  Over the past few years it has been demonstrated that "coarse timesteppers"
establish a link between traditional numerical analysis and microscopic/
stochastic simulation. The underlying assumption of the associated
lift-run-restrict-estimate procedure is that macroscopic models exist and close
in terms of a few governing moments of microscopically evolving distributions,
but they are unavailable in closed form. This leads to a system identification
based computational approach that sidesteps the necessity of deriving explicit
closures. Two-level codes are constructed; the outer code performs macroscopic,
continuum level numerical tasks, while the inner code estimates -through
appropriately initialized bursts of microscopic simulation- the quantities
required for continuum numerics. Such quantities include residuals, time
derivatives, and the action of coarse slow Jacobians. We demonstrate how these
coarse timesteppers can be applied to perform equation-free computations of a
kinetic Monte Carlo simulation of E. coli chemotaxis. Coarse-grained
contraction mappings, system level stability analysis as well as acceleration
of the direct simulation, are enabled through this computational multiscale
enabling technology.
