Faber approximation to the Mori-Zwanzig equation

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.