The multigrid V-cycle method is a popular method for solving systems of linear equations. It computes an approximate solution by using smoothing on fine levels and solving a system of linear equations on the coarsest level. Solving on the coarsest level depends on the size and difficulty of the problem. If the size permits, it is typical to use a direct method based on LU or Cholesky decomposition. In the settings with large coarsest-level problems approximate solvers such as iterative Krylov subspace methods, or direct methods based on low-rank approximation, are often used. The accuracy of the coarsest-level solver is typically determined based on the experience of the users with the concrete problems and methods. In this paper we present an approach to analyzing the effects of approximate coarsest-level solves on the convergence of the V-cycle method for symmetric positive definite problems. Using this approach we discuss how the convergence of the V-cycle method may be affected by (1) the choice of the tolerance in a stopping criterion based on the relative residual norm for an iterative coarsest-level solver or (2) by the choices of the low-rank threshold parameter and finite precision arithmetic for a block low-rank direct coarsest-level solver.Furthermore we present new coarsest-level stopping criteria tailored to the multigrid method and suggest a heuristic strategy for their effective use in practical computations.

翻译：多重网格V-cycle方法是求解线性方程组的一种流行方法。它通过在细网格层进行平滑处理并在最粗网格层求解线性方程组来获得近似解。最粗网格层的求解取决于问题的规模和难度。若规模允许，通常使用基于LU或Cholesky分解的直接求解法。当最粗网格问题规模较大时，常采用近似求解器，如迭代Krylov子空间方法或基于低秩近似的直接求解法。最粗网格求解器的精度通常由用户根据具体问题和方法的经验确定。本文提出一种分析对称正定问题中近似最粗层求解对V-cycle方法收敛性影响的方法。利用该方法，我们讨论了V-cycle方法的收敛性如何受到以下因素影响：(1)基于相对残差范数的迭代最粗层求解器终止准则中容差的选择，或(2)块低秩直接最粗层求解器中低秩阈值参数和有限精度算术的选择。此外，我们提出了针对多重网格方法定制的新型最粗层终止准则，并给出在实际计算中有效使用这些准则的启发式策略。