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Compact Brownian surfaces I. Brownian disks
No full textInternational audienceWe show that, under certain natural assumptions, large random plane bipartite maps with a boundary converge after rescaling to a one-parameter family (BD_L, 0 < L < infinity) of random metric spaces homeomorphic to the closed unit disk of R^2, the space BD_L being called the Brownian disk of perimeter L and unit area. These results can be seen as an extension of the convergence of uniform plane quadrangulations to the Brownian map, which intuitively corresponds to the limit case where L = 0. Similar results are obtained for maps following a Boltzmann distribution, in which the perimeter is fixed but the area is random
Tightness and duality of martingale transport on the Skorokhod space *
No full textInternational audienceThe martingale optimal transport aims to optimally transfer a probability measure to another along the class of martingales. This problem is mainly motivated by the robust superhedging of exotic derivatives in financial mathematics, which turns out to be the corresponding Kantorovich dual. In this paper we consider the continuous-time martingale transport on the Skorokhod space of c`adì ag paths. Similar to the classical setting of optimal transport, we introduce different dual problems and establish the corresponding dualities by a crucial use of the S−topology and the dynamic programming principle 1
Certified Descent Algorithm for shape optimization driven by fully-computable a posteriori error estimators
No full textInternational audienceIn this paper we introduce a novel certified shape optimization strategy-named Certified Descent Algorithm (CDA)-to account for the numerical error introduced by the Finite Element approximation of the shape gradient. We present a goal-oriented procedure to derive a certified upper bound of the error in the shape gradient and we construct a fully-computable, constant-free a posteriori error estimator inspired by the complementary energy principle. The resulting CDA is able to identify a genuine descent direction at each iteration and features a reliable stopping criterion. After validating the error estimator, some numerical simulations of the resulting certified shape optimization strategy are presented for the well-known inverse identification problem of Electrical Impedance Tomography.Dans cet article, on introduit une nouvelle stratégie d'optimisation de forme certifiée - Algorithme de Descente Certifiée (CDA) - qui prend en compte l'erreur introduite par l'approximation de la dérivée de forme par l'application de la méthode des éléments finis. On présente une procédure permettant d'obtenir une borne supérieure sur l'erreur réalisée lors du calcul de la dérivée de forme. L'estimation à posteriori est basée sur le principe de l'énergie complémentaire et ne dépend pas de constantes non explictement calculables. L'Algorithme de Descente Certifiée pour l'optimisation de forme identifie une véritable direction de descente à chaque itération et permet d'établir un critère d'arrêt fiable. Après avoir validé l'estimation de l'erreur, on applique l'Algorithme de Descente Certifiée à un problème inverse issu de la tomographie d'impédance électrique
Coupling schemes for the FSI forward prediction challenge: comparative study and validation
No full textInternational audienceThis paper presents a numerical study in which several partitioned solution procedures for incompressible fluid-structure interaction are compared and validated against the results of an experimental FSI benchmark. The numerical methods discussed cover the three main families of coupling schemes: strongly coupled, semi-implicit and loosely coupled. Very good agreement is observed between the numerical and experimental results. The comparisons confirm that strong coupling can be efficiently avoided, via semi-implicit and loosely coupled schemes, without compromising stability and accuracy.Cet article présente une étude numérique dans laquelle plusieurs algorithmes partitionnés pour l'interaction fluide structure sont comparés et validés avec des résultats expérimentaux. Les méthodes numériques discutées couvrent les trois familles principales de schémasde couplage: fortement couplés, semi implicites et faiblement couplés. Un très bon accord est obtenu entre les résultats numériques et expérimentaux. Les comparaisons confirment que le couplage fort peut être contourné efficacement, par l'intermédiaire de schémas semi implicites et faiblement couplés, sans compromettre la stabilité et la précision
A characterization of switched linear control systems with finite L 2 -gain
No full textInternational audienceMotivated by an open problem posed by J.P. Hespanha, we extend the notion of Barabanov norm and extremal trajectory to classes of switching signals that are not closed under concatenation. We use these tools to prove that the finiteness of the L2-gain is equivalent, for a large set of switched linear control systems, to the condition that the generalized spectral radius associated with any minimal realization of the original switched system is smaller than one
Intrinsic random walks and sub-Laplacians in sub-Riemannian geometry
No full textInternational audienceOn a sub-Riemannian manifold we define two type of Laplacians. The macroscopic Laplacian ∆ω, as the divergence of the horizontal gradient, once a volume ω is fixed, and the microscopic Laplacian, as the operator associated with a geodesic random walk. We consider a general class of random walks, where all sub-Riemannian geodesics are taken in account. This operator depends only on the choice of a complement c to the sub-Riemannian distribution, and is denoted L c. We address the problem of equivalence of the two operators. This problem is interesting since, on equiregular sub-Riemannian manifolds, there is always an intrinsic volume (e.g. Popp's one P) but not a canonical choice of complement. The result depends heavily on the type of structure under investigation: • On contact structures, for every volume ω, there exists a unique complement c such that ∆ω = L c. • On Carnot groups, if H is the Haar volume, then there always exists a complement c such that ∆H = L c. However this complement is not unique in general. • For quasi-contact structures, in general, ∆P = L c for any choice of c. In particular, L c is not symmetric w.r.t. Popp's measure. This is surprising especially in dimension 4 where, in a suitable sense, ∆P is the unique intrinsic macroscopic Laplacian. A crucial notion that we introduce here is the N-intrinsic volume, i.e. a volume that depends only on the set of parameters of the nilpotent approximation. When the nilpotent approximation does not depend on the point, a N-intrinsic volume is unique up to a scaling by a constant and the corresponding N-intrinsic sub-Laplacian is unique. This is what happens for dimension smaller or equal than 4, and in particular in the 4-dimensional quasi-contact structure mentioned above
Tropicalization of facets of polytopes
No full textPreprint arXiv:1408.6176International audienceIt is known that any tropical polytope is the image under the valuation map of ordinary polytopes over the Puiseux series field. The latter polytopes are called lifts of the tropical polytope. We prove that any pure tropical polytope is the intersection of the tropical half-spaces given by the images under the valuation map of the facet-defining half-spaces of a certain lift. We construct this lift explicitly, taking into account geometric properties of the given polytope. Moreover, when the generators of the tropical polytope are in general position, we prove that the above property is satisfied for any lift. This solves a conjecture of Develin and Yu
Normalization in Lie algebras via mould calculus and applications
No full textInternational audienceWe establish Ecalle's mould calculus in an abstract Lie-theoretic setting and use it to solve a normalization problem, which covers several formal normal form problems in the theory of dynamical systems. The mould formalism allows us to reduce the Lie-theoretic problem to a mould equation, the solutions of which are remarkably explicit and can be fully described by means of a gauge transformation group. The dynamical applications include the construction of Poincaré-Dulac formal normal forms for a vector field around an equilibrium point, a formal infinite-order multiphase averaging procedure for vector fields with fast angular variables (Hamiltonian or not), or the construction of Birkhoff normal forms both in classical and quantum situations. As a by-product we obtain, in the case of harmonic oscillators, the convergence of the quantum Birkhoff form to the classical one, without any Diophantine hypothesis on the frequencies of the unperturbed Hamiltonians
Successive enlargement of filtrations and application to insider information *
No full textInternational audienceWe model in a dynamic way an insider's private information flow which is successively augmented by a family of initial enlargement of filtrations. According to the a priori available information, we propose several density hypotheses which are presented in hierarchical order from the weakest one to the stronger ones. We compare these hypotheses, in particular, with the Jacod's one, and deduce conditional expectations under each of them by providing consistent expressions with respect to the common reference filtration. Finally, this framework is applied to a default model with insider information on the default threshold and some numerical illustrations are performed
On the pointwise mean squared error of a multidimensional term-by-term thresholding wavelet estimator
No full textInternational audienceIn this paper we provide a theoretical contribution to the point-wise mean squared error of an adaptive multidimensional term-by-term thresholding wavelet estimator. A general result exhibiting fast rates of convergence under mild assumptions on the model is proved. It can be applied for a wide range of nonparametric models including possible dependent observations. We give applications of this result for the nonpara-metric regression function estimation problem (with random design) and the conditional density estimation problem