1 Department of Informatics and Mathematical Modeling, Technical University of Denmark2 Scientific Computing, Department of Informatics and Mathematical Modeling, Technical University of Denmark3 Department of Applied Mathematics and Computer Science, Technical University of Denmark
Wavelet analysis is a relatively new mathematical discipline which has generated much interest in both theoretical and applied mathematics over the past decade. Crucial to wavelets are their ability to analyze different parts of a function at different scales and the fact that they can represent polynomials up to a certain order exactly. As a consequence, functions with fast oscillations, or even discontinuities, in localized regions may be approximated well by a linear combination of relatively few wavelets. In comparison, a Fourier expansion must use many basis functions to approximate such a function well. These properties of wavelets have lead to some very successful applications within the field of signal processing. This dissertation revolves around the role of wavelets in scientific computing and it falls into three parts: Part I gives an exposition of the theory of orthogonal, compactly supported wavelets in the context of multiresolution analysis. These wavelets are particularly attractive because they lead to a stable and very efficient algorithm, namely the fast wavelet transform (FWT). We give estimates for the approximation characteristics of wavelets and demonstrate how and why the FWT can be used as a front-end for efficient image compression schemes. Part II deals with vector-parallel implementations of several variants of the Fast Wavelet Transform. We develop an efficient and scalable parallel algorithm for the FWT and derive a model for its performance. Part III is an investigation of the potential for using the special properties of wavelets for solving partial differential equations numerically. Several approaches are identified and two of them are described in detail. The algorithms developed are applied to the nonlinear Schrödinger equation and Burgers' equation. Numerical results reveal that good performance can be achieved provided that problems are large, solutions are highly localized, and numerical parameters are chosen appropriately, depending on the problem in question.