With the emergence of mixed precision capabilities in hardware, iterative refinement schemes for solving linear systems $Ax=b$ have recently been revisited and reanalyzed in the context of three or more precisions. These new analyses show that under certain constraints on condition number, the LU factorization of the matrix can be computed in low precision without affecting the final accuracy. Another promising technique is GMRES-based iterative refinement, which, in contrast to the standard approach, use GMRES preconditioned by the low-precision triangular factors to solve for the approximate solution update in each refinement step. This more accurate solution method extends the range of problems which can be solved with a given combination of precisions. However, in certain settings, GMRES may require too many iterations per refinement step, making it potentially more expensive than simply recomputing the LU factors in a higher precision. Krylov subspace recycling is a well-known technique for reusing information across sequential invocations of a Krylov subspace method on systems with the same or a slowly changing coefficient matrix. In this work, we incorporate the idea of Krylov subspace recycling into a mixed precision GMRES-based iterative refinement solver. The insight is that in each refinement step, we call preconditioned GMRES on a linear system with the same coefficient matrix $A$, with only the right-hand side changing. In this way, the GMRES solves in subsequent refinement steps can be accelerated by recycling information obtained from the first step. We perform extensive numerical experiments on various random dense problems, Toeplitz problems (prolate matrices), and problems from real applications, which confirm the benefits of the recycling approach.
翻译:随着硬件精密能力混杂的出现,最近对解决线性系统的迭代精炼计划进行了重新审视,并在三个或三个以上精准的范畴内重新分析。这些新的分析表明,在某些条件数目的限制下,可以低精确度计算矩阵的LU因子化,而不会影响最终精确度。另一个有希望的技术是基于GMRES的迭代精炼,与标准方法不同,利用低精度三角因素的GMRES,解决每个改进步骤的近似解决方案更新。这一更准确的解决方案方法扩大了可以通过精确度组合解决的一系列问题。然而,在某些环境下,GMRES可能需要在条件数目上进行过多的迭代,从而可能比简单地将LU的因子因子化因素重新精确度计算出更高的精确度。 Krylov 子空间再循环是一种广为人所熟知的方法,用来在每组使用Krylov 子空间方法进行相同的或缓慢变化的系数矩阵更新。在这项工作中,我们把Krylov 子空间循环循环循环循环循环的回收概念概念的概念扩大到一个混合的精细化方法。在每步级的GMRES的精炼方法上,我们只能在不断精确的精细化的根基的精确的根基的根基的根基的根基的根基的根基的根基的根基的根基的根基的根基的根基的根基的根基的根基的根基的根基的根基的根基的根的根的根的根的根基的根基的根的根基的根基的根基的根基的根基炼。