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：由于其最优复杂度，多重网格（MG）方法是求解由偏微分方程离散化产生的大规模线性系统最流行的方法之一。然而，标准MG方法本质上是乘法性的，其并行实现受到通信复杂度增加的影响。在这种情况下，MG方法的加法变体由于其固有的并行性提供了一个良好的替代方案，尽管其收敛速度较慢。本研究将加法多重网格方法与多重预处理共轭梯度（MPCG）方法相结合。在所提出的方法中，MPCG方法将来自MG层次结构不同层级的校正作为独立的预处理搜索方向。在此方法中，MPCG方法通过使用预处理搜索方向的线性组合来更新当前迭代，其中线性组合的最优系数通过利用CG方法的能量范数最小化来计算。我们方法背后的思想是结合MPCG方法搜索方向的$A$-共轭性与MG层级校正的拟$H_1$-正交性。在数值部分，我们研究了所提出方法与作为CG方法预处理器的标准加法及乘法MG方法相比的性能。