  A common problem in the sciences is that a signal of interest is observed
only indirectly, through smooth functionals of the signal whose values are then
obscured by noise. In such inverse problems, the functionals dampen or entirely
eliminate some of the signal's interesting features. This makes it difficult or
even impossible to fully reconstruct the signal, even without noise. In this
paper, we develop methods for handling sequences of related inverse problems,
with the problems varying either systematically or randomly over time. Such
sequences often arise with automated data collection systems, like the data
pipelines of large astronomical instruments such as the Large Synoptic Survey
Telescope (LSST). The LSST will observe each patch of the sky many times over
its lifetime under varying conditions. A possible additional complication in
these problems is that the observational resolution is limited by the
instrument, so that even with many repeated observations, only an approximation
of the underlying signal can be reconstructed. We propose an efficient
estimator for reconstructing a signal of interest given a sequence of related,
resolution-limited inverse problems. We demonstrate our method's effectiveness
in some representative examples and provide theoretical support for its
adoption.
