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Uniform multi-penalty regularization for linear ill-posed inverse problems

Villiam Bortolotti, Germana Landi, Fabiana Zama

TL;DR

This work addresses linear ill-posed inverse problems by formulating a uniform multi-penalty regularization framework that uses multiple pointwise penalties under a Uniform Penalty Principle. It introduces two Majorization-Minimization algorithms, UpenMM and GUpenMM, to jointly compute the unknown $u$ and the penalty parameters $\lambda_i$, and proves their convergence to solutions that satisfy the Uniform Penalty Principle. The derivation rests on a Bayesian view via Augmented-Tikhonov regularization, establishing a Balancing Principle for many penalties and showing that regularization parameters are critical points of the function $\Phi_\gamma(\boldsymbol{\lambda})$. Numerical experiments in 1D and 2D NMR-relaxometry-like settings demonstrate the effectiveness and robustness of the pointwise $L^2$-based penalties (with a small $\epsilon$) and a global $\ell_1$ term, improving reconstruction accuracy and tail behavior compared to single-penalty approaches. Overall, the paper extends UPEN from its NMR origins to general linear inverse problems and provides convergence guarantees for the proposed multi-parameter regularization algorithms.

Abstract

This study examines, in the framework of variational regularization methods, a multi-penalty regularization approach which builds upon the Uniform PENalty (UPEN) method, previously proposed by the authors for Nuclear Magnetic Resonance (NMR) data processing. The paper introduces two iterative methods, UpenMM and GUpenMM, formulated within the Majorization-Minimization (MM) framework. These methods are designed to identify appropriate regularization parameters and solutions for linear inverse problems utilizing multi-penalty regularization. The paper demonstrates the convergence of these methods and illustrates their potential through numerical examples in one and two-dimensional scenarios, showing the practical utility of point-wise regularization terms in solving various inverse problems.

Uniform multi-penalty regularization for linear ill-posed inverse problems

TL;DR

This work addresses linear ill-posed inverse problems by formulating a uniform multi-penalty regularization framework that uses multiple pointwise penalties under a Uniform Penalty Principle. It introduces two Majorization-Minimization algorithms, UpenMM and GUpenMM, to jointly compute the unknown and the penalty parameters , and proves their convergence to solutions that satisfy the Uniform Penalty Principle. The derivation rests on a Bayesian view via Augmented-Tikhonov regularization, establishing a Balancing Principle for many penalties and showing that regularization parameters are critical points of the function . Numerical experiments in 1D and 2D NMR-relaxometry-like settings demonstrate the effectiveness and robustness of the pointwise -based penalties (with a small ) and a global term, improving reconstruction accuracy and tail behavior compared to single-penalty approaches. Overall, the paper extends UPEN from its NMR origins to general linear inverse problems and provides convergence guarantees for the proposed multi-parameter regularization algorithms.

Abstract

This study examines, in the framework of variational regularization methods, a multi-penalty regularization approach which builds upon the Uniform PENalty (UPEN) method, previously proposed by the authors for Nuclear Magnetic Resonance (NMR) data processing. The paper introduces two iterative methods, UpenMM and GUpenMM, formulated within the Majorization-Minimization (MM) framework. These methods are designed to identify appropriate regularization parameters and solutions for linear inverse problems utilizing multi-penalty regularization. The paper demonstrates the convergence of these methods and illustrates their potential through numerical examples in one and two-dimensional scenarios, showing the practical utility of point-wise regularization terms in solving various inverse problems.
Paper Structure (3 sections, 1 theorem, 29 equations)

This paper contains 3 sections, 1 theorem, 29 equations.

Table of Contents

  1. Introduction \newlabelsec:intro0
  2. Contributions
  3. Derivation of the uniform multi-penalty regularization

Key Result

Theorem 2.1

\newlabelteo:J0 There exists at least one minimizer to the Augmented-Tikhonov function $\mathcal{J}$: with $\alpha_0'=\left (\frac{M}{2} + \alpha_0-1 \right )$, $\alpha_i'=\left (\frac{p}{2} + \alpha_i-1 \right )$, $i=0, \ldots, p$.

Theorems & Definitions (2)

  • Theorem 2.1
  • Proof 1