  In this paper, we investigate iterative methods that are based on sampling of
the data for computing Tikhonov-regularized solutions. We focus on very large
inverse problems where access to the entire data set is not possible all at
once (e.g., for problems with streaming or massive datasets). Row-access
methods provide an ideal framework for solving such problems, since they only
require access to "blocks" of the data at any given time. However, when using
these iterative sampling methods to solve inverse problems, the main challenges
include a proper choice of the regularization parameter, appropriate sampling
strategies, and a convergence analysis. To address these challenges, we first
describe a family of sampled iterative methods that can incorporate data as
they become available (e.g., randomly sampled). We consider two sampled
iterative methods, where the iterates can be characterized as solutions to a
sequence of approximate Tikhonov problems. The first method requires the
regularization parameter to be fixed a priori and converges asymptotically to
an unregularized solution for randomly sampled data. This is undesirable for
inverse problems. Thus, we focus on the second method where the main benefits
are that the regularization parameter can be updated during the iterative
process and the iterates converge asymptotically to a Tikhonov-regularized
solution. We describe adaptive approaches to update the regularization
parameter that are based on sampled residuals, and we describe a limited-memory
variant for larger problems. Numerical examples, including a large-scale
super-resolution imaging example, demonstrate the potential for these methods.

Numerical Analysis