ME 252 B

**Computational
Fluid Dynamics:**

**Marie
Farge ^{1} & Kai Schneider^{2}**

Winter 2004

University of California, Santa Barbara

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

2t ruE Liomond 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 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.

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.

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 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.

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).

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.

Lecture on Fourier transform and sampling : Fourier transform

Lectures on continuous wavelet transform: Continuous wavelet transform 1d

Continuous wavelet transform 2d

Lectures on orthogonal wavelet transform: Discrete wavelet transform 1d

Lectures on wavelet packet transform: Wavelet packet transform

Excercices

Sheet
1 : ucsb_ex1.pdf
*due to January 14th*

Sheet
2 : ucsb_ex2.pdf
*due to January 30th*

Sheet
3 : ucsb_ex3.pdf
*due to February 13th*

Additional material

Marie
Farge and Kai Schneider, 2002**Analysing and compressing
turbulent fields with wavelets***Note IPSL, n°20, April
2002 *pdf-file

Marie Farge, 1992

**Wavelet
transforms and their applications to turbulence.**

*Ann.
Rev. Fluid Mech*, 24:395-457, 1992 pdf-file

Barbara Burke, 1996

**The world according to wavelets.**

Stephane Mallat, 1999

**A wavelet tour of signal processing.**

Paul S. Addison, 2002

**The illustrated wavelet transform handbook.**

*Institute of Physics (IOP)*

Stephane Jaffard, Yves Meyer and Robert D. Ryan, 2001

**Wavelets: tools for science and technology.**

M.V. Wickerhauser, 1994

**Adapted wavelet analysis from theory to software.**

Wavelet application in cosmetics

*Last
update 31.1.2004 Webmaster*