Lattice H-matrices for Massively Parallel Micromagnetic Simulations of Current-induced Domain Wall Motion

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.