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


Ole Møller Nielsen - go to my home page


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.