  A common approach to studying high-dimensional systems with emergent
low-dimensional behavior is based on lift-evolve-restrict maps (called
equation-free methods): first, a user-defined lifting operator maps a set of
low-dimensional coordinates into the high-dimensional phase space, then the
high-dimensional (microscopic) evolution is applied for some time, and finally
a user-defined restriction operator maps down into a low-dimensional space
again. We prove convergence of equation-free methods for finite time-scale
separation with respect to a method parameter, the so-called healing time. Our
convergence result justifies equation-free methods as a tool for performing
high-level tasks such as bifurcation analysis on high-dimensional systems.
  More precisely, if the high-dimensional system has an attracting invariant
manifold with smaller expansion and attraction rates in the tangential
direction than in the transversal direction (normal hyperbolicity), and
restriction and lifting satisfy some generic transversality conditions, then an
implicit formulation of the lift-evolve-restrict procedure generates an
approximate map that converges to the flow on the invariant manifold for
healing time going to infinity. In contrast to all previous results, our result
does not require the time scale separation to be large. A demonstration with
Michaelis-Menten kinetics shows that the error estimates of our theorem are
sharp.
  The ability to achieve convergence even for finite time scale separation is
especially important for applications involving stochastic systems, where the
evolution occurs at the level of distributions, governed by the Fokker-Planck
equation. In these applications the spectral gap is typically finite. We
investigate a low-dimensional stochastic differential equation where the ratio
between the decay rates of fast and slow variables is 2.
