  For large-scale discrete ill-posed problems, LSQR, a Lanczos
bidiagonalization process based Krylov method, is most often used. It is well
known that LSQR has natural regularizing properties, where the number of
iterations plays the role of the regularization parameter. In this paper, for
severely and moderately ill-posed problems, we establish quantitative bounds
for the distance between the $k$-dimensional Krylov subspace and the subspace
spanned by $k$ dominant right singular vectors. They show that the
$k$-dimensional Krylov subspace may capture the $k$ dominant right singular
vectors for severely and moderately ill-posed problems, but it seems not the
case for mildly ill-posed problems. These results should be the first step
towards to estimating the accuracy of the rank-$k$ approximation generated by
Lanczos bidiagonalization. We also derive some other results, which help
further understand the regularization effects of LSQR. We draw to a conclusion
that a hybrid LSQR should generally be used for mildly ill-posed problems. We
report numerical experiments to confirm our theory. We present more definitive
and general observed phenomena, which will derive more research.
