Some Recent Progress in Multiscale Modeling

In recent years we have seen an explosive growth of activities in multiscale modeling and computation, with applications in many areas including material science, fluid mechanics, chemistry, and biology. It is widely recognized that multiscale techniques will become an essential part of computational science and engineering. The main purpose of the present paper is to review some (not all) of these progresses, with a view toward building systematic, reliable and controlled multiscale techniques for a variety of applications. The basic set-up will be the following. We are given a system whose microscopic behavior, with state variable u, is described by a given microscopic model. This microscopic model is too inefficient to be used in full detail. On the other hand, we are only interested in the macroscopic behavior of the system, described by the state variable U . U and u are linked together by a compression operator Q: U = Qu.