  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.
