ME 252 B

Computational Fluid Dynamics:

Wavelet transforms and their applications to turbulence

Marie Farge¹ & Kai Schneider²

Winter 2004

University of California, Santa Barbara

¹ LMD-CNRS, Ecole Normale Supérieure ² CMI, Université de Provence

24 rue Lhomond 39 rue Joliot-Curie

75231 Paris Cedex 05, France 13453 Marseille Cedex 13, France

Email mailto:farge@lmd.ens.fr Email : mailto:kschneid@cmi.univ-mrs.fr

http://wavelets.ens.fr/ http://www.l3m.univ-mrs.fr/schneider.htm

Courses: Monday / Wednesday / Friday 9.00 am – 9.50 am, GIRVETZ 2123

Office hours: Tuesday / Thursday 4.00 pm – 5.00 pm, Engineering II, room 2332

Objectives of the course

Our goal is to bring the students to a level of understanding which allows them to apply wavelet methods to solve their own problems. To guarantee a good assimilation of the wavelet theory, we will first recall the Fourier transform and the most important mathematical theorems related to it. We will then provide the students with a basic knowledge of wavelets, for both the continuous wavelet transform, the orthogonal wavelet transform and the wavelet packet transform, without going too far into their mathematical background. We will give the basics concerning their numerical implementation and illustrate their use with academic examples. We will illustrate the course with applications to signal and image processing, data compression, denoising and resolution of PDEs, focusing in particular to applications in turbulence.

Content of the course

Fourier transform

Fourier integral and Fourier series, academic examples.

Properties, Parseval's theorem, convolution. Uncertainty principle,

Brillouin's information plane.

Auto-correlation function, spectrum, Wiener-Khinchin's theorem.

Discrete Fourier transform, Shannon's sampling theorem, fast Fourier transform.

Applications for analyzing the global regularity of a function, for filtering and denoising signals and images.

Continuous wavelet transform

Definitions, academic examples.

Information plane, choice of the mother wavelet.

Properties, Parseval's theorem, reproducing kernel.

Boundary effects, algorithms.

Scalogram and its relation to spectrum and structure functions.

Extension to two and three dimensions.

Orthogonal wavelet transform

Discretization of the wavelet space, quasi-orthogonal representations, wavelet frames.

Orthogonal wavelet bases, properties, academic examples.

Quadrature mirror filters, multi-resolution analysis.

Fast wavelet transform algorithm.

Biorthogonal wavelets, wavelets on the interval.

Extension to two and three dimensions.

Applications for compressing and denoising signals and images.

Wavelet packet transform

Wavelet packets and Malvar wavelets orthogonal bases.

Information plane, information cost, information entropy.

Theoretical dimension of the representation, choice of the best basis.

Fast wavelet packet transform algorithm.

Extension to two and three dimensions.

Applications for compressing and denoising signals and images,
comparison with wavelets.

Applications numerical analysis

Wavelets and operator equations.

Norm equivalences and preconditioning of matrices.

Nonlinear approximation, adaptive grids, and error estimates.

Adaption strategy for evolution problems.

Compression of operators (BCR algorithm).

Evaluation of nonlinear terms, conncection coefficients, collocation on adaptive grids.

Operator adapted wavelets (biorthognal decompositions and vaguelettes).

Applications to turbulence

Definition and properties of turbulence.

Statistical theory of two-dimensional and three-dimensional turbulence.

Intermittency and coherent structures.

Wavelet analysis of turbulent signals from both laboratory and numerical
experiments using continuous and orthogonal wavelets.

Extraction of coherent structures in one, two and three dimensional turbulent flows,
and comparison between wavelets, wavelet packets and Malvar wavelets.

Resolution of the incompressible Navier-Stokes equations using orthogonal wavelets.

Coherent Vortex Simulation (CVS), based on nonlinear wavelet filtering,
and comparison with Large Eddy Simulation (LES), based on linear Fourier filtering.