  The Douglas-Rachford (DR) method is a widely used method for finding a point
in the intersection of two closed convex sets (feasibility problem). However,
the method converges weakly and the associated rate of convergence is hard to
analyze in general. In addition, the direct extension of the DR method for
solving more-than-two-sets feasibility problems, called the $r$-sets-DR method,
is not necessarily convergent. To improve the efficiency of the optimization
algorithms, the introduction of randomization and the momentum technique has
attracted increasing attention. In this paper, we propose the randomized
$r$-sets-DR (RrDR) method for solving the feasibility problem derived from
linear systems, showing the benefit of the randomization as it brings linear
convergence in expectation to the otherwise divergent $r$-sets-DR method.
Furthermore, the convergence rate does not depend on the dimension of the
coefficient matrix. We also study RrDR with heavy ball momentum and establish
its accelerated rate. Numerical experiments are provided to confirm our results
and demonstrate the notable improvements in accuracy and efficiency of the DR
method, brought by the randomization and the momentum technique.
