  We propose a strategy for large volume non-perturbative renormalization which
alleviates the window problem by reducing cut-off effects. We perform a
proof-of-concept study using position space renormalization scheme and the CLS
$N_f=2+1$ ensembles generated at 5 different lattice spacings. We show that in
the advocated strategy results for the renormalization constants are to a large
extend independent of the specific lattice direction used to define the
renormalization condition. Hence, very short lattice distances become
accessible even on coarse lattices and the contact with perturbation theory can
be performed at much higher energy scales. Our results include
non-perturbatively estimated renormalization constants for quark bilinear
operators in the scalar, pseudoscalar and axial-vector channels using position
space renormalization scheme which we subsequently translate to the
$\overline{\mathrm{MS}}$ scheme perturbatively at $1.5$ GeV. Our proposal is
applicable to other non-perturbative large volume renormalization schemes such
as RI-MOM and its variants.
