A Nested Krylov Method Using Half-Precision Arithmetic

TL;DR

This paper presents a nested Krylov method that employs half-precision arithmetic to significantly accelerate sparse linear solvers while maintaining convergence, demonstrating notable speedups over traditional methods.

Contribution

It introduces a novel nested Krylov approach integrating GMRES and Richardson methods with a multi-precision strategy, effectively leveraging fp16 for improved performance.

Findings

Achieves up to 2.42x speedup over double-precision implementations.

Maintains convergence while using fp16 arithmetic.

Outperforms standard Krylov solvers like GMRES, CG, and BiCGStab.

Abstract

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.