Multiresolution in Multiscale: A New Role for Wavelets

In the emerging field of multiscale systems, many practical problems in the physical and biological sciences require the simulation of signals at the molecular level or the merging of signals generated over large and different spatial and time scales. Computationally, the former is intractable for large-sized systems (hundreds of millions of atoms) over large time scales (micro to milliseconds) while the compounding of information over different scales presents unique challenges and opportunities for wavelet-based as also other techniques. While the multiresolution aspect of wavelets has been explored in this context, the generation of a formal, rigorous methodology that would govern their use in the understanding of interscale phenomena still exists as an open problem. The purpose here is to provide a brief elementary introduction to the multiscale problem, to introduce the role that wavelets and in particular the Compound Wavelet Matrix have played in the bridging of information across scales, and to suggest related areas for exploration. A few simple illustrative multiscale simulation examples where wavelets are used to bridge scales are shown.