  Stochastic Primal-Dual Hybrid Gradient (SPDHG) is an algorithm proposed by
Chambolle et al. (2018) to efficiently solve a wide class of nonsmooth
large-scale optimization problems. In this paper we contribute to its
theoretical foundations and prove its almost sure convergence for convex but
neither necessarily strongly convex nor smooth functionals, as well as for any
random sampling. In addition, we study SPDHG for parallel Magnetic Resonance
Imaging reconstruction, where data from different coils are randomly selected
at each iteration. We apply SPDHG using a wide range of random sampling methods
and compare its performance across a range of settings, including mini-batch
size and step size parameters. We show that the sampling can significantly
affect the convergence speed of SPDHG and for many cases an optimal sampling
can be identified.
