论文标题
混合精度迭代精致和稀疏的近相预处理
Mixed Precision Iterative Refinement with Sparse Approximate Inverse Preconditioning
论文作者
论文摘要
借助混合精度硬件的商业可用性，基于混合精度的迭代精制精制方案已成为求解稀疏线性系统的流行方法。但是，对这些方法的现有分析是基于使用完整的LU因素化来构建预处理程序，以在每个细化步骤中在GMRE中使用。在实际应用中，经常出于性能原因，通常使用不精确的预处理技术，例如不完整的LU或稀疏近似倒置。 在这项工作中，我们研究了基于Frobenius Norm最小化基于GMRES的迭代改进的稀疏近似反向调节器的使用。我们以有限的精度分析了稀疏近似逆的计算，并得出了满足用户指定停止标准的约束。然后，我们分析了使用稀疏的近似逆预处理的五个精确循环的改进方案的行为和收敛约束的行为，我们称为SPAI-GMRES-IR。我们的数值实验证实了理论分析，并说明了预处理器的稀疏性与GMRES-IR收敛速率之间的权衡。
With the commercial availability of mixed precision hardware, mixed precision GMRES-based iterative refinement schemes have emerged as popular approaches for solving sparse linear systems. Existing analyses of these approaches, however, are based on using full LU factorizations to construct preconditioners for use within GMRES in each refinement step. In practical applications, inexact preconditioning techniques, such as incomplete LU or sparse approximate inverses, are often used for performance reasons. In this work, we investigate the use of sparse approximate inverse preconditioners based on Frobenius norm minimization within GMRES-based iterative refinement. We analyze the computation of sparse approximate inverses in finite precision and derive constraints under which user-specified stopping criteria will be satisfied. We then analyze the behavior of and convergence constraints for a five-precision GMRES-based iterative refinement scheme that uses sparse approximate inverse preconditioning, which we call SPAI-GMRES-IR. Our numerical experiments confirm the theoretical analysis and illustrate the resulting tradeoffs between preconditioner sparsity and GMRES-IR convergence rate.
