This talk mainly concerns the mathematical justification of a viscous compressible multi-fluid model linked to the Baer-Nunziato model used by engineers, see for instance [M., Eyrolles (1975)]. More precisely, we show that some built approximate finite-energy weak solutions of the isentropic compressible Navier-Stokes equations converge, on a short time interval, to the strong solution of this viscous compressible multi-fluid model provided the initial density sequence is uniformly bounded with a corrresponding Young measure which is a linear convex combination of m Dirac measures.
It is well-known that one-dimensional isentropic gas dynamics has two elementary waves, i.e., shock wave and rarefaction wave. Among the two waves, only the rarefaction wave can be connected with vacuum. Given a rarefaction wave with one-side vacuum state to the compressible Euler equations, we can construct a sequence of solutions to one-dimensional compressible isentropic Navier-Stokes equations which converge to the above rarefaction wave with vacuum as the viscosity tends to zero. Moreover, the uniform convergence rate is obtained. The proof consists of a scaling argument and elementary energy analysis, based on the underlying rarefaction wave structures.
Compressed sensing (CS) is a new strategy to sample complicated data such as audio signals or natural images. Instead of performing a pointwise evaluation using localized sensors, signals are projected on a small number of delocalized random vectors. This talk is intended to give an overview of this emerging technology. It will cover both theoritical guarantees and practical applications in image processing and numerical analysis. The initial theory of CS was jointly developed by Donoho [1] and Candès, Romberg and Tao [2]. It makes use of the sparsity of signals to minimize the number of random measurements. Natural images are for instance well approximated using a few number of wavelets, and this sparsity is at the heart of the non-linear reconstruction process. I will discuss the extend to which the current theory captures the practical success of CS. I will pay a particular attention to the worse case analysis of the recovery, and perform a non-asymptotic evaluation of the performances [3]. To obtain better recovery guarantees, I propose a probabilistic analysis of the recovery of the sparsity support of the signal, which leads to constants that are explicit and small [4]. CS ideas have the potential to revolutionize other fields beyond signal processing. In particular, the resolution of large scale problems in numerical analysis could beneficiate from random projections. This performs a dimensionality reduction while simplifying the structure of the problem if the projection is well designed. As a proof of concept, I will present a new compressive wave equation solver, that use projections on random Laplacian eigenvectors [5]. [1] D. Donoho, Compressed sensing, IEEE Trans. Info. Theory, vol. 52, no. 4, pp. 1289-1306, 2006. [2] E. Candès, J. Romberg, and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Info. Theory, vol. 52, no. 2, pp. 489-509, 2006. [3] C. Dossal, G. Peyré and J. Fadili, A Numerical Exploration of Compressed Sampling Recovery, Linear Algebra and its Applications, Vol. 432(7), p.1663-1679, 2010. [4] C. Dossal, M.L. Chabanol, G. Peyré and J. Fadili, Sparse Support Identi
The ADER approach (Toro et al. 2001 and many others) allows the construction of non-linear one step fully discrete numerical schemes of arbitrary order of accuracy in space and time, for solving evolutionary partial differential equations. The ADER approach operates in the frameworks of finite volume and DG finite element methods and is applicable to multidimensional problems on unstructured meshes. The schemes have two basic ingredients: (a) a non-linear spatial reconstruction operator and (b) the solution of a generalized (or high-order) Riemann problem that links spatial data distribution and time evolution. After describing the main ideas of the methodology I will also show some applications involving hyperbolic and parabolic equations.
We extend the well-known Serrin's blowup criterion for the three-dimensional incompressible Navier-Stokes equations to the 3D compressible Navier-Stokes equations with vacuum. In other words, in addition to Serrin's condition on the velocity, the L^1(0,T;L^{infty}) norm of the divergence of the velocity is also needed to control the possible breakdown of strong (or smooth) solutions for the three-dimensional compressible Navier-Stokes equations. Moreover, under some additional constraint on the viscosity coefficients, either the L^1(0,T;L^{infty}) norm of the divergence of the velocity or the upper bound of the density will be enough to guarantee the global existence of classical (or strong) solutions.``
Nous présentons les méthodes d'optimisation de structure par la méthode des courbes de niveaux (level set). Nous montrons ensuite comment le modèle de Francfort-Marigo pour l'endommagement peut se traiter numériquement de façon efficace par ce type de méthode dès lors que l'on a calculé la dérivé de forme pour un problème à deux matériaux.
We consider the variational problem which consists in minimizing the compliance of a prescribed amount of elastic material, placed into a given design region, and sumbitted to an exterior balanced load. We discuss the asymptotic analysis of this problem when the design region is either a cylinder of infinitesimal height (case of thin plates) or a cylinder of infinitesimal cross section (case of thin rods). The results are contained in some recent papers in collaboration with Guy Bouchitte' and Pierre Seppecher.
Dans le cas des EDP stochastique, les solutions sont définies sur un espace de dimension infinie et les techniques utilisées pour des équations stochastiques ordinaires - fonction de Lyapunov, hypoellipticité, compacité du semi groupe de transition etc.- ne peuvent pas être appliquées ou nécessitent d'être adaptées. Dans cet exposé j'illustrerai des méthodes utilisées pour l'étude des mesures invariantes pour les EDP stochastiques et leurs applications à des cas spécifiques: dynamique de populations, équation de Burgers, équations de Navier-Stokes etc.
We present some numerical methods to solve control problems in the coefficients where the cost functional may depend on the gradient of the state non linearly. The main difficulty comes from the fact that the relaxed functional cost is not explicitly known. We prove some convergence results just using an upper or a lower approximation of this relaxed functional.
We consider a control problem in the coefficients for an elliptic linear equation where the cost functional is non-linear in the gradient of the function state. The control variables are the coefficients of the diffusion matrix. This type of problems arises in Optimal Design of Composite Materials. It is well known that they have not a solution in general. Here we use the homogenization method to obtain a relaxed formulation.
La simulation numérique des écoulements turbulents est délicate. En effet, lorsque le pas d'espace du maillage est plus grand que l'échelle dissipative, le maillage ne permet pas la représentation des plus petites échelles de l'écoulement réel. L'énergie transférée depuis les grandes échelles vers les petites échelles, par l'action des termes d'interaction non linéaires, n'est pas dissipée correctement. On constate alors une augmentation anormale de l'énergie au niveau des échelles qui correspondent à la taille de la maille de calcul. En conséquence, la réalisation d'une simulation numérique directe (résolution de toutes les échelles physiques sans modélisation de la turbulence) pour des écoulements caractérisés par un nombre de Reynolds élevé est très coûteuse en ressources informatiques. Plusieurs méthodes ont été développées pour permettre la simulation numérique de tels écoulements. La méthode multi-niveaux que nous proposons consiste à appliquer un traitement spécifique à chaque échelle, en considérant les propriétés physiques de l'écoulement. La décomposition des échelles du champ de vitesse est utilisée pour imposer une décroissance correcte du spectre d'énergie. La dynamique des grandes échelles est améliorée par le contrôle de l'accumulation de l'énergie sur les modes élevés.
On donne une condition géométrique nécessaire et suffisante sur un domaine borné arbitraire pour que l'opérateur divergence possède un inverse à droite continu dans des espaces de Lebesgue et de Sobolev à poids. On relie aussi cette question à des inégalités de Poincaré. On retrouve en particulier des résultats connus lorsque le domaine est lipschitzien ou plus généralement est un domaine de John.
The complete water wave problem remains a difficult task despite recent progresses in this field (Clamond & Grue, 2001). Its intrinsic complexity and stiffness prevent from efficient simulations in complex and large domains. Consequently, a number of approximative models have been proposed. In the present work we consider weakly nonlinear/weakly dispersive wave regime which is modelled by the family of Boussinesq type equations. Mathematically these models are expressed as dispersive nonlinear PDEs. In the present study we apply some finite volumes methods to these models. Our numerical schemes are tested on various practical problems. First, we consider some classical questions of soliton dynamics: solitary wave propagation, conservation of invariants, interactions, dispersive shock formation. A comparison with experiments on solitons head-on collision is performed (J. Hammack et al, 2004). Finally, we pay a lot of attention to the problem of the wave run-up onto a beach. This problem is very challenging from physical point of view (triple point) and numerical techniques have to treat wet/dry interface transition. Our algorithm is validated against experimental data of Synolakis and Zelt on breaking and nonbreaking solitary waves run-up onto a plane beach. This is a joint work with D. Dutykh and Th. Katsaounis.