  In this paper we propose a novel class of methods for high order accurate
integration of multirate systems of ordinary differential equation
initial-value problems. The proposed methods construct multirate schemes by
approximating the action of matrix $\varphi$-functions within explicit
exponential Rosenbrock (ExpRB) methods, thereby called Multirate Exponential
Rosenbrock (MERB) methods. They consist of the solution to a sequence of
modified "fast" initial-value problems, that may themselves be approximated
through subcycling any desired IVP solver. In addition to proving how to
construct MERB methods from certain classes of ExpRB methods, we provide
rigorous convergence analysis of these methods and derive efficient MERB
schemes of orders two through six (the highest order ever constructed
infinitesimal multirate methods). We then present numerical simulations to
confirm these theoretical convergence rates, and to compare the efficiency of
MERB methods against other recently-introduced high order multirate methods.
