This article discusses parallel hierarchical-matrices ( $\mathcal {H}$ -matrices) to compute a demagnetizing field, which is the most time-consuming part in the micromagnetic simulation of current-induced domain wall motion (CDWM). Although normal $\mathcal {H}$ -matrices exhibit high efficiencies for small numbers of message passing interface (MPI) processes, the performance rapidly decays due to load imbalance and the MPI communication costs as the number of processes increases. We introduce lattice $\mathcal {H}$ -matrices to improve the parallel scalability, when using a large number of processes. The applicability of lattice $\mathcal {H}$ -matrices to CDWM simulations is confirmed and proper lattice block sizes and process grid shapes of the lattice $\mathcal {H}$ -matrices for $\mathcal {H}$ -matrix-vector products (HMVPs) are investigated using practical data sets. Under appropriate settings, the lattice $\mathcal {H}$ -matrices exhibit almost linear complexity in memory usage and calculation time of HMVPs. Our implementation continues to accelerate at least up to about 3600 MPI processes, even in a small problem with several tens of thousands of unknowns.

中文翻译：

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用于电流诱导畴壁运动的大规模平行微磁模拟的晶格 H 矩阵

本文讨论并行分层矩阵（ $\mathcal {H}$ -矩阵）来计算退磁场，这是电流诱导畴壁运动 (CDWM) 微磁模拟中最耗时的部分。虽然正常 $\mathcal {H}$ -matrices 对于少量消息传递接口 (MPI) 进程表现出很高的效率，由于负载不平衡和 MPI 通信成本随着进程数量的增加，性能会迅速下降。我们介绍格子 $\mathcal {H}$ -matrices 以提高并行可扩展性，当使用大量进程时。格子的适用性 $\mathcal {H}$ -CDWM 模拟的矩阵被确认，适当的晶格块尺寸和晶格的处理网格形状 $\mathcal {H}$ - 矩阵 $\mathcal {H}$ 使用实际数据集研究矩阵向量产品 (HMVP)。在适当的设置下，晶格 $\mathcal {H}$ - 矩阵在 HMVP 的内存使用和计算时间方面表现出几乎线性的复杂性。我们的实施继续加速至少约 3600 个 MPI 进程，即使是在具有数万个未知数的小问题中也是如此。