  We consider the problem of minimizing the sum of a convex function and a
convex function composed with an injective linear mapping. For such problems,
subject to a coercivity condition at fixed points of the corresponding Picard
iteration, iterates of the alternating directions method of multipliers
converge locally linearly to points from which the solution to the original
problem can be computed. Our proof strategy uses duality and strong metric
subregularity of the Douglas--Rachford fixed point mapping. Our analysis does
not require strong convexity and yields error bounds to the set of model
solutions. We show in particular that convex piecewise linear-quadratic
functions naturally satisfy the requirements of the theory, guaranteeing
eventual linear convergence of both the Douglas--Rachford algorithm and the
alternating directions method of multipliers for this class of objectives under
mild assumptions on the set of fixed points. We demonstrate this result on
quantitative image deconvolution and denoising with multiresolution statistical
constraints.
