Efficient TV regularization of large-scale linear inverse problems via the SCD semismooth* Newton method with applications in tomography

TL;DR

This paper introduces an efficient semismooth* Newton method for large-scale TV regularized inverse problems, offering superlinear convergence and strong mathematical guarantees, demonstrated on tomographic imaging tasks.

Contribution

The paper develops a novel semismooth* Newton approach tailored for TV regularization in large-scale inverse problems, improving efficiency and convergence over existing methods.

Findings

The method achieves locally superlinear convergence.

It performs well on large-scale tomographic problems.

It outperforms other state-of-the-art TV regularization techniques.

Abstract

In this paper, we consider the efficient numerical minimization of Tikhonov functionals resulting from total-variation (TV) regularization of linear inverse problems. Since the TV penalty is non-smooth, this is typically done either via smooth approximations, which are inexact, or using non-smooth optimization techniques, which can often be numerically expensive, in particular for large-scale problems. Here, we present a numerically efficient minimization approach based on the recently proposed semismooth* Newton method, which employs a novel concept of graphical derivatives and exhibits locally superlinear convergence. The proposed approach is specifically tailored to TV regularization, suitable for large-scale inverse problems, and supported by strong mathematical convergence guarantees. Furthermore, we demonstrate its performance on two (large-scale) tomographic imaging problems and compare our results to those obtained via other state-of-the-art TV regularization approaches.