  This paper is concerned with the regularization of large-scale discrete
inverse problems by means of inexact Krylov methods. Specifically, we derive
two new inexact Krylov methods that can be efficiently applied to unregularized
or Tikhonov-regularized least squares problems, and we study their theoretical
properties, including links with their exact counterparts and strategies to
monitor the amount of inexactness. We then apply the new methods to separable
nonlinear inverse problems arising in blind deblurring. In this setting
inexactness stems from the uncertainty in the parameters defining the blur,
which may be recovered using a variable projection method leading to an
inner-outer iteration scheme (i.e., one cycle of inner iterations is performed
to solve one linear deblurring subproblem for any intermediate values of the
blurring parameters computed by a nonlinear least squares solver). The new
inexact solvers can naturally handle varying inexact blurring parameters while
solving the linear deblurring subproblems, allowing for a much reduced number
of total iterations and substantial computational savings with respect to their
exact counterparts.
