On SCD Semismooth$^*$ Newton methods for the efficient minimization of Tikhonov functionals with non-smooth and non-convex penalties

TL;DR

This paper introduces a new class of SCD semismooth$^*$ Newton methods for efficiently minimizing Tikhonov functionals with non-smooth, non-convex penalties, demonstrating superlinear convergence and effectiveness in tomographic imaging.

Contribution

The paper presents a novel SCD semismooth$^*$ Newton framework based on graphical derivatives, with explicit algorithms for sparsity and total variation penalties, improving optimization in variational regularization.

Findings

Methods exhibit locally superlinear convergence.

Numerical results show improved efficiency in tomographic imaging.

Algorithms effectively handle non-smooth, non-convex penalties.

Abstract

We consider the efficient numerical minimization of Tikhonov functionals with nonlinear operators and non-smooth and non-convex penalty terms, which appear for example in variational regularization. For this, we consider a new class of SCD semismooth$^*$ Newton methods, which are based on a novel concept of graphical derivatives, and exhibit locally superlinear convergence. We present a detailed description of these methods, and provide explicit algorithms in the case of sparsity and total-variation penalty terms. The numerical performance of these methods is then illustrated on a number of tomographic imaging problems.