This study presents a novel mixed-precision iterative refinement algorithm, GADI-IR, within the general alternating-direction implicit (GADI) framework, designed for efficiently solving large-scale sparse linear systems. By employing low-precision arithmetic, particularly half-precision (FP16), for computationally intensive inner iterations, the method achieves substantial acceleration while maintaining high numerical accuracy. Key challenges such as overflow in FP16 and convergence issues for low precision are addressed through careful backward error analysis and the application of a regularization parameter $\alpha$. Furthermore, the integration of low-precision arithmetic into the parameter prediction process, using Gaussian process regression (GPR), significantly reduces computational time without degrading performance. The method is particularly effective for large-scale linear systems arising from discretized partial differential equations and other high-dimensional problems, where both accuracy and efficiency are critical. Numerical experiments demonstrate that the use of FP16 and mixed-precision strategies not only accelerates computation but also ensures robust convergence, making the approach advantageous for various applications. The results highlight the potential of leveraging lower-precision arithmetic to achieve superior computational efficiency in high-performance computing.

翻译：本研究提出了一种新颖的混合精度迭代精化算法——GADI-IR，该算法基于通用交替方向隐式（GADI）框架，旨在高效求解大规模稀疏线性系统。通过采用低精度算术（特别是半精度FP16）执行计算密集的内迭代，该方法在保持高数值精度的同时实现了显著加速。针对FP16中的溢出问题及低精度下的收敛难题，本研究通过细致的后向误差分析并结合正则化参数$\alpha$的应用予以解决。此外，将低精度算术集成至基于高斯过程回归（GPR）的参数预测过程中，可在不降低性能的前提下显著减少计算时间。该方法对于离散化偏微分方程及其他高维问题产生的大规模线性系统尤为有效，其中精度与效率均为关键考量。数值实验表明，采用FP16及混合精度策略不仅能加速计算，还能确保鲁棒收敛，使得该方法在多种应用场景中具有优势。研究结果凸显了利用低精度算术在高性能计算中实现卓越计算效率的潜力。