Wavelet publications (ps-files) and software (tar.gz-files)

Wavelets in Scientific Computing - Ph.D. dissertation 1998
Wavelets in Scientific Computing - Slides from Ph.D. defence 1998
Wavelet Package Wavelet software for Matlab 5.0 (UNIX)
High Dimensional Wavelet Smoothing - Proceedings from the CTAC99 conference
High Dimensional Wavelet Smoothing - Preprint
Parallel Performance of Fast Wavelet Transforms
Slides from Data Mining Talk, October 2000
Slides from talk at Ninth International Python Conference, Long Beach, California, March 2001
Wavelets and Data Mining - Slides from CTAC99 conference held at the ANU September 1999
HISURF High dimensional smoothing using wavelets. (Matlab 5.0)
Wavelets in Numerical analysis - Lecture at DTU, May 1998
Wavelets in Power engineering
Fast 2D Wavelet Transform of Circulant Matrices - Technical Report 1997
Vector-Parallel Fast Wavelet Transforms. Proceedings for Seventh Parallel Computing Workshop held at Australian National University, September 1997. Also available at Area4 working note 22.
Wavelets and the nonlinear Schrodinger equation, 1997
Parallel Wavelet Transforms. Proceedings for One Day Workshop on Basic Parallel Numerical Algorithms and Their Applications held at Technical University of Denmark, December 1997. You can download my slides here .
A scalable parallel 2D wavelet transform algorithm.. Also available at Technical report TR-CS-97-21
Poster at NGSWCSE 29/9-1/10 1997

Wavelets

Wavelets are basis function which are particularly well suited for representing piecewise smooth functions. The reason is that wavelets are well localized in space as well as in scale and that they can represent polynomials up to a certain order exactly. These properties are used in signal analysis for compression of digital images, denoising and edge detection for example..

I am investigating the potential for using these desirable properties of wavelets for solving partial differential equations. One approach is to work with an adaptive grid; The grid density must then be high where the solution changes rapidly while it can be low where the solution is smooth. The wavelets are then used to detect where steep gradients (for example shocks) are forming. The picture below shows a numerical solution to Burgers' equation. The circles on the x-axis indicate grid points as chosen by the wavelet method. It is seen that the grid density is high where the shock is forming. Click here or on the picture to see an animation of this process.