Overview

In fact, advancements in the area (of wavelets) are occurring at such a rate that the very meaning of wavelet analysis keeps changing to incorporate new ideas.

Bj rn Jawerth and Wim Sweldens SIAM Review,Vol. 36 (1994)

As indicated in Chapter 8, it is often viable to apply a DWT to a sparse linear system to derive a wavelet type preconditioner. In this chapter we consider anew wayofobtaining the same wavelet preconditioner without applying a DWT. The assumption is that the sparse representation of A in a single scale finite element basis is already available and the corresponding wavelet is less sparse than A.

Our idea is to work with this sparse matrix A in order to implicitly compute the representation of A and its preconditioner in the wavelet basis. Thus the main advantage is that the new strategy removes the costs associated with forming the wavelet matrix explicitly and works with a sparse matrix A directly while making full use of the robust preconditioning property of wavelets. The general difficulty of specifying a suitable pattern (Chapter 5) is resolved in the new setting. The obtained preconditioners are good sparse approximations to the inverse of A computed by taking advantage of the compression obtained by working in a wavelet basis. In fact, efficient application to both sparse and dense A can be considered as shown in the following.

Section 10.1 Introduction

Section 10.2...