  Multirate methods have been used for decades to temporally evolve
initial-value problems in which different components evolve on distinct time
scales, and thus use of different step sizes for these components can result in
increased computational efficiency. Generally, such methods select these
different step sizes based on experimentation or stability considerations. For
problems that evolve on a single time scale, adaptivity approaches that strive
to control local temporal error are widely used to achieve numerical results of
a desired accuracy with minimal computational effort, while alleviating the
need for manual experimentation with different time step sizes. However, there
is a notable gap in the publication record on the development of adaptive
time-step controllers for multirate methods. In this paper, we extend the
single-rate controller work of Gustafsson (1994) to the multirate method
setting. Specifically, we develop controllers based on polynomial
approximations to the principal error functions for both the "fast" and "slow"
time scales within multirate infinitesimal (MRI) methods. We additionally
investigate a variety of approaches for estimating the errors arising from each
time scale within MRI methods. We then numerically evaluate the proposed
multirate controllers and error estimation strategies on a range of multirate
test problems, comparing their performance against an estimated optimal
performance. Through this work, we combine the most performant of these
approaches to arrive at a set of multirate adaptive time step controllers that
robustly achieve desired solution accuracy with minimal computational effort.
