  We develop a new effective approximation of the Mori-Zwanzig equation based
on operator series expansions of the orthogonal dynamics propagator. In
particular, we study the Faber series, which yields asymptotically optimal
approximations converging at least $R$-superlinearly with the polynomial order
for linear dynamical systems. We provide a through theoretical analysis of the
new method and present numerical applications to random wave propagation and
harmonic chains of oscillators interacting on the Bethe lattice and on graphs
with arbitrary topology.
