Multiscale Modeling

Most discussions of modeling in the philosophy of science consider the relationship between a single model, picked out by a set of equations, and a physical system. While this is appropriate for many purposes, there are also modeling contexts in which the challenge is to relate several different kinds of models to a system. One such case can be grouped under the heading of 'multiscale modeling'. Multiscale modeling involves considering two or more models that represent a system at different scales. An intuitive example is a model that represents basic particle-to-particle interactions and a continuum model involving larger scale variables for things like pressure and temperature.

In my own work on multiscale modeling I had always assumed that the larger-scale, macro models would be more accessible, and that the challenge lay in seeing how successful macro models relate to underlying micro models. From this perspective, Batterman's work shows how certain macro models of a system can be vindicated without the need of a micro model for that system.

It seems, though, that applied mathematicians have also developed techniques for working exclusively with a micro model due to the intractable nature of some macro modeling tasks. The recently posted article by E and Vanden-Eijnden, "Some Critical Issues for the 'Equation-free' Approach to Multiscale Modeling", challenges one such technique. As developed by Kevrekidis, Gear, Hummer and others, the equation-free approach aims to model the macro evolution of a system using only the micro model and its equations:

We assume that we do not know how to write simple model equations at the right macroscopic scale for their collective, coarse grained behavior. We will argue that, in many cases, the derivation of macroscopic equations can be circumvented: by using short bursts of appropriately initialized microscopic simulation, one can effectively solve the macroscopic equations without ever writing them down, and build a direct bridge between microscopic simulation and traditional continuum numerical analysis. It is, thus, possible to enable microscopic simulators to directly perform macroscopic systems level tasks (1347).

At an intuitive level, the techniques involve using a sample of microscopic calculations to estimate the development of the system at the macroscopic level. E and Vanden-Eijnden question both the novelty of this approach and its application to simple sorts of problems. One challenge is that the restriction to the micro level may not be any more tractable than a brute force numerical solution to the original macro level problem.