Multigrid methods are popular iterative methods for solving large-scale sparse systems of linear equations. We present a mixed precision formulation of the multigrid V-cycle with general assumptions on the finite precision errors coming from the application of coarsest-level solver and smoothing. Inspired by existing analysis, we derive a bound on the relative finite precision error of the V-cycle which gives insight into how the finite precision errors from the individual components of the method may affect the overall finite precision error. We use the result to study V-cycle methods with smoothing based on incomplete Cholesky factorization. The results imply that in certain settings the precisions used for applying the IC smoothing can be significantly lower than the precision used for computing the residual, restriction, prolongation and correction on the concrete level. We perform numerical experiments using simulated floating point arithmetic with the MATLAB Advanpix toolbox as well as experiments computed on GPUs using the Ginkgo library. The experiments illustrate the theoretical findings and show that in the considered settings the IC smoothing can be applied in relatively low precisions, resulting in significant speedups (up to 1.43x) and energy savings (down to 71%) in comparison with the uniform double precision variant.

翻译：多重网格方法是求解大规模稀疏线性方程组的常用迭代方法。本文提出了一种混合精度的多重网格V循环格式，对来自最粗层求解器与光滑化应用的有限精度误差作了广义假设。受现有分析启发，我们推导了V循环相对有限精度误差的界，揭示了方法各组成部分的有限精度误差如何影响整体有限精度误差。利用该结果，我们研究了基于不完全Cholesky分解光滑化的V循环方法。结果表明，在某些设定下，应用IC光滑化所需的精度可显著低于具体层级上计算残差、限制算子、延拓算子及校正量所需的精度。我们使用MATLAB Advanpix工具箱的模拟浮点运算进行数值实验，并利用Ginkgo库在GPU上开展计算实验。实验验证了理论发现，表明在所考虑设定下，IC光滑化可在相对较低的精度下实施，与统一双精度版本相比，能实现显著的加速（最高达1.43倍）与能耗节约（最低至71%）。