标题： 适用于任意奇异系统和最小二乘问题的NR-SSOR右预处理RRGMRES 显示英文标题

标题： NR-SSOR right preconditioned RRGMRES for arbitrary singular systems and least squares problems

摘要： GMRES 被认为可以确定$ A x = b $的最小二乘解，其中$ A \in R^{n \times n} $在任意$ b \in R^n $和初始迭代$ x_0 \in R^n $下不会出现中断，当且仅当$ A $是范围对称的，即 $ R(A^T) = R(A) $, where $ A $ may be singular and $ b $ may not be in the range space $ R(A) $ of $ A $. In this paper, we propose applying the Range Restricted GMRES (RRGMRES) to $ A C A^T z = b $, where $ C \in R^{n \times n} $ is symmetric positive definite. 这确定了在任意（奇异）矩阵$ A \in R^{n \times n} $和$ b, x_0 \in R^n $的情况下，$ A x = b $的最小二乘解$ x = C A^T z $不会中断，并且与将 GMRES、RRGMRES 和 MINRES-QLP 应用于$ A x = b $的不一致问题相比更加稳定和准确，当$ b \notin R(A) $时。 特别是，我们建议将NR-SSOR作为内部迭代右预条件器应用，这在最小二乘问题 $ \min_{x \in R^n} \| b - A x\|_2 $ 对于 $ A \in R^{m \times n} $ 和任意 $ b \in R^m $ 时也表现出高效的性能。数值实验验证了所提出方法的有效性。 显示英文摘要

摘要： GMRES is known to determine a least squares solution of $ A x = b $ where $ A \in R^{n \times n} $ without breakdown for arbitrary $ b \in R^n $, and initial iterate $ x_0 \in R^n $ if and only if $ A $ is range-symmetric, i.e. $ R(A^T) = R(A) $, where $ A $ may be singular and $ b $ may not be in the range space $ R(A) $ of $ A $. In this paper, we propose applying the Range Restricted GMRES (RRGMRES) to $ A C A^T z = b $, where $ C \in R^{n \times n} $ is symmetric positive definite. This determines a least squares solution $ x = C A^T z $ of $ A x = b $ without breakdown for arbitrary (singular) matrix $ A \in R^{n \times n} $ and $ b, x_0 \in R^n $, and is much more stable and accurate compared to GMRES, RRGMRES and MINRES-QLP applied to $ A x = b $ for inconsistent problems when $ b \notin R(A) $. In particular, we propose applying the NR-SSOR as the inner iteration right preconditioner, which also works efficiently for least squares problems $ \min_{x \in R^n} \| b - A x\|_2 $ for $ A \in R^{m \times n} $ and arbitrary $ b \in R^m $. Numerical experiments demonstrate the validity of the proposed method.