Numerical simulations of quantum chromodynamics (QCD) on a lattice require the frequent solution of linear systems of equations with large, sparse and typically ill-conditioned matrices. Algebraic multigrid methods are meanwhile the standard for these difficult solves. Although the linear systems at the coarsest level of the multigrid hierarchy are much smaller than the ones at the finest level, they can be severely ill-conditioned, thus affecting the scalability of the whole solver. In this paper, we investigate different novel ways to enhance the coarsest-level solver and demonstrate their potential using DD-$\alpha$AMG, one of the publicly available algebraic multigrid solvers for lattice QCD. We do this for two lattice discretizations, namely clover-improved Wilson and twisted mass. For both the combination of two of the investigated enhancements, deflation and polynomial preconditioning, yield significant improvements in the regime of small mass parameters. In the clover-improved Wilson case we observe a significantly improved insensitivity of the solver to conditioning, and for twisted mass we are able to get rid of a somewhat artificial increase of the twisted mass parameter on the coarsest level used so far to make the coarsest level solves converge more rapidly.

翻译：格点量子色动力学(QCD)数值模拟需要频繁求解大规模稀疏且通常病态的线性方程组。代数多重网格方法已成为这类困难求解问题的标准算法。尽管多重网格层级中最粗层的线性系统规模远小于最精细层，但前者可能呈现严重病态，从而影响整个求解器的可扩展性。本文研究了多种增强粗层求解器的新方法，并利用公开的格点QCD代数多重网格求解器DD-$\alpha$AMG展示了其潜力。我们针对两种格点离散化方案（即clover-improved Wilson和twisted mass）开展研究。对于这两种方案，两种增强策略（即消去法和多项式预条件）的组合在小质量参数区间带来了显著改进。在clover-improved Wilson情况下，我们观察到求解器对条件数的敏感性显著降低；而对于twisted mass，我们成功消除了目前为使粗层求解更快收敛而人为增加粗层扭曲质量参数的现象。