Linear systems in applications are typically well-posed, and yet the coefficient matrices may be nearly singular in that the condition number $\kappa(\boldsymbol{A})$ may be close to $1/\varepsilon_{w}$, where $\varepsilon_{w}$ denotes the unit roundoff of the working precision. It is well known that iterative refinement (IR) can make the forward error independent of $\kappa(\boldsymbol{A})$ if $\kappa(\boldsymbol{A})$ is sufficiently smaller than $1/\varepsilon_{w}$ and the residual is computed in higher precision. We propose a new iterative method, called Forward-and-Backward Stabilized Minimal Residual or FBSMR, by conceptually hybridizing right-preconditioned GMRES (RP-GMRES) with quasi-minimization. We develop FBSMR based on a new theoretical framework of essential-forward-and-backward stability (EFBS), which extends the backward error analysis to consider the intrinsic condition number of a well-posed problem. We stabilize the forward and backward errors in RP-GMRES to achieve EFBS by evaluating a small portion of the algorithm in higher precision while evaluating the preconditioner in lower precision. FBSMR can achieve optimal accuracy in terms of both forward and backward errors for well-posed problems with unpolluted matrices, independently of $\kappa(\boldsymbol{A})$. With low-precision preconditioning, FBSMR can reduce the computational, memory, and energy requirements over direct methods with or without IR. FBSMR can also leverage parallelization-friendly classical Gram-Schmidt in Arnoldi iterations without compromising EFBS. We demonstrate the effectiveness of FBSMR using both random and realistic linear systems.

翻译：实际应用中的线性系统通常是适定的，然而其系数矩阵可能近乎奇异，条件数$\kappa(\boldsymbol{A})$可能接近$1/\varepsilon_{w}$，其中$\varepsilon_{w}$表示工作精度的单位舍入误差。众所周知，若$\kappa(\boldsymbol{A})$充分小于$1/\varepsilon_{w}$且残差以更高精度计算，则迭代精化（IR）可使前向误差独立于$\kappa(\boldsymbol{A})$。我们提出一种新型迭代方法——前后向稳定极小残差法（FBSMR），其核心思想是概念性地将右预处理GMRES（RP-GMRES）与拟最小化相结合。FBSMR基于本质前后向稳定性（EFBS）的新理论框架构建，该框架扩展了后向误差分析，将适定问题的固有条件数纳入考量。我们通过以更高精度执行算法中的小部分运算，同时以更低精度计算预处理子，从而稳定RP-GMRES中的前向和后向误差，实现EFBS。对于含未污染矩阵的适定问题，FBSMR可在前向和后向误差两方面均达到最优精度，且该精度独立于$\kappa(\boldsymbol{A})$。采用低精度预处理时，FBSMR相比直接法（无论是否结合IR）可降低计算、内存及能耗需求。此外，FBSMR可在不牺牲EFBS的前提下，利用支持并行化的经典Gram-Schmidt算法进行Arnoldi迭代。我们通过随机和实际线性系统验证了FBSMR的有效性。