  Subspace recycling techniques have been used quite successfully for the
acceleration of iterative methods for solving large-scale linear systems. These
methods often work by augmenting a solution subspace generated iteratively by a
known algorithm with a fixed subspace of vectors which are ``useful'' for
solving the problem. Often, this has the effect of inducing a projected version
of the original linear system to which the known iterative method is then
applied, and this projection can act as a deflation preconditioner,
accelerating convergence. Most often, these methods have been applied for the
solution of well-posed problems. However, they have also begun to be considered
for the solution of ill-posed problems.
  In this paper, we consider subspace augmentation-type iterative schemes
applied to linear ill-posed problems in a continuous Hilbert space setting,
based on a recently developed framework describing these methods. We show that
under suitable assumptions, a recycling method satisfies the formal definition
of a regularization, as long as the underlying scheme is itself a
regularization. We then develop an augmented subspace version of the gradient
descent method and demonstrate its effectiveness, both on an academic Gaussian
blur model and on problems arising from the adaptive optics community for the
resolution of large sky images by ground-based extremely large telescopes.
