Convergence theory for adaptive smoothed aggregation multigrid methods used in lattice QCD
In recent years adaptive smoothed aggregation algebraic multigrid ($\alpha$SA-AMG) methods have been developed and subsequently adapted for use in lattice quantum chromodynamics (QCD). The purpose of these efforts has been to reduce the critical slowdown that occurs in lattice QCD algorithms when working on state-of-the-art problems. Convergence theorems can establish the robustness of such methods and lead to ways to improve them. Most convergence theorems for $\alpha$SA-AMG methods require the underlying matrix to be Hermitian, which the lattice Wilson-Dirac matrix is not. A general two-level $\alpha$SA-AMG convergence theorem for non-Hermitian matrices is known, and it suggests convergence in O(k$^2$) iterations, where k is an error scaling constant. However, the general theory does not consider the $\Gamma_5$-symmetry of the Wilson-Dirac matrix, or the use of spin symmetry in adaptive smoothed aggregation procedures. This paper considers both $\Gamma_5$ symmetry and spin symmetry in analyzing the convergence of two recently introduced lattice QCD algebraic multigrid methods. One of these is an $\alpha$SA-AMG method, and the other is a bootstrap multigrid method. The conclusion reached is that both of these methods should converge in O(k) iterations, the same order of magnitude that would be expected for Hermitian matrices. The question of whether it is preferable to use singular vectors or eigenvectors to build intergrid interpolation operators for multigrid methods is reexamined. The investigation here suggests that both may lead to similar convergence results.
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Algorithms and Machines