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arXiv · 0807.0778

A forward-backward splitting algorithm for the minimization of non-smooth convex functionals in Banach space

Kristian Bredies
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Abstract

We consider the task of computing an approximate minimizer of the sum of a smooth and non-smooth convex functional, respectively, in Banach space. Motivated by the classical forward-backward splitting method for the subgradients in Hilbert space, we propose a generalization which involves the iterative solution of simpler subproblems. Descent and convergence properties of this new algorithm are studied. Furthermore, the results are applied to the minimization of Tikhonov-functionals associated with linear inverse problems and semi-norm penalization in Banach spaces. With the help of Bregman-Taylor-distance estimates, rates of convergence for the forward-backward splitting procedure are obtained. Examples which demonstrate the applicability are given, in particular, a generalization of the iterative soft-thresholding method by Daubechies, Defrise and De Mol to Banach spaces as well as total-variation based image restoration in higher dimensions are presented.

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Kristian Bredies. 2008-07-04. A forward-backward splitting algorithm for the minimization of non-smooth convex functionals in Banach space. https://doi.org/10.1088/0266-5611%2F25%2F1%2F015005

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