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

TL;DR

The paper tackles the efficiency of solving linear systems with the Wilson-Dirac operator in lattice QCD by refining the multigrid setup. It generates a large, smoother-based test-vector basis and then applies a chiral rank-$k$ truncation via singular value decomposition to form the prolongation and restriction operators, preserving the chiral structure. Across two lattice ensembles—an anisotropic Hadron Spectrum configuration and an isotropic MILC configuration—the approach yields faster solves, identifies near-optimal basis sizes and truncation levels, and demonstrates robust volume scaling. The method reduces sensitivity to setup iterations, offering a practical improvement for large-scale Wilson fermion simulations relevant to exascale computing.

Abstract

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 $m_π \approx 239$ MeV from the Hadron Spectrum Collaboration and an isotropic lattice with $m_π \approx 220$ MeV from the MILC Collaboration. The lattice volume dependence of the method is also examined.

Figures (7)

• Figure 1: The mean solve time as a function of the initial basis size for Ensemble A (left) and Ensemble B (right). The bottom and top $y$-axis displays $m$ for levels $\ell = 0,1$, respectively.
• Figure 2: The singular spectrum for the chirally split test vectors on the first domain for level $\ell = 0$ (top) and $\ell = 1$ (bottom) of Ensemble A (left) and Ensemble B (right).
• Figure 3: The mean execution time for the system of linear equations when the degree of the truncation is varied using an initial basis size of $m = 96$ for Ensemble A (left) and Ensemble B (right). The bottom and top $y$-axis displays $k$ for levels $\ell = 0,1$, respectively.
• Figure 4: The total number of iterations on $\ell = 0$ (upper left), $\ell = 1$ (upper right), $\ell = 2$ (lower left) and the mean solve time of the system of linear equations for Ensemble A.
• Figure 5: As Fig. \ref{['fig::iters_aniso']} for Ensemble B.