  Next generation radio-interferometers, like the Square Kilometre Array, will
acquire tremendous amounts of data with the goal of improving the size and
sensitivity of the reconstructed images by orders of magnitude. The efficient
processing of large-scale data sets is of great importance. We propose an
acceleration strategy for a recently proposed primal-dual distributed
algorithm. A preconditioning approach can incorporate into the algorithmic
structure both the sampling density of the measured visibilities and the noise
statistics. Using the sampling density information greatly accelerates the
convergence speed, especially for highly non-uniform sampling patterns, while
relying on the correct noise statistics optimises the sensitivity of the
reconstruction. In connection to CLEAN, our approach can be seen as including
in the same algorithmic structure both natural and uniform weighting, thereby
simultaneously optimising both the resolution and the sensitivity. The method
relies on a new non-Euclidean proximity operator for the data fidelity term,
that generalises the projection onto the $\ell_2$ ball where the noise lives
for naturally weighted data, to the projection onto a generalised ellipsoid
incorporating sampling density information through uniform weighting.
Importantly, this non-Euclidean modification is only an acceleration strategy
to solve the convex imaging problem with data fidelity dictated only by noise
statistics. We showcase through simulations with realistic sampling patterns
the acceleration obtained using the preconditioning. We also investigate the
algorithm performance for the reconstruction of the 3C129 radio galaxy from
real visibilities and compare with multi-scale CLEAN, showing better
sensitivity and resolution. Our MATLAB code is available online on GitHub.
