The heterogeneous multiscale method (HMM)

There has been a lot of interest on HMM (the heterogeneous multiscale method) and the ``equation-free'' approach to multiscale modeling. Many people have asked the questions: What are they exactly? What are the similarities and differences? Are they really useful and what are their potential uses? To address these questions, we have to put ourselves in the context of the releveant literature that already existed before HMM and ``equation-free'' were developed. Most people do not seem to be aware of the fact that there had been many successful algorithms for capturing the macroscale behavior of a system with the help of microscale models before HMM and ``equation-free''. Achi Brandt also proposed general strategies for developing such algorithms, by going between macro and micro states/models using interpolation and restriction operators, and performing microscopic simulations on small windows for short times. These histories are rarely discussed. In addition, it has been difficult to isolate what exactly ``equation-free'' is.

Several years have passed. There is now a large volume of literature on both HMM and ``equation-free''. It is perhaps a good time to start addressing the questions raised above.

This page provides some of the background materials as well as some personal perspectives on these matters. Major players on the ``equation-free'' approach have also been invited to provide their perspectives. These will be posted once they are received.