Multiscale and Wavelet Methods for Operator Equations

Abstract

These notes are to bring out some basic mechanisms governing wavelet methods for the numerical treatment of differential and integral equations. Some introductory examples illustrate the quasi–sparsity of wavelet representations of functions and operators. This leads us to identify the key features of general wavelet bases in the present context, namely locality, cancellation properties and norm equivalences. Some analysis and construction principles regarding these properties are discussed next. The scope of problems to which these concepts apply is outlined along with a brief discsussion of the principal obstructions to an efficient numerical treatment. This covers elliptic boundary value problems as well as saddle point problems. The remainder of these notes is concerned with a new paradigm for the adaptive solution of such problems. It is based on an equivalent formulation of the original variational problem in wavelet coordinates. Combining the well-posedness of the original problem with the norm equivalences induced by the wavelet basis, the transformed problem can be arranged to be well–posed in the Euclidean metric. This in turn allows one to devise convergent iterative schemes for the infinite dimensional problem over l ₂. The numerical realization consists then of the adaptive application of the wavelet representations of the involved operators. Such application schemes are described and the basic concepts for analyzing their computational complexity, rooted in nonlinear approximation, are outlined. We conclude with an outlook on possible extensions, in particular, to nonlinear problems.