   Discrete-orthogonal wavelets Contents Index

## Discrete-orthogonal wavelets

For the purpose of decomposing flow fields with respect to space and scale we make use of the wavelet formalism. We concentrate upon discrete-orthogonal wavelets for various reasons:

• A Parseval relation lets us argument in terms of energy contributions from scales/positions.
• Filtering/manipulation in wavelet space is possible.
• The size of the data is not increased as it would be if the continuous transform were applied in breadth.

Here we document the technical points of the different types of transform which we have used in the course of our project. Note that there has been a strong involvement of J. Fröhlich from the U. of Karlsruhe.

• A simple example of wavelet compression, i.e. non-linear filtering in wavelet space according to coefficient magnitude, applied to data from homogeneous-isotropic turbulence at modest grid size. Go to this page.
• A parallel algorithm for the discrete orthogonal wavelet transform, using the slice data model and the MPI library. Adapted for long filters, i.e. work in Fourier space, like those associated with spline wavelets. This is PIK Report No. 68 as of September 2000. It is available as pdf document (1.3MB). The images can be viewed online.
• Wavelets based upon Legendre polynomials, useful for analyzing data on the interval (as opposed to the classic wavelets for the real line or the torus). PIK Report No. 72 as of July 2001 by J. Fröhlich and M. Uhlmann (29MB).
• Analysis of data on non-uniform grid in bounded domain. A realistic example of fluid dynamics data: vortex-dipole rebound from a no-slip wall at circulation-based Reynolds number of ReGamma=771.
• A short report describing the simulation which uses a Fourier/B-splines method.
• An animated visualization of the vorticity in MPEG format (800KB).
• Its 2D wavelet coefficient scalogram, using spline wavelets in the horizontal direction and Legendre wavelets vertically. Animations in MPEG format (approx. 1MB) are available for two different multi-resolution analyses (MRA): "classical", i.e. square 2D MRA; rectangular 2D MRA.
• Wavelets & Turbulent Flow Analysis: an overview in form of a slide show (pdf, 5MB).
• "Local Spectra in Plane Channel Flow Using Wavelets Designed for the Interval" by M. Uhlmann and J. Fröhlich (Proc. 9th ETC, Southampton, 2002, pp. 111-114). This conference contribution shows some applications of the Legendre wavelet basis to turbulent plane channel flow (local spectra and intermittency index) as well as outlining some perspectives for achieving a better space localization. The slides and the text of the conference presentation are accessible.
• "Analysis of channel flow using improved polynomial wavelets for the interval" by M. Uhlmann and J. Fröhlich (Proc. 3rd TSFP, Sendai, 2003, pp. 841-846). This conference contribution shows applications of the new localized Legendre wavelet basis to turbulent plane channel flow.

markus.uhlmann AT kit.edu   Discrete-orthogonal wavelets Contents Index