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

TL;DR

This paper introduces two iterative algorithms, UpenMM and GUpenMM, within the MM framework for multi-penalty regularization in linear inverse problems, demonstrating their convergence and practical effectiveness through numerical examples.

Contribution

The paper develops and analyzes two novel iterative methods, UpenMM and GUpenMM, for multi-penalty regularization in linear inverse problems, extending the UPEN approach.

Findings

Methods converge reliably in numerical tests

Algorithms effectively identify regularization parameters

Point-wise regularization improves solution quality

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.