Key points are not available for this paper at this time.
In the companion article, a framework for structural multiscale geometric organization of subsets of 12ptminimal amsmath wasysym amsfonts amssymb amsbsy mathrsfs -69pt document equation*R^nequation*document and of graphs was introduced. Here, diffusion semigroups are used to generate multiscale analyses in order to organize and represent complex structures. We emphasize the multiscale nature of these problems and build scaling functions of Markov matrices (describing local transitions) that lead to macroscopic descriptions at different scales. The process of iterating or diffusing the Markov matrix is seen as a generalization of some aspects of the Newtonian paradigm, in which local infinitesimal transitions of a system lead to global macroscopic descriptions by integration. This article deals with the construction of fast-order N algorithms for data representation and for homogenization of heterogeneous structures.
Coifman et al. (Tue,) studied this question.