Hardware trends have motivated the development of mixed precision algo-rithms in numerical linear algebra, which aim to decrease runtime while maintaining acceptable accuracy. One recent development is the development of an adaptive precision sparse matrix-vector produce routine, which may be used to accelerate the solution of sparse linear systems by iterative methods. This approach is also applicable to the application of inexact preconditioners, such as sparse approximate inverse preconditioners used in Krylov subspace methods. In this work, we develop an adaptive precision sparse approximate inverse preconditioner and demonstrate its use within a five-precision GMRES-based iterative refinement method. We call this algorithm variant BSPAI-GMRES-IR. We then analyze the conditions for the convergence of BSPAI-GMRES-IR, and determine settings under which BSPAI-GMRES-IR will produce similar backward and forward errors as the existing SPAI-GMRES-IR method, the latter of which does not use adaptive precision in preconditioning. Our numerical experiments show that this approach can potentially lead to a reduction in the cost of storing and applying sparse approximate inverse preconditioners, although a significant reduction in cost may comes at the expense of increasing the number of GMRES iterations required for convergence.

翻译：硬件发展趋势推动了数值线性代数中混合精度算法的发展，旨在降低运行时间的同时保持可接受的精度。近期的一项重要进展是开发了自适应精度稀疏矩阵-向量乘积例程，该例程可用于通过迭代方法加速稀疏线性系统的求解。该方法同样适用于非精确预条件子的应用，例如Krylov子空间方法中使用的稀疏近似逆预条件子。本文开发了一种自适应精度稀疏近似逆预条件子，并展示了其在基于五精度GMRES的迭代精化方法中的应用，我们称该算法变体为BSPAI-GMRES-IR。随后分析了BSPAI-GMRES-IR的收敛条件，确定了其在何种设置下能够与现有不使用自适应精度预条件的SPAI-GMRES-IR方法产生相似的后向误差和前向误差。数值实验表明，该方法可能降低存储和应用稀疏近似逆预条件子的成本，但成本的显著降低可能以增加收敛所需GMRES迭代次数为代价。