一种使用半精度算术的嵌套Krylov方法
A Nested Krylov Method Using Half-Precision Arithmetic
AI总结：
提出一种深度嵌套的Krylov方法，集成灵活GMRES与Richardson，逐步降至fp16精度，在保持收敛的同时实现高达1.65倍和2.42倍的加速。
摘要：
低精度计算对于高效利用内存带宽和计算核心至关重要。尽管许多混合精度算法已被开发用于迭代稀疏线性求解器，但有效利用半精度（fp16）算术仍然具有挑战性。本研究提出了一种新颖的嵌套Krylov方法，该方法在深度嵌套结构中集成了灵活GMRES和Richardson方法，从双精度逐步降低精度至最内层求解器的fp16。为避免超出精度极限的无意义计算，低精度内层求解器每次调用仅执行少量迭代，而嵌套结构确保其频繁执行。数值实验表明，在该方法中使用fp16直接提升求解器性能而不损害收敛性，相较于双精度和双单精度混合实现，分别实现了高达1.65倍和2.42倍的加速。此外，所提方法优于或匹配其他标准Krylov求解器，包括重启的GMRES、CG和BiCGStab方法。
英文摘要：
Low-precision computing is essential for efficiently utilizing memory bandwidth and computing cores. While many mixed-precision algorithms have been developed for iterative sparse linear solvers, effectively leveraging half-precision (fp16) arithmetic remains challenging. This study introduces a novel nested Krylov approach that integrates the flexible GMRES and Richardson methods in a deeply nested structure, progressively reducing precision from double-precision to fp16 toward the innermost solver. To avoid meaningless computations beyond precision limits, the low-precision inner solvers perform only a few iterations per invocation, while the nested structure ensures their frequent execution. Numerical experiments show that using fp16 in the approach directly enhances solver performance without compromising convergence, achieving speedups of up to 1.65x and 2.42x over double-precision and double-single mixed-precision implementations, respectively. Moreover, the proposed method outperforms or matches other standard Krylov solvers, including restarted GMRES, CG, and BiCGStab methods.
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