  We present and discuss a framework for computer-aided multiscale analysis,
which enables models at a "fine" (microscopic/stochastic) level of description
to perform modeling tasks at a "coarse" (macroscopic, systems) level. These
macroscopic modeling tasks, yielding information over long time and large space
scales, are accomplished through appropriately initialized calls to the
microscopic simulator for only short times and small spatial domains. Our
equation-free (EF) approach, when successful, can bypass the derivation of the
macroscopic evolution equations when these equations conceptually exist but are
not available in closed form. We discuss how the mathematics-assisted
development of a computational superstructure may enable alternative
descriptions of the problem physics (e.g. Lattice Boltzmann (LB), kinetic Monte
Carlo (KMC) or Molecular Dynamics (MD) microscopic simulators, executed over
relatively short time and space scales) to perform systems level tasks
(integration over relatively large time and space scales,"coarse" bifurcation
analysis, optimization, and control) directly. In effect, the procedure
constitutes a systems identification based, "closure on demand" computational
toolkit, bridging microscopic/stochastic simulation with traditional continuum
scientific computation and numerical analysis. We illustrate these ideas
through examples from chemical kinetics (LB, KMC), rheology (Brownian
Dynamics), homogenization and the computation of "coarsely self-similar"
solutions, and discuss various features, limitations and potential extensions
of the approach.
