Overlap valence on<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:math>flavor domain wall fermion configurations with deflation and low-mode substitution

The overlap fermion propagator is calculated on $2+1$ flavor domain-wall fermion gauge configurations on ${16}^{3}\ifmmode\times\else\texttimes\fi{}32$, ${24}^{3}\ifmmode\times\else\texttimes\fi{}64$ and ${32}^{3}\ifmmode\times\else\texttimes\fi{}64$ lattices. With hyper-cubic (HYP) smearing and low eigenmode deflation, it is shown that the inversion of the overlap operator can be expedited by $\ensuremath{\sim}20$ times for the ${16}^{3}\ifmmode\times\else\texttimes\fi{}32$ lattice and $\ensuremath{\sim}80$ times for the ${32}^{3}\ifmmode\times\else\texttimes\fi{}64$ lattice. The overhead cost for calculating eigenmodes ranges from 4.5 to 7.9 propagators for the above lattices. Through the study of hyperfine splitting, we found that the $O({m}^{2}{a}^{2})$ error is small and these dynamical fermion lattices can adequately accommodate quark mass up to the charm quark. A preliminary calculation of the low-energy constant ${\ensuremath{\Delta}}_{\mathrm{mix}}$ which characterizes the discretization error of the pion made up of a pair of sea and valence quarks in this mixed-action approach is carried out via the scalar correlator with periodic and antiperiodic boundary conditions. It is found to be small which shifts a 300 MeV pion mass by $\ensuremath{\sim}10$ to 19 MeV on these sets of lattices. We have studied the signal-to-noise issue of the noise source for the meson and baryon. We introduce a new algorithm with ${Z}_{3}$ grid source and low eigenmode substitution to study the many-to-all meson and baryon correlators. It is found to be efficient in reducing errors for the correlators of both mesons and baryons. With 64-point ${Z}_{3}$ grid source and low-mode substitution, it can reduce the statistical errors of the light quark (${m}_{\ensuremath{\pi}}\ensuremath{\sim}200--300\text{ }\text{ }\mathrm{MeV}$) meson and nucleon correlators by a factor of $\ensuremath{\sim}3--4$ as compared to the point source. The ${Z}_{3}$ grid source itself can reduce the errors of the charmonium correlators by a factor of $\ensuremath{\sim}3$.