive)

We consider a linear parabolic transmission problem across an interface of codimension one in a bounded domain, where the transmission conditions involve a parabolic operator on the interface. This system is an idealization of a three layer model, in which the central layer has a small thickness. We prove a Carleman estimate in the neighborhood of the interface for an associated elliptic operator by means of partial estimates in several microlocal regions. In turn, from the Carleman estimate, we obtain a spectral inequality that yields the null-controllability of the parabolic system. These results are uniform with respect to the small thickness parameter. This is a joint work with Jérôme Le Rousseau and Luc Robbiano.

Dans ce travail, en collaboration avec Jérôme Le Rousseau et Luc Robbiano, on considère un problème de transmission à travers une interface de codimension un dans un domaine borné, pour lequel la condition de transmission implique un opérateur parabolique dans l'interface. Ce système est une idéalisation d'un modèle à trois couches, dans lequel la couche centrale est de petite épaisseur. On montre une inégalité de Carleman dans un voisinage de l'interface pour un opérateur elliptique associé, grâce à des estimées partielles dans différentes régions microlocales. De cette inégalité de Carleman, on déduit une inégalité spectrale, qui implique la contrôlabilité à zéro du système parabolique. Tout les résultats obtenus sont uniformes par rapport au petit paramètre d'épaisseur.

Thomas Lepoutre (INRIA Rhône Alpes et Institut Camille Jordan - Université Lyon 1)

PDF Relaxed cross diffusion models: a general well-posedness result (Modèles de diffusion croisée régularisés : existence et unicité)

Cross diffusion models try to represent the trend of two (or more) species to avoid another. We present here a model in which the pressure is nonlocal. For this model we give a global existence theorem for continuous coefficients together with uniqueness in the Lipschitz coefficients cases. Joint work with Michel Pierre and Guillaume Rolland.

Les modèles de diffusion croisée cherchent à tenir compte du fait que des espèces cherchent à s'éviter mutuellement. Nous présentons ici un modèle de diffusion croisée non locale. Nous donnons pour ce modèle un résultat d'existence pour des coefficients continus et d'unicité pour des coefficients lipschitziens. Ce travail est le fruit d'une collaboration avec Michel Pierre et Guillaume Rolland.

Andrea Manzoni (Modelling and Scientific Computing - Ecole Polytechnique Fédérale de Lausanne)

PDF A reduced computational and geometrical framework for inverse problems in haemodynamics

The solution of inverse problems in cardiovascular mathematics is computationally expensive. Model reduction techniques might be needed to enable the solution of inverse problems also in real-life applications. In this presentation we apply a domain parametrization technique to reduce both the geometrical and computational complexity of the forward problem, and replace the finite element solution of the incompressible Navier-Stokes equations by a computationally less expensive reduced basis approximation. This greatly reduces the cost of simulating the forward problem. We then consider the solution of inverse problems in both the deterministic sense, by solving a least-squares problem, and in the statistical sense, by using a Bayesian framework for quantifying uncertainty. Two inverse problems in haemodynamics are considered: (i) a simplified stenosed artery model for quantifying the risk of atherosclerosis by identifying the material parameters of the arterial wall based on pressure measurements; (ii) a simplified femoral bypass graft model for robust shape design under uncertain residual flow in the main arterial branch identified from pressure measurements.

Evelyne Miot (Université Paris Sud - Orsay)

PDF On the Vlasov-Poisson system with point charges in 2 and 3 dimensions (Le système de Vlasov-Poisson avec charges ponctuelles en 2 ou 3 dimensions)

In this talk, we will consider a modified Vlasov-Poisson system describing the evolution of a two or three-dimensional bounded distribution of electrical particles, a plasma, interacting with a finite number of charged point particles. For the three-dimensional case, I will present a global existence and uniqueness result in the case of repulsive interaction between the plasma and the charges. In two dimensions and for an attractive interaction, global existence holds but uniqueness is an open problem. This is joint work with Silvia Caprino, Carlo Marchioro and Mario Pulvirenti.

Dans cet exposé, on considérera un système de Vlasov-Poisson décrivant l'évolution, en dimensions deux ou trois, d'une distribution bornée de particules chargées (plasma) en interaction avec des charges ponctuelles. En dimension trois et dans le cas d'une interaction répulsive, on établira l'existence et l'unicité d'une solution globale. En dimension deux et pour une interaction attractive, on présentera un résultat d'existence globale. L'unicité est un problème ouvert. Il s'agit d'un travail en collaboration avec Silvia Caprino, Carlo Marchioro et Mario Pulvirenti.

Lucia Mirabella (Cardiovascular Fluid Mechanics Laboratory - Georgia Institute of Technology - Atlanta)

PDF Surgical mathematics: from non-homogeneous boundary value problems to the operating room

Mathematical models and numerical simulations of phenomena occurring in the cardiovascular system represent a widespread tool to improve the understanding of healthy and pathological conditions of human subjects. Moreover they are becoming an important complement to the clinical practice. This application demands for both accurate mathematical models and efficient numerical algorithms. In addition, computational experiments need to be reliable and to maintain a strict connection with in-vivo measurements.
In this talk we will present examples of integration between mathematics and medicine, with the final goal of performing reliable and fast numerical simulations of patient specific conditions. We will show numerical techniques developed to speed up the solution of the PDE system that models the electrical activity of the heart (Bidomain system). We will also describe a methodology developed to simulate blood flow in moving domains, based on a strong integration of numerical methods and medical imaging, to include the actual motion of biological structures in subject-specific simulations while limiting the computational cost. The impact of this methodology on planning surgical procedures in patients with congenital heart defects will be presented.

Jean-Marie Mirebeau (Ceremade - Université Paris Dauphine)

PDF Mesh adaptation for approximation by finite elements of arbitrary degree (Adaptation de maillage pour l'approximation par éléments finis de degré arbitraire)

The solutions of numerous PDEs present strongly anisotropic features. Their regularity, measured through the norms of some of their derivatives, is highly inhomogeneous and dependent on the position and spatial direction, which suggests and often imposes to adapt the computational methods. In the context of finite elements discretisation of PDEs, on unstructured meshes, a first objective is to identify locally the appropriate scale and direction of the (future) adapted mesh, using estimates of the derivatives of the functions involved in the simulation. The second objective, which is not the least, is to produce a mesh satisfying these requirements. I will present two recent results tied to these objectives, together with numerical illustrations based on the software FreeFEM++. These results are more precisely tied to anisotropic finite elements of degree higher than one, and to the construction of anisotropic meshes in which the largest interior angle is controlled a priori.

Les solutions de nombreuses EDP présentent des comportements fortement anisotropes. Leur régularité, mesurée par les normes de certaines dérivées, varie fortement selon les directions spatiales, ce qui suggère voire impose d'utiliser des méthodes de calcul adaptées. Dans le contexte de la simulation d'EDP par éléments finis sur des maillages non-structurés, un premier objectif est de déterminer localement les directions et les echelles souhaitables du maillage, à l'aide d'estimations locales des dérivées des fonctions intervenant dans la simulation. La seconde étape, non des moindres, est de construire un maillage répondant à cette description. Je présenterai des résultats récents portant sur ces deux problèmes, accompagnés de simulations numériques utilisant FreeFEM++. Ces résultats sont liés respectivement à l'utilisation d'éléments finis de degré supérieur à un, et à la construction de maillages anisotropes dont l'angle maximal des éléments est borné à priori.

Nicolas Rougerie (LPMMC - Université Joseph Fourier - Grenoble)

PDF On the third critical speed in Gross-Pitaevskii theory (Sur la troisième vitesse critique en théorie de Gross-Pitaevskii)

Gross-Pitaevskii theory is the most common framework for the description of rotating Bose-Einstein condensates. It is of great importance to be able to characterize using this variationnal model the properties of the vortices that the rotation nucleates in the condensate. In this talk I will briefly review the physics of rotating superfluids before focusing on the problem of identifying a certain critical rotation speed where a major phase transition occurs. It is characterized by the appearance of a giant vortex at the center of the condensate. I will present recent mathematical results regarding the evaluation of the critical speed. Joint work with Michele Correggi, Florian Pinsker and Jakob Yngvason.

La théorie de Gross-Pitaevskii est l'outil le plus communément utilisé pour la description des condensats de Bose-Einstein en rotation. En particulier, déterminer dans le cadre de ce modèle variationnel les caractéristiques des tourbillons quantiques que la rotation génère dans le condensat est un enjeu important. Dans cet exposé je survolerai la physique des superfluides en rotation avant de me concentrer sur le problème de l'identification d'une certaine vitesse de rotation marquant une importante transition de phase. Celle-ci est caractérisée par l'apparition d'un vortex (tourbillon) géant au centre du condensat. Je présenterai des résultats récents donnant une évaluation rigoureuse de la vitesse critique pour la transition de phase. Collaboration avec Michele Correggi, Florian Pinsker et Jakob Yngvason.

Gianluigi Rozza (Modelling and Scientific Computing - Ecole Polytechnique Fédérale de Lausanne)

PDF Reduced basis method for parametrized optimal control problems governed by PDEs

We present the development, analysis and computer implementation of efficient numerical methods for the solution of optimal control problems based on parametrized partial differential equations. Our goal is to develop a new approach based on suitable model reduction paradigms for the rapid and reliable solution of optimal control problems usually characterized by large computational costs already in the non-parametric case. Since the characterization of a complex system in terms of physical quantities (like source terms, boundary conditions, material properties) and/or geometrical configuration usually depends on a set of parameters to be identified and/or assimilated, the system response will be parameter dependent as well, and so will be the optimal control. In all these cases, we are required to solve parametrized optimal control problems, where the prediction of optimal control inputs and the optimization of given output of interests is required for each different value of the parameters. Therefore, when performing the optimization process for many different parameter values (many-query context) or when, for a given new configuration, we want to compute the solution in a rapid way (real-time context), the computational effort may be unacceptably high and, often, unaffordable. For this reason we aim at reducing the complexity of the original problem by means of suitable model order reduction techniques, yet preserving its main features and the same input-output behaviour. The general framework based on the reduced basis method for the efficient numerical solution of parametrized optimal control problems can be used in fields like parameter identification, data assimilation and quantification of uncertainty.

Antonio Segatti (Dipartimento di Matematica F. Casorati - Università di Pavia)

PDF On a variational principle for gradient flows in metric spaces (Sur un principe variationnel pour les équations de flot de gradient dans les espaces métriques)

In this talk I will report on an ongoing project with R. Rossi, G. Savaré and U. Stefanelli regarding a novel variational approach to gradient flow evolutions in metric spaces. In particular, we propose a functional defined on entire trajectories, whose minimizers converge to curves of maximal slope under proper convexity assumptions on the driving energies. A crucial step of the argument is the reformulation of the variational approach in terms of a dynamic programming principle. The result is applicable to a large class of nonlinear evolution PDEs including nonlinear drift-diffusion, Fokker-Planck, and heat flows on metric-measure spaces.

Didier Smets (LJLL - Université Pierre et Marie Curie - Paris 6)

PDF The flow of a curve by its binormal curvature (Le flot d'une courbe par sa courbure binormale)

The motion of vortex filaments in incompressible fluids was first studied by Helmholtz and Kelvin in the second half of the nineteenth century. In 1906, Da Rios formally derived an asymptotic geometrical flow for those filaments, which was later termed the LIA flow, for Local Induction Approximation. Even though the model has some known limitations at finite vortex core (stretching is not taken into account), its validity in the asymptotic of indefinitely small section has never been proved or disproved. In the talk, we will focus on the geometrical flow and show how to extend the notion of solutions to situations where the curve(s) are barely Lipschitz and may recombine (the so-called vortex reconnection) and change topology. In this framework, we have proved with R.L. Jerrard (Toronto) a global existence theorem as well as a weak-strong uniqueness one, valid before self-intersections. We will also present numerical simulations suggesting intriguing features of the flow, in particular for non-smooth curves.

L'étude de la dynamique des filaments de vorticité dans les fluides incompressibles remonte à Helmholtz et Kelvin dans la seconde moitié du 19ème siècle. En 1906, Da Rios dérive formellement un flot géométrique asymptotique pour ces filaments ; ce flot est ensuite baptisé LIA, pour Local Induction Approximation. Bien qu'il présente certaines limitations (notoirement il ne tient pas compte de l'effet d'étirement), sa validité comme asymptotique dans la limite de section indéfiniment petite des filaments n'a jamais été ni établie ni contredite. Dans l'exposé, on présentera un cadre permettant d'étendre la notion de solutions du flot par LIA à des courbes lipschitziennes, pouvant se recombiner ou changer de topologie. En collaboration avec R.L. Jerrard (Toronto), nous avons établi un résultat d'existence globale dans ce cadre, ainsi qu'un résultat d'unicité fort-faible avant apparition d'auto-intersections. On présentera aussi quelques simulations numériques intrigantes, en particulier pour des courbes peu régulières.

Lorenzo Tamellini (MOX - Dipartimento di Matematica - Politecnico di Milano)

PDF Polynomial approximation of elliptic PDEs with stochastic coefficients

Partial differential equations with stochastic coefficients are a suitable tool to describe systems whose parameters are not completely determined, either because of measurement errors or intrinsic lack of knowledge on the system. Since in the case of elliptic PDEs the state variables usually exhibits high regularity in their dependence with respect to the random parameters, we consider here a global polynomial approximation over the stochastic space for the state variables and their statistical moments ("Sparse Grid Stochastic Collocation Method"). When the number of parameters is moderate, this method can be remarkably more effective than classical sampling methods. However, contrary to the latter, the performance of the sparse grid approximation deteriorates as the number of random variables increases ("Curse of Dimensionality"); to prevent this, care has to be put in the construction of the approximating sparse grid. In this talk we will show that the construction of a sparse grid can be reformulated as a classical knapsack problem over the tensor interpolation operators forming the sparse grid. We will solve the knapsack problem thanks to a-priori estimates of the profit of each operator, and show numerically the effectiveness of the sparse grids thus obtained.