Abstract

Co-author: Hongpeng Sun (Renmin University of China)

We present preconditioned and accelerated versions of the Douglas-Rachford (DR) splitting method for the solution of convex-concave saddle-point problems which often arise in variational imaging. The methods enable to replace the solution of a linear system in each iteration step in the corresponding DR iteration by approximate solvers without the need of controlling the error. These iterations are shown to converge in Hilbert space under minimal assumptions on the preconditioner and for any step-size. Moreover, ergodic sequences associated with the iteration admit at least a convergence rate in terms of restricted primal-dual gaps. Further, strong convexity of one or both of the involved functionals allow for acceleration strategies that yield improved rates of and for , respectively.

The methods are applied to non-smooth and convex variational imaging problems. We discuss denoising and deconvolution with and discrepancy and total variation (TV) as well as total generalized variation (TGV) penalty. Preconditioners which are specific to these problems are presented, the results of numerical experiments are shown and the benefits of the respective preconditioned iterations are discussed.