Resolution of the 1D regularized Burgers equation using a spatial wavelet approximation

The Burgers equation with a small viscosity term, initial and periodic boundary conditions is resolved using a spatial approximation constructed from an orthonormal basis of wavelets. The algorithm is directly derived from the notions of multiresolution analysis and tree algorithms. Before the numerical algorithm is described these notions are first recalled. The method uses extensively the localization properties of the wavelets in the physical and Fourier spaces. Moreover, the authors take advantage of the fact that the involved linear operators have constant coefficients. Finally, the algorithm can be considered as a time marching version of the tree algorithm. The most important point is that an adaptive version of the algorithm exists: it allows one to reduce in a significant way the number of degrees of freedom required for a good computation of the solution. Numerical results and description of the different elements of the algorithm are provided in combination with different mathematical comments on the method and some comparison with more classical numerical algorithms

An endpoint Littlewood-Paley inequality for BVP associated with the Laplacian on Lipschitz domains

We prove a commutator inequality of Littlewood-Paley type between partial derivatives and functions of the Laplacian on a Lipschitz domain which gives interior energy estimates for some BVP. It can be seen as an endpoint inequality for a family of energy estimates

Time Scale Approach for Chirp Detection

International audienceTwo different approaches for joint detection and estimation of signals embedded in stationary random noise are considered and compared, for the subclass of amplitude and frequency modulated signals. Matched filter approaches are compared to time-frequency and time scale based approaches. Particular attention is paid to the case of the so-called " power-law chirps " , characterized by monomial and polynomial amplitude and frequency functions. As target application, the problem of gravitational waves at interferometric detectors is considered

Conjecture de Kato sur les ouverts de R.

We prove Kato's conjecture for second order elliptic differential operators on an open set in dimension 1 with arbitrary boundary conditions. The general case reduces to studying the operator T = - d/dx a(x) d/dx on an interval, when a(x) is a bounded and accretive function. We show for the latter situation that the domain of T is spanned by an unconditional basis of wavelets with cancellation properties that compensate the action of the non-regular function a(x)

Bases d'ondelettes sur les courbes corde-arc, Noyau de Cauchy et Espaces de Hardy Associés.

Se construyen dos bases incondicionales de L2(R) adaptadas al estudio de la integral de Cauchy sobre una curva cuerda-arco, y se extiende la construcción a L2(Rd). Esto permite obtener una prueba simple del "Teorema T(b)" de G. David, J.L. Journé u S. Semmes. Se define un espacio de Hardy ponderado Hb1(Rd) caracterizado por las bases anteriores. Finalmente se aplican estos métodos al estudio del potencial de doble capa sobre una superficie lipschitziana

TRANSMISSION À TRAVERS UN DIOPTRE ET RECONSTRUCTION DU SIGNAL SOURCE

Nous considérons dans l'espace tridimensionnel, deux milieux fluides homogènes séparés par une interface plane. Le milieu de plus faible célérité contient une source ponctuelle qui émet un signal dépendant arbitrairement du temps. Pour une distance radiale fixe, et un temps d'observation donné, on montre que le milieu où se trouve le point d'observation, peut être approché par une série de filtres à Ɗf/f = Cst. Par analogie à la formule de reconstitution simple de la transformée en ondelettes, nous établissons pour de grandes distances radiales, une formule de reconstruction du signal-source. La pression transmise joue alors un rôle équivalent à celui d'un coefficient d'ondelettes et la profondeur à celui du paramètre de dilatation. En vue d'une réalisation expérimentale, on s'intéressera au cas du dioptre air-eau.In three-dimensional space, we consider two homogeneous media separated by plane interface. In the lowest velocity media is located a point-source, which emits in time an arbitrary signal. The second medium contains the observation point. By analogy to a reconstruction formula of wavelet transform, we have obtained a formula for the reconstruction of the time dependence of the source-signal. This reconstruction involves an integration over depth (dilation parameter) of the transmitted pressure (wavelet coefficient) at the observation points. To link in future theoretical results to results obtained from an experiment, we consider the air-water interface