This study investigates the iterative regularization properties of two Krylov methods for solving large-scale ill-posed problems: the changing minimal residual Hessenberg method (CMRH) and a novel hybrid variant called the hybrid changing minimal residual Hessenberg method (H-CMRH). Both methods share the advantages of avoiding inner products, making them efficient and highly parallelizable, and particularly suited for implementations that exploit randomization and mixed precision arithmetic. Theoretical results and extensive numerical experiments suggest that H-CMRH exhibits comparable performance to the established hybrid GMRES method in terms of stabilizing semiconvergence, but H-CMRH has does not require any inner products, and requires less work and storage per iteration.

翻译：本研究探讨了两种用于求解大规模不适定问题的Krylov方法的迭代正则化特性：变化最小残差Hessenberg方法（CMRH）及其新型混合变体——混合变化最小残差Hessenberg方法（H-CMRH）。两种方法均具有避免内积运算的优点，因而计算高效、高度可并行化，尤其适合利用随机化与混合精度算术的实现方式。理论结果与大量数值实验表明，H-CMRH在稳定半收敛性方面与成熟的混合GMRES方法具有相当的性能，但H-CMRH完全不需要内积运算，且每次迭代所需的工作量和存储空间更少。