  For large scale symmetric discrete ill-posed problems, MINRES and MR-II are
often used iterative regularization solvers. We call a regularized solution
best possible if it is at least as accurate as the best regularized solution
obtained by the truncated singular value decomposition (TSVD) method. In this
paper, we analyze their regularizing effects and establish the following
results: (i) the filtered SVD expression are derived for the regularized
solutions by MINRES; (ii) a hybrid MINRES that uses explicit regularization
within projected problems is needed to compute a best possible regularized
solution to a given ill-posed problem; (iii) the $k$th iterate by MINRES is
more accurate than the $(k-1)$th iterate by MR-II until the semi-convergence of
MINRES, but MR-II has globally better regularizing effects than MINRES; (iv)
bounds are obtained for the 2-norm distance between an underlying
$k$-dimensional Krylov subspace and the $k$-dimensional dominant eigenspace.
They show that MR-II has better regularizing effects for severely and
moderately ill-posed problems than for mildly ill-posed problems, and a hybrid
MR-II is needed to get a best possible regularized solution for mildly
ill-posed problems; (v) bounds are derived for the entries generated by the
symmetric Lanczos process that MR-II is based on, showing how fast they decay.
Numerical experiments confirm our assertions. Stronger than our theory, the
regularizing effects of MR-II are experimentally shown to be good enough to
obtain best possible regularized solutions for severely and moderately
ill-posed problems.
