  We discuss all-to-all quark propagator techniques in two (related) contexts
within Lattice QCD: the computation of closed quark propagators, and
applications to the so-called "eye diagrams" appearing in the computation of
non-leptonic kaon decay amplitudes. Combinations of low-mode averaging and
diluted stochastic volume sources that yield optimal signal-to-noise ratios for
the latter problem are developed. We also apply a recently proposed probing
algorithm to compute directly the diagonal of the inverse Dirac operator, and
compare its performance with that of stochastic methods. At fixed computational
cost the two procedures yield comparable signal-to-noise ratios, but probing
has practical advantages which make it a promising tool for a wide range of
applications in Lattice QCD.
