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

TL;DR

This paper introduces a generalized forward-backward splitting algorithm for minimizing sums of smooth and non-smooth convex functionals in Banach spaces, extending existing methods from Hilbert spaces.

Contribution

It develops a new iterative algorithm with proven convergence and convergence rates for Banach space problems, including applications to inverse problems and image restoration.

Findings

Proposed a generalized splitting algorithm for Banach spaces.

Proved convergence and convergence rates of the algorithm.

Demonstrated applicability to inverse problems and image processing.

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.