Based on current trends in computer architectures, faster compute speeds must come from increased parallelism rather than increased clock speeds, which are currently stagnate. This situation has created the well-known bottleneck for sequential time-integration, where each individual time-value (i.e., time-step) is computed sequentially. One approach to alleviate this and achieve parallelism in time is with multigrid. In this work, we consider multigrid-reduction-in-time (MGRIT), a multilevel method applied to the time dimension that computes multiple time-steps in parallel. Like all multigrid methods, MGRIT relies on the complementary relationship between relaxation on a fine-grid and a correction from the coarse grid to solve the problem. All current MGRIT implementations are based on unweighted-Jacobi relaxation; here we introduce the concept of weighted relaxation to MGRIT. We derive new convergence bounds for weighted relaxation, and use this analysis to guide the selection of relaxation weights. Numerical results then demonstrate that non-unitary relaxation weights consistently yield faster convergence rates and lower iteration counts for MGRIT when compared with unweighted relaxation. In most cases, weighted relaxation yields a 10%-20% saving in iterations. For A-stable integration schemes, results also illustrate that under-relaxation can restore convergence in some cases where unweighted relaxation is not convergent.

翻译：根据计算机架构当前趋势,计算速度更快的速度必须来自目前停滞的更多平行现象,而不是更高的时钟速度。这种情况造成了众所周知的连续时间整合的瓶颈,每个单个时间值(即时间步)都是按顺序计算的。缓解这一问题并实现时间平行的方法之一是多格化。在这项工作中,我们考虑多格里减少时间(MGRIIT)这一适用于同时计算多个时间步骤的时间趋同层面的多层次方法。与所有多格方法一样,GRIIT依靠在细格放松和粗格中纠正解决问题之间的互补关系。目前所有MGRIIT的实施都以非加权-Jacobi放松为基础;这里我们引入了MGRIT的加权放松概念。我们从加权放松中得出新的趋同界限,并使用这一分析来指导放松重量的选择。随后的量化结果表明,不统一放松权重会持续提高一些不趋同率,而在最粗格的网格中,在最大幅度的趋同后,A级调整后,在最大幅度的压力调整后,A级调整后,在10调重的压压下,也算。