论文标题
线性离散不足的问题的Krylov子空间，低等级近似值和LSQR的RITZ值：多重奇异值情况
The Krylov Subspaces, Low Rank Approximations and Ritz Values of LSQR for Linear Discrete Ill-Posed Problems: the Multiple Singular Value Case
论文作者
论文摘要
For the large-scale linear discrete ill-posed problem $\min\|Ax-b\|$ or $Ax=b$ with $b$ contaminated by white noise, the Golub-Kahan bidiagonalization based LSQR method and its mathematically equivalent CGLS, the Conjugate Gradient (CG) method applied to $A^TAx=A^Tb$, are most commonly used.它们具有内在的正规化效果，其中迭代数字$ k $扮演正规化参数的作用。长期以来的基本问题是：{\ em LSQR和CGL可以找到2-Norm过滤最佳的正则化解决方案}？当$ a $的单数值很简单时，作者就此问题给出了确定的答案。本文将结果扩展到了多个单数值情况，并研究了Krylov子空间的近似准确性，Golub-Kahan Biidiagonalization产生的低级近似值的质量以及RITZ值的收敛性。对于这两种问题，我们证明LSQR在半连接时发现了最佳的正规化解决方案。特别是，我们考虑了一些最佳，最佳和一般排名$ k $ a $ a $ a $ a $的重要和未触及的问题。数值实验证实了我们的理论。一般排名$ k $近似的结果及其非零奇异值的属性适用于多个Krylov求解器，包括LSQR，CGME，Minres，Minres，MR-II，MR-II，GMRES和RRGMRES。
For the large-scale linear discrete ill-posed problem $\min\|Ax-b\|$ or $Ax=b$ with $b$ contaminated by white noise, the Golub-Kahan bidiagonalization based LSQR method and its mathematically equivalent CGLS, the Conjugate Gradient (CG) method applied to $A^TAx=A^Tb$, are most commonly used. They have intrinsic regularizing effects, where the iteration number $k$ plays the role of regularization parameter. The long-standing fundamental question is: {\em Can LSQR and CGLS find 2-norm filtering best possible regularized solutions}? The author has given definitive answers to this question for severely and moderately ill-posed problems when the singular values of $A$ are simple. This paper extends the results to the multiple singular value case, and studies the approximation accuracy of Krylov subspaces, the quality of low rank approximations generated by Golub-Kahan bidiagonalization and the convergence properties of Ritz values. For the two kinds of problems, we prove that LSQR finds 2-norm filtering best possible regularized solutions at semi-convergence. Particularly, we consider some important and untouched issues on best, near best and general rank $k$ approximations to $A$ for the ill-posed problems with the singular values $σ_k=\mathcal{O}(k^{-α})$ with $α>0$, and the relationships between them and their nonzero singular values. Numerical experiments confirm our theory. The results on general rank $k$ approximations and the properties of their nonzero singular values apply to several Krylov solvers, including LSQR, CGME, MINRES, MR-II, GMRES and RRGMRES.
