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For the large-scale linear discrete ill-posed problem $\min\|Ax-b\|$ or$Ax=b$ with $b$ contaminated by a white noise, the Lanczos bidiagonalizationbased LSQR method and its mathematically equivalent Conjugate Gradient (CG)method for $A^TAx=A^Tb$ are most commonly used. They have intrinsicregularizing effects, where the number $k$ of iterations plays the role ofregularization parameter. However, there has been no answer to thelong-standing fundamental concern by Bj\"{o}rck and Eld\'{e}n in 1979: forwhich kinds of problems LSQR and CGLS can find best possible regularizedsolutions? Here a best possible regularized solution means that it is at leastas accurate as the best regularized solution obtained by the truncated singularvalue decomposition (TSVD) method or standard-form Tikhonov regularization. Inthis paper, assuming that the singular values of $A$ are simple, we analyze theregularization of LSQR for severely, moderately and mildly ill-posed problems.We establish accurate estimates for the 2-norm distance between the underlying$k$-dimensional Krylov subspace and the $k$-dimensional dominant right singularsubspace of $A$. For the first two kinds of problems, we then prove that LSQRfinds a best possible regularized solution at semi-convergence occurring atiteration $k_0$ and that, for $k=1,2,\ldots,k_0$, (i) the $k$-step Lanczosbidiagonalization always generates a near best rank $k$ approximation to $A$;(ii) the $k$ Ritz values always approximate the first $k$ large singular valuesin natural order; (iii) the $k$-step LSQR always captures the $k$ dominant SVDcomponents of $A$. For the third kind of problem, we prove that LSQR generallycannot find a best possible regularized solution. Numerical experiments confirmour theory.

The regularization theory of the Krylov iterative solvers LSQR and CGLS  for linear discrete ill-posed problems, part I: the simple singular value  case