Efficient solutions of large-scale, ill-conditioned and indefinite algebraic equations are ubiquitously needed in numerous computational fields, including multiphysics simulations, machine learning, and data science. Because of their robustness and accuracy, direct solvers are crucial components in building a scalable solver toolchain. In this article, we will review recent advances of sparse direct solvers along two axes: 1) reducing communication and latency costs in both task- and data-parallel settings, and 2) reducing computational complexity via low-rank and other compression techniques such as hierarchical matrix algebra. In addition to algorithmic principles, we also illustrate the key parallelization challenges and best practices to deliver high speed and reliability on modern heterogeneous parallel machines.

翻译：大规模、病态及不定代数方程的高效求解在众多计算领域中普遍存在需求，包括多物理场仿真、机器学习与数据科学。由于其鲁棒性与精确性，直接求解器在构建可扩展求解器工具链中至关重要。本文将从两个维度综述稀疏直接求解器的最新进展：1）在任务并行与数据并行环境中降低通信与延迟开销；2）通过低秩及其他压缩技术（如层次矩阵代数）降低计算复杂度。除算法原理外，本文还将阐述关键并行化挑战及在现代异构并行机器上实现高速度与可靠性的最佳实践。