  Next generation radio interferometric telescopes are entering an era of big
data with extremely large data sets. While these telescopes can observe the sky
in higher sensitivity and resolution than before, computational challenges in
image reconstruction need to be overcome to realize the potential of
forthcoming telescopes. New methods in sparse image reconstruction and convex
optimization techniques (cf. compressive sensing) have shown to produce higher
fidelity reconstructions of simulations and real observations than traditional
methods. This article presents distributed and parallel algorithms and
implementations to perform sparse image reconstruction, with significant
practical considerations that are important for implementing these algorithms
for Big Data. We benchmark the algorithms presented, showing that they are
considerably faster than their serial equivalents. We then pre-sample gridding
kernels to scale the distributed algorithms to larger data sizes, showing
application times for 1 Gb to 2.4 Tb data sets over 25 to 100 nodes for up to
50 billion visibilities, and find that the run-times for the distributed
algorithms range from 100 milliseconds to 3 minutes per iteration. This work
presents an important step in working towards computationally scalable and
efficient algorithms and implementations that are needed to image observations
of both extended and compact sources from next generation radio interferometers
such as the SKA. The algorithms are implemented in the latest versions of the
SOPT (https://github.com/astro-informatics/sopt) and PURIFY
(https://github.com/astro-informatics/purify) software packages {(Versions
3.1.0)}, which have been released alongside of this article.
