This article presents a fast and approximate multifrontal solver for large sparse linear systems. In a recent work by Liu et al., we showed the efficiency of a multifrontal solver leveraging the butterfly algorithm and its hierarchical matrix extension, HODBF (hierarchical off-diagonal butterfly) compression to compress large frontal matrices. The resulting multifrontal solver can attain quasi-linear computation and memory complexity when applied to sparse linear systems arising from spatial discretization of high-frequency wave equations. To further reduce the overall number of operations and especially the factorization memory usage to scale to larger problem sizes, in this article we develop a composite multifrontal solver that employs the HODBF format for large-sized fronts, a reduced-memory version of the nonhierarchical block low-rank format for medium-sized fronts, and a lossy compression format for small-sized fronts. This allows us to solve sparse linear systems of dimension up to 2.7 × larger than before and leads to a memory consumption that is reduced by 70% while ensuring the same execution time. The code is made publicly available in GitHub.

本文提出了一种适用于大型稀疏线性系统的快速近似多前沿求解器。在 Liu 等人最近的一项工作中，我们展示了利用蝴蝶算法及其分层矩阵扩展 HODBF（分层非对角蝴蝶）压缩来压缩大型额叶矩阵的多额求解器的效率。当应用于高频波动方程的空间离散化产生的稀疏线性系统时，所得的多前沿求解器可以获得准线性计算和存储复杂性。为了进一步减少操作总数，特别是分解内存使用量以扩展到更大的问题规模，在本文中，我们开发了一种复合多前沿求解器，该求解器针对大型前沿采用 HODBF 格式，用于中型前端的非分层块低等级格式的减少内存版本，以及用于小型前端的有损压缩格式。这使我们能够解决维度比以前大 2.7 倍的稀疏线性系统，并在保证相同执行时间的同时，内存消耗减少 70%。该代码在 GitHub 上公开发布。