  In many modern imaging applications the desire to reconstruct high resolution
images, coupled with the abundance of data from acquisition using ultra-fast
detectors, have led to new challenges in image reconstruction. A main challenge
is that the resulting linear inverse problems are massive. The size of the
forward model matrix exceeds the storage capabilities of computer memory, or
the observational dataset is enormous and not available all at once. Row-action
methods that iterate over samples of rows can be used to approximate the
solution while avoiding memory and data availability constraints. However,
their overall convergence can be slow. In this paper, we introduce a sampled
limited memory row-action method for linear least squares problems, where an
approximation of the global curvature of the underlying least squares problem
is used to speed up the initial convergence and to improve the accuracy of
iterates. We show that this limited memory method is a generalization of the
damped block Kaczmarz method, and we prove linear convergence of the
expectation of the iterates and of the error norm up to a convergence horizon.
Numerical experiments demonstrate the benefits of these sampled limited memory
row-action methods for massive 2D and 3D inverse problems in tomography
applications.
