论文标题
稳定的GMRE方法，用于线性方程式的单数和严重条件系统
A stabilized GMRES method for singular and severely ill-conditioned systems of linear equations
论文作者
论文摘要
考虑使用右键基本的GMRE（AB-GMRE）来获得线性方程不一定的不一定系统的最小值解决方案。 Morikuni（博士学位论文，2013年）表明，对于一些不一致且条件不足的问题，迭代可能会分歧。这主要是因为GMRES方法中的Hessenberg矩阵变得非常不良条件，因此所产生的三角形系统的向后取代在数值上变得不稳定。我们建议使用标准的Cholesky分解来求解与上述三角系统相对应的正常方程，提出了稳定的GMRE。这具有将赫森伯格基质的微小奇异值向上移动的效果，这导致了不准确的溶液。我们分析了该方法为什么起作用。数值实验表明，该提出的方法不仅用于将AB-GMRE应用于不确定的系统，而且还用于将GMRE应用于严重的线性线性方程的严重条件范围对称系统。
Consider using the right-preconditioned GMRES (AB-GMRES) for obtaining the minimum-norm solution of inconsistent underdetermined systems of linear equations. Morikuni (Ph.D. thesis, 2013) showed that for some inconsistent and ill-conditioned problems, the iterates may diverge. This is mainly because the Hessenberg matrix in the GMRES method becomes very ill-conditioned so that the backward substitution of the resulting triangular system becomes numerically unstable. We propose a stabilized GMRES based on solving the normal equations corresponding to the above triangular system using the standard Cholesky decomposition. This has the effect of shifting upwards the tiny singular values of the Hessenberg matrix which lead to an inaccurate solution. We analyze why the method works. Numerical experiments show that the proposed method is robust and efficient, not only for applying AB-GMRES to underdetermined systems, but also for applying GMRES to severely ill-conditioned range-symmetric systems of linear equations.
