  In recent years, it has become increasingly popular to construct
coarse-grained models with non-Markovian dynamics to account for an incomplete
separation of time scales. One challenge of a systematic coarse-graining
procedure is the extraction of the dynamical properties, namely, the memory
kernel, from equilibrium all-atom simulations. In this article, we propose an
iterative method for memory reconstruction from dynamical correlation
functions. Compared to previously proposed noniterative techniques, it ensures
by construction that the target correlation functions of the original
fine-grained systems are reproduced accurately by the coarse-grained system,
regardless of time step and discretization effects. Furthermore, we also
propose a new numerical integrator for generalized Langevin equations that is
significantly more accurate than the more commonly used generalization of the
velocity Verlet integrator. We demonstrate the performance of the
above-described methods using the example of backflow-induced memory in the
Brownian diffusion of a single colloid. For this system, we are able to
reconstruct realistic coarse-grained dynamics with time steps about 200 times
larger than those used in the original molecular dynamics simulations.
