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.

翻译：本研究在变分正则化方法框架下，探讨了一种基于均匀罚（UPEN）方法的多罚正则化策略。UPEN方法由作者先前提出并应用于核磁共振（NMR）数据处理。本文介绍了两种迭代方法——UpenMM和GUpenMM，它们均基于Majorization-Minimization（MM）框架构建。这些方法旨在利用多罚正则化策略为线性逆问题确定合适的正则化参数与解。本文证明了这两类方法的收敛性，并通过一维和二维数值算例展示了其潜力，揭示了逐点正则化项在解决各类逆问题中的实际应用价值。