  The generalized Langevin equation is a model for the motion of coarse-grained
particles where dissipative forces are represented by a memory term. The
numerical realization of such a model requires the implementation of a
stochastic delay-differential equation and the estimation of a corresponding
memory kernel. Here we develop a new approach for computing a data-driven
Markov model for the motion of the particles, given equidistant samples of
their velocity autocorrelation function. Our method bypasses the determination
of the underlying memory kernel by representing it via up to about twenty
auxiliary variables. The algorithm is based on a sophisticated variant of the
Prony method for exponential interpolation and employs the Positive Real Lemma
from model reduction theory to extract the associated Markov model. We
demonstrate the potential of this approach for the test case of anomalous
diffusion, where data are given analytically, and then apply our method to
velocity autocorrelation data of molecular dynamics simulations of a colloid in
a Lennard-Jones fluid. In both cases, the VACF and the memory kernel can be
reproduced very accurately. Moreover, we show that the algorithm can also
handle input data with large statistical noise. We anticipate that it will be a
very useful tool in future studies that involve dynamic coarse-graining of
complex soft matter systems.
