Chiral rank-$k$ truncations for the multigrid preconditioner of Wilson fermions in lattice QCD

We present a modification to the setup algorithm for the multigrid preconditioner of Wilson fermions in lattice QCD. A larger number of test vectors than that used in conventional multigrid is generated by the smoother. This set of test vectors is then truncated by a singular value decomposition on the chiral components of the test vectors, which are subsequently used to form the prolongation and restriction matrices of the multigrid hierarchy. This modification is demonstrated to improve the convergence of linear equations on an anisotropic lattice with $m_{\pi} \approx 239$ MeV from the Hadron Spectrum Collaboration and an isotropic lattice with $m_{\pi} \approx 220$ MeV from the MILC Collaboration. The lattice volume dependence of the method is also examined.