Wavelet analysis and parallel numerical algorithms

Abstract

Multiresolution wavelet and wavelet packet decomposition has recently found a wide range of application fields. In this work we show that its localization property in the frequency domain together with the corresponding orthogonal splitting of the multiresolution spaces can be used to build up new parallel algorithms. It is then shown that with the construction of semiorthogonal wavelets and wavelet packet packets, which in some cases can be $adapted$ to certain differential and integral operator, the corresponding numerical problems split into indipendet subproblems according to the orthogonality of the multiresolution spaces. Parallelism is therefore inherent in this basis change: the solutions of the subproblems obtained concurrently by different processors are defined on the whole physical domain but are local in the frequency domain ad they correspond to different frequency bands. The final solution, which is global in both frequency and physical spaces, is then easily obtained by means of the usual wavelet packet reconstruction algorithm