We present a new class of statistical error reduction techniques for Monte-Carlo simulations. Using covariant symmetries, we show that correlation functions can be constructed from inexpensive approximations without introducing any systematic bias in
the final result. We introduce a new class of covariant approximation averaging techniques, known as all-mode averaging (AMA), in which the approximation takes account of contributions of all eigenmodes through the inverse of the Dirac operator computed from the conjugate gradient method with a relaxed stopping condition. In this paper we compare the performance and computational cost of our new method with traditional methods using correlation functions and masses of the pion, nucleon, and vector meson in $N_f=2+1$ lattice QCD using domain-wall fermions. This comparison indicates that AMA significantly reduces statistical errors in Monte-Carlo calculations over conventional methods for the same cost.
A method for computing renormalization constants in the Rome Southampton scheme with volume sources and arbitrary momenta is described. This new method is found to enable controlled and precise continuum extrapolations and opens the way to compute th
e running of operators nonperturbatively in the Rome Southampton scheme. We describe this in detail and exhibit several examples of lattice step scaling functions.
