  Recent developments in multiscale computation allow the solution of ``coarse
equations'' for the expected macroscopic behavior of
microscopically/stochastically evolving particle distributions without ever
obtaining these coarse equations in closed form. The closure is obtained ``on
demand'' through appropriately initialized bursts of microscopic simulation.
The effective coupling of microscopic simulators with macrosocopic behavior
embodied in this approach requires certain decisions about the nature of the
unavailable ``coarse equation''. Such decisions include (a) the determination
of the highest spatial derivative active in the equation, (b) whether the
coarse equation satisfies certain conservation laws, and (c) whether the coarse
dynamics is Hamiltonian. These decisions affect the number and type of boundary
conditions as well as the nature of the algorithms employed in good solution
practice. In the absence of an explicit formula for the temporal derivative, we
propose, implement and validate a simple scheme for deciding these and other
similar questions about the coarse equation using only the microscopic
simulator. Microscopic simulations under periodic boundary conditions are
carried out for appropriately chosen families of random initial conditions;
evaluating the sample variance of certain statistics over the simulation
ensemble allows us to infer the highest order of spatial derivatives active in
the coarse equation. In the same spirit we show how to determine whether a
certain coarse conservation law exists or not, and we discuss plausibility
tests for the existence of a coarse Hamiltonian or integrability. We argue that
such schemes constitute an important part of the equation-free approach to
multiscale computation.
