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

Helmut Gfrerer, Simon Hubmer, Ronny Ramlau

TL;DR

A new class of SCD semismooth Newton methods, which are based on a novel concept of graphical derivatives, and exhibit locally superlinear convergence are considered, which are based on a novel concept of graphical derivatives.

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.

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

TL;DR

A new class of SCD semismooth Newton methods, which are based on a novel concept of graphical derivatives, and exhibit locally superlinear convergence are considered, which are based on a novel concept of graphical derivatives.

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.

Paper Structure

This paper contains 15 sections, 13 theorems, 108 equations, 4 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Background on variational analysis
  3. The SCD semismooth* Newton method
  4. The SCD semismooth* Newton method for inclusions
  5. The SCD semismooth* Newton method in optimization
  6. SCD semismooth* Newton and Tikhonov
  7. SCD semismooth* Newton for Tikhonov functionals
  8. Globalizing the SCD semismooth* Newton method
  9. Application to sparsity and TV regularization
  10. Application to sparsity regularization
  11. Application to TV regularization
  12. Numerical Examples
  13. Conclusion
  14. Support
  15. The SCD derivative of $\ell_p$ norms with $0\leq p \leq 1$

Key Result

Theorem 3.1

Let $D\subset\mathbb{R}^n$ be open and let $H:D\to\mathbb{R}^n$ be locally Lipschitz continuous. Furthermore, assume that $\bar{x}\in D$ is a solution of EqNonlEqH with EqSS, and assume that all matrices $A\in{\rm conv\,}\overline\nabla H(\bar{x})$ are nonsingular. Then, there exists a neighborhood is well defined and the resulting sequence ${x}^{(k)}$ converges superlinearly to $\bar{x}$.

Figures (4)

  • Figure 6.1: Test data: sinograms and reference reconstructions adapted from FIP_Lotus_2016FIP_Walnut_2015.
  • Figure 6.2: SCD semismooth* Newton reconstructions for the walnut data.
  • Figure 6.3: SCD semismooth* Newton reconstructions for the lotus data.
  • Figure 6.4: Illustration of the computational efficiency of the SCD semismooth* Newton method for the case of sparsity regularization, compared with classic FISTA (for $\ell_1$) Beck_Teboulle_2009.

Theorems & Definitions (28)

  • Definition 3.1
  • Theorem 3.1: QiSun93
  • Definition 3.2
  • Definition 3.3
  • Example 3.1
  • Definition 3.4
  • Remark
  • Definition 3.5
  • Theorem 3.3
  • proof
  • ...and 18 more