Statistics for Spatial Data.

Wavelet transforms are recent mathematical techniques, based on group theory and square integrable representations, which allows one to unfold a signal, or a field, into both space and scale, and possibly directions. They use analyzing functions, called wavelets, which are localized in space. The scale decomposition is obtained by dilating or contracting the chosen analyzing wavelet before convolving it with the signal. The limited spatial support of wavelets is important because then the behavior of the signal at infinity does not play any role. Therefore the wavelet analysis or syn­ thesis can be performed locally on the signal, as opposed to the Fourier transform which is inherently nonlocal due to the space-filling nature of the trigonometric functions. Wavelet transforms have been applied mostly to signal processing, image coding, and numerical analysis, and they are still evolving. So far there are only two complete presentations of this topic, both written in French, one for engineers (Gasquet & Witomski 1 990) and the other for mathematicians (Meyer 1 990a), and two conference proceedings, the first in English (Combes et al 1 989), the second in French (Lemarie 1 990a). In preparation are a textbook (Holschneider 199 1 ), a course (Dau­ bee hies 1 99 1), three conference procecdings (Mcyer & Paul 199 1 , Beylkin et al 199 1b, Farge et al 1 99 1), and a special issue of IEEE Transactions

Wavelet Transforms and their Applications to Turbulence

An efficient ghost-cell immersed boundary method (GCIBM) for simulating turbulent flows in complex geometries is presented. A boundary condition is enforced through a ghost cell method. The reconstruction procedure allows systematic development of numerical schemes for treating the immersed boundary while preserving the overall second-order accuracy of the base solver. Both Dirichlet and Neumann boundary conditions can be treated. The current ghost cell treatment is both suitable for staggered and non-staggered Cartesian grids. The accuracy of the current method is validated using flow past a circular cylinder and large eddy simulation of turbulent flow over a wavy surface. Numerical results are compared with experimental data and boundary-fitted grid results. The method is further extended to an existing ocean model (MITGCM) to simulate geophysical flow over a three-dimensional bump. The method is easily implemented as evidenced by our use of several existing codes.

A Ghost-Cell Immersed Boundary Method for Flow in Complex Geometry

Second-moment closure: present… and future?

Recent advances in wavelet analyses: Part 1. A review of concepts

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.

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

A nonlinear multiresolution scheme within Harten's framework is presented, based on a new nonlinear, centered piecewise polynomial interpolation technique. Analytical properties of the resulting subdivision scheme, such as convergence, smoothness, and stability, are studied. The stability and the compression properties of the associated multiresolution transform are demonstrated on several numerical experiments on images.

Analysis of a New Nonlinear Subdivision Scheme. Applications in Image Processing

On the stability of the PPH nonlinear multiresolution

The basic definitions and the most attractive properties of the wavelet transform are reviewed and explained using the classical language of turbulence The wavelet transform appears to be a natural alternative to the decompositions commonly used in fluid dynamics and turbulence (mainly the Fourier decomposition)

The wavelet transform - Some applications to fluid dynamics and turbulence

Laser doppler anemometry (LDA) measurements and numerical calculations have been made for a laminar boundary layer on triangular riblets. Calculated mean velocity distributions along the riblet contour are in good agreement with the measured ones. The results show that no transversal motion exists above and within the riblet valleys (e.g., no secondary motion). It is found that despite the large wetted area increase, the frictional drag is not increased on riblets relative to a smooth wall. This result suggests that the viscous effects are at play in the drag reduction for a turbulent boundary layer, in the sense that they compensate for the increase in wetted area.

Jacques Liandrat

Papers

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

Analysis of a New Nonlinear Subdivision Scheme. Applications in Image Processing

On the stability of the PPH nonlinear multiresolution

The wavelet transform - Some applications to fluid dynamics and turbulence

Laminar boundary layer over riblets