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We present preconditioned and accelerated versions of the Douglas-Rachford (DR) splitting method. The methods enable to replace the solution of a linear system in each step in the DR iteration by approximate solvers without the need of controlling the error. These iterations are shown to converge in Hilbert space under minimal assumptions. Further, strong convexity of one or both of the functionals allow for accelerations that yield improved rates of $O(1/k^2)$ and $O(w^k)$, $0<w<1$, respectively.
This presentation is part of Minisymposium “MS59 - Approaches for fast optimisation in imaging and inverse problems (3 parts)”
organized by: Jingwei Liang (University of Cambridge) , Carola-Bibiane Schönlieb (University of Cambridge) , Mila Nikolova (CMLA - CNRS ENS Cachan, University Paris-Saclay) .