Wavelets in Scientific Computing
Résumé
Waveletteori er en forholdsvis ny matematisk disciplin, som har vakt stor interesse indenfor b°ade teoretisk og anvendt matematik i løbet af det seneste °arti. De altafgørende egenskaber ved wavelets er at de kan analysere forskellige dele af en funktion p°a forskellige skalatrin, samt at de kan repræsentere polynomier nøjagtigt op til en given grad. Dette fører til, at funktioner med hurtige oscillationer eller singulariteter indenfor lokaliserede omr°ader kan approksimeres godt med en linearkombination af forholdsvis f°a wavelets. Til sammenligning skal man medtage mange led i en Fourierrække for at opn°a en god tilnærmelse til den slags funktioner. Disse egenskaber ved wavelets har med held været anvendt indenfor signalbehandling. Denne afhandling omhandler wavelets rolle indenfor scientific computing og den best°ar af tre dele: Del I giver en gennemgang af teorien for ortogonale, kompakt støttede wavelets med udgangspunkt i multiskala analyse. S°adanne wavelets er særligt attraktive, fordi de giver anledning til en stabil og særdeles effektiv algoritme, kaldet den hurtige wavelet transformation (FWT). Vi giver estimater for approksimationsegenskaberne af wavelets og demonstrerer, hvordan og hvorfor FWT-algoritmen kan bruges som første led i en effektiv billedkomprimerings metode. Del II omhandler forskellige implementeringer af FWT algoritmen p°a vektorcomputere og parallelle datamater. Vi udvikler en effektiv og skalerbar parallel FWT algoritme og angiver en model for dens ydeevne. Del III omfatter et studium af mulighederne for at bruge wavelets særlige egenskaber til at løse partielle differentialligninger numerisk. Flere forskellige tilgange identificeres og to af dem beskrives detaljeret. De udviklede algoritmer anvendes p°a den ikke-lineære Schr¨odinger ligning og Burgers ligning. Numeriske undersøgelser viser, at algoritmerne kan være effektive under forudsætning af at problemerne er store, at løsningerne er stærkt lokaliserede og at de forskellige numeriske metode-parametre kan vælges p°a passende vis afhængigt af det p°agældende problem.
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.
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