  One-particle Green's functions obtained from the self-consistent solution of
the Dyson equation can be employed in evaluation of spectroscopic and
thermodynamic properties for both molecules and solids. However, typical
acceleration techniques used in the traditional quantum chemistry
self-consistent algorithms cannot be easily deployed for the Green's function
methods, because of non-convex grand potential functional and non-idempotent
density matrix. Moreover, the inclusion of correlation effects in the form of
the self-energy matrix and changing chemical potential or fluctuations in the
number of particles can make the optimization problem more difficult. In this
paper, we study acceleration techniques to target the self-consistent solution
of the Dyson equation directly. We use the direct inversion in the iterative
subspace (DIIS), the least-squared commutator in the iterative subspace
(LCIIS), and the Krylov space accelerated inexact Newton method (KAIN). We
observe that the definition of the residual has a significant impact on the
convergence of the iterative procedure. Based on the Dyson equation, we
generalize the concept of the commutator residual used in DIIS (CDIIS) and
LCIIS, and compare it with the difference residual used in DIIS and KAIN. The
commutator residuals outperform the difference residuals for all considered
molecular and solid systems within both GW and GF2. The generalized CDIIS and
LCIIS methods successfully converged restricted GF2 calculations for a number
of strongly correlated systems, which could not be converged before. We also
provide practical recommendations to guide convergence in such pathological
cases.
