  We introduce a micro-macro parareal algorithm for the time-parallel
integration of multiscale-in-time systems. The algorithm first computes a
cheap, but inaccurate, solution using a coarse propagator (simulating an
approximate slow macroscopic model), which is iteratively corrected using a
fine-scale propagator (accurately simulating the full microscopic dynamics).
This correction is done in parallel over many subintervals, thereby reducing
the wall-clock time needed to obtain the solution, compared to the integration
of the full microscopic model. We provide a numerical analysis of the algorithm
for a prototypical example of a micro-macro model, namely singularly perturbed
ordinary differential equations. We show that the computed solution converges
to the full microscopic solution (when the parareal iterations proceed) only if
special care is taken during the coupling of the microscopic and macroscopic
levels of description. The convergence rate depends on the modeling error of
the approximate macroscopic model. We illustrate these results with numerical
experiments.
