Convergence of the alternating split Bregman algorithm in infinite-dimensional Hilbert spaces

TL;DR

This paper establishes weak convergence results for the alternating split Bregman algorithm in infinite-dimensional Hilbert spaces, including an approximate version with errors, by leveraging Douglas-Rachford splitting theory.

Contribution

It extends convergence analysis of the split Bregman algorithm to infinite-dimensional spaces and introduces an approximate version with error tolerance, using dual problem focus.

Findings

Proves weak convergence in infinite-dimensional Hilbert spaces.

Shows convergence of an approximate algorithm with errors.

Utilizes Douglas-Rachford splitting and dual problem analysis.

Abstract

We prove results on weak convergence for the alternating split Bregman algorithm in infinite dimensional Hilbert spaces. We also show convergence of an approximate split Bregman algorithm, where errors are allowed at each step of the computation. To be able to treat the infinite dimensional case, our proofs focus mostly on the dual problem. We rely on Svaiter's theorem on weak convergence of the Douglas-Rachford splitting algorithm and on the relation between the alternating split Bregman and Douglas-Rachford splitting algorithms discovered by Setzer. Our motivation for this study is to provide a convergent algorithm for weighted least gradient problems arising in the hybrid method of imaging electric conductivity from interior knowledge (obtainable by MRI) of the magnitude of one current.