Preconditioned alternating direction method of multipliers for inverse problems with constraints. (9th January 2017)

• Record Type: Journal Article
• Authors: Jiao, Yuling; Jin, Qinian; Lu, Xiliang; Wang, Weijie
• Is Part Of: Inverse problems. Volume 33:Number 2(2017:Feb.)
• Publication Date: 2017-01-09
• Subjects: inverse problems with constraints -- preconditioned alternating direction method of multipliers -- regularization -- deterministic stopping rule -- heuristic rule; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/33/2/025004
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: We propose a preconditioned alternating direction method of multipliers (ADMM) to solve linear inverse problems in Hilbert spaces with constraints, where the feature of the sought solution under a linear transformation is captured by a possibly non-smooth convex function. During each iteration step, our method avoids solving large linear systems by choosing a suitable preconditioning operator. In case the data is given exactly, we prove the convergence of our preconditioned ADMM without assuming the existence of a Lagrange multiplier. In case the data is corrupted by noise, we propose a stopping rule using information on noise level and show that our preconditioned ADMM is a regularization method; we also propose a heuristic rule when the information on noise level is unavailable or unreliable and give its detailed analysis. Numerical examples are presented to test the performance of the proposed method.