可扩展的基于单层不完全$LDL^T$分解与谱修正的大规模稀疏系统近似选定求逆
Scalable Approximate Selected Inversion Based on Single-Level Incomplete $LDL^T$ Factorization and Spectral Corrections for Large Sparse Systems
- TU Braunschweig（布伦瑞克工业大学）
- Università della Svizzera italiana（瑞士意大利语大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出四种基于不完全LDL^T分解的并行近似求逆技术（SelInv、NInv、Mix和Mix-SPAI），并结合特征向量低秩修正，用于高效计算大规模稀疏对称系统逆矩阵元素，实验验证了其性能。
AI中文摘要：
本文介绍了四种用于计算大规模稀疏对称系统逆矩阵元素的并行数值技术，所有这些技术均基于不完全$LDL^T$（ILDL）分解：（1）选定求逆方法（SelInv），该方法利用$LDL^T$分解在计算因子的稀疏模式内恢复矩阵逆的元素；（2）一种基于对L因子逆应用截断Neumann级数展开的近似求逆方法（NInv），以精度降低为代价提供了一种替代方案；（3）一种融合SelInv和NInv两者优点的Mix近似方法；（4）Mix-SPAI，该方法对Mix方法的输出应用稀疏近似逆（SPAI）细化以提高逐元素精度。为了在保持稳定稀疏模式的同时进一步提高精度，我们还采用了基于特征向量更新的低秩修正，作为收紧丢弃容差的替代方案。我们报告了所提出的数值技术在一组来自科学和工业应用的综合稀疏矩阵上的性能。
英文摘要：
This article introduces four parallel numerical techniques for computing entries of the inverse of large sparse symmetric systems, all grounded in incomplete $LDL^T$ (ILDL) factorizations: (1) the selected inversion method (SelInv), which applies the $LDL^T$ factorization to recover entries of the matrix inverse within the sparsity pattern of the computed factors; (2) an approximate inversion method based on a truncated Neumann series expansion applied to the inverse of the L factor (NInv), providing an alternative at the cost of reduced accuracy; (3) a Mix approximation that merges the best of both SelInv and NInv; and (4) Mix-SPAI, which applies sparse approximate inverse (SPAI) refinement on the output of the Mix method to improve entry-level accuracy. To further improve accuracy while maintaining a stable sparsity pattern, we additionally employ a low-rank correction based on eigenvector updates, providing an alternative to tightening the drop tolerance. We report the performance of the proposed numerical techniques on a comprehensive collection of sparse matrices from scientific and industrial applications.
