  Tikhonov regularization for projected solutions of large-scale ill-posed
problems is considered. The Golub-Kahan iterative bidiagonalization is used to
project the problem onto a subspace and regularization then applied to find a
subspace approximation to the full problem. Determination of the regularization
parameter for the projected problem by unbiased predictive risk estimation,
generalized cross validation and discrepancy principle techniques is
investigated. It is shown that the regularized parameter obtained by the
unbiased predictive risk estimator can provide a good estimate for that to be
used for a full problem which is moderately to severely ill-posed. A similar
analysis provides the weight parameter for the weighted generalized cross
validation such that the approach is also useful in these cases, and also
explains why the generalized cross validation without weighting is not always
useful. All results are independent of whether systems are over or
underdetermined. Numerical simulations for standard one dimensional test
problems and two dimensional data, for both image restoration and tomographic
image reconstruction, support the analysis and validate the techniques. The
size of the projected problem is found using an extension of a noise revealing
function for the projected problem Hn\u etynkov\'a, Ple\u singer, and Strako\u
s, [\textit{BIT Numerical Mathematics} {\bf 49} (2009), 4 pp. 669-696.].
Furthermore, an iteratively reweighted regularization approach for edge
preserving regularization is extended for projected systems, providing
stabilization of the solutions of the projected systems and reducing dependence
on the determination of the size of the projected subspace.
