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.

Last modified June 10, 1998