Structured dense matrices result from boundary integral problems in electrostatics and geostatistics, and also Schur complements in sparse preconditioners such as multi-frontal methods. Exploiting the structure of such matrices can reduce the time for dense direct factorization from $O(N^3)$ to $O(N)$. The Hierarchically Semi-Separable (HSS) matrix is one such low rank matrix format that can be factorized using a Cholesky-like algorithm called ULV factorization. The HSS-ULV algorithm is highly parallel because it removes the dependency on trailing sub-matrices at each HSS level. However, a key merge step that links two successive HSS levels remains a challenge for efficient parallelization. In this paper, we use an asynchronous runtime system PaRSEC with the HSS-ULV algorithm. We compare our work with STRUMPACK and LORAPO, both state-of-the-art implementations of dense direct low rank factorization, and achieve up to 2x better factorization time for matrices arising from a diverse set of applications on up to 128 nodes of Fugaku for similar or better accuracy for all the problems that we survey.

中文翻译：

---

使用运行时系统对结构化密集矩阵进行 $O(N)$ 分布式直接分解

结构化密集矩阵源自静电学和地统计学中的边界积分问题，以及稀疏预处理器（例如多前沿方法）中的 Schur 补充。利用此类矩阵的结构可以将密集直接分解的时间从 $O(N^3)$ 减少到 $O(N)$。分层半可分离 (HSS) 矩阵就是一种低秩矩阵格式，可以使用称为 ULV 分解的类 Cholesky 算法进行分解。HSS-ULV 算法是高度并行的，因为它消除了对每个 HSS 级别的尾随子矩阵的依赖。然而，链接两个连续 HSS 级别的关键合并步骤仍然是高效并行化的挑战。在本文中，我们使用带有 HSS-ULV 算法的异步运行时系统 PaRSEC。我们将我们的工作与 STRUMPACK 和 LORAPO 进行比较，这两种方法都是最先进的密集直接低秩分解的实现，并且对于由 Fugaku 多达 128 个节点上的不同应用程序集产生的矩阵，实现了高达 2 倍的更好分解时间我们调查的所有问题的准确性相似或更高。