  We present an approach for GW calculations of quasiparticle energies with
quasi-quadratic scaling by approximating high-energy contributions to the
Green's function in its Lehmann representation with effective stochastic
vectors. The method is easy to implement without altering the GW code,
converges rapidly with stochastic parameters, and treats systems of various
dimensionality and screening response. Our calculations on a 5.75$^\circ$
twisted MoS$_2$ bilayer show how large-scale GW methods include geometry
relaxations and electronic correlations on an equal basis in structurally
nontrivial materials.
