2020/03/19 by Zhongxiao Jia · 1 citation
Mathematics · Computer Science · #math.NA #cs.NA #msc:65F22 #msc:65R32 #msc:15A18 #msc:65J20 #msc:65F10 #msc:65F20
paper · pdf · doi:10.1088/1361-6420/ab9c45
published as Inverse Problems, 2020 · 31 pages, 5 figures. arXiv admin note: text overlap with arXiv:1608.05907, arXiv:1805.10132, arXiv:1701.05708, arXiv:1811.03454
arxiv created 2020/03/19 · arxiv updated 2020/07/21
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 ATAx=ATb, 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 σk=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.