The Krylov Subspaces, Low Rank Approximations and Ritz Values of LSQR for Linear Discrete Ill-Posed Problems: the Multiple Singular Value Case
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For the large-scale linear discrete ill-posed problem minAx-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: 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=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.
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