Due to its optimal complexity, the multigrid (MG) method is one of the most popular approaches for solving large-scale linear systems arising from the discretization of partial differential equations. However, the parallel implementation of standard MG methods, which are inherently multiplicative, suffers from increasing communication complexity. In such cases, the additive variants of MG methods provide a good alternative due to their inherently parallel nature, although they exhibit slower convergence. This work combines the additive multigrid method with the multipreconditioned conjugate gradient (MPCG) method. In the proposed approach, the MPCG method employs the corrections from the different levels of the MG hierarchy as separate preconditioned search directions. In this approach, the MPCG method updates the current iterate by using the linear combination of the preconditioned search directions, where the optimal coefficients for the linear combination are computed by exploiting the energy norm minimization of the CG method. The idea behind our approach is to combine the $A$-conjugacy of the search directions of the MPCG method and the quasi $H_1$-orthogonality of the corrections from the MG hierarchy. In the numerical section, we study the performance of the proposed method compared to the standard additive and multiplicative MG methods used as preconditioners for the CG method.

翻译：由于具有最优复杂度，多重网格法是最常用于求解偏微分方程离散化所产生大规模线性系统的方法之一。然而，标准多重网格方法（本质上是乘法型）的并行实现面临日益增长的通信复杂度。在这种情况下，加法型多重网格变体因其固有的并行特性成为良好的替代方案，尽管其收敛速度较慢。本研究将加法多重网格方法与多预处理共轭梯度（MPCG）方法相结合。在所提出的方法中，MPCG方法将多重网格层级中不同层次产生的修正项作为独立的预条件搜索方向。该方法通过利用共轭梯度法的能量范数最小化来计算线性组合的最优系数，从而使用预条件搜索方向的线性组合对当前迭代解进行更新。本方法的核心思想在于融合MPCG方法搜索方向的$A$-共轭性与多重网格层级修正项的拟$H_1$-正交性。在数值实验部分，我们研究了所提方法与作为CG方法预条件子的标准加法型和乘法型多重网格方法相比的性能表现。