Affine nonexpansive operators, Attouch-Th\'era duality and the   Douglas-Rachford algorithm
  The Douglas-Rachford splitting algorithm was originally proposed in 1956 to solve a system of linear equations arising from the discretization of a partial differential equation. In 1979, Lions and Mercier brought forward a very powerful extension of this method suitable to solve optimization problems.   In this paper, we revisit the original affine setting. We provide a powerful convergence result for finding a zero of the sum of two maximally monotone affine relations. As a by product of our analysis, we obtain results concerning the convergence of iterates of affine nonexpansive mappings as well as Attouch-Th\'era duality. Numerous examples are presented. 
Heinz H. Bauschke, Brett Lukens, Walaa M. Moursi
math.OC