带过光滑惩罚项的希尔伯特尺度下非线性Tikhonov正则化：平衡原理的考察
Nonlinear Tikhonov regularization in Hilbert scales with oversmoothing penalty: inspecting balancing principles
AI总结：
该研究针对希尔伯特尺度下带过光滑惩罚的非线性Tikhonov正则化问题，推导了覆盖多光滑性条件的误差分解式，提出平衡原理参数选择方法并分析其阶最优条件，辅以数值案例补充了反问题理论的相关研究。
摘要：
针对带光滑性促进惩罚项的非线性不适定方程的Tikhonov正则化分析，是反问题理论中的重要课题。聚焦希尔伯特尺度模型，过光滑惩罚项的情形（即惩罚项在真解处取无穷值的情况）受到了越来越多的关注。本文所考虑的非线性结构与B. Hofmann和P. Mathé的研究《希尔伯特尺度下非线性不适定问题的过光滑惩罚Tikhonov正则化》（发表于《Inverse Problems》2018年）一致。这类分析可解决两个基本问题：何时能实现阶最优重构？如何选择正则化参数？本研究从两个主要方面补充了已有研究：首先，推导了将误差分解为依赖光滑性的项和（不依赖光滑性的）噪声传播项的分解式，可覆盖大范围的光滑性条件；其次，提出了基于平衡原理的参数选择方法。通过对平衡法参数选择的历史与变体的详细讨论，明确了这类平衡原理在何种条件下可实现阶最优重构。基于某指数增长模型的数值案例研究提供了更多见解。
英文摘要：
The analysis of Tikhonov regularization for nonlinear ill-posed equations with smoothness promoting penalties is an important topic in inverse problem theory. With focus on Hilbert scale models, the case of oversmoothing penalties, i.e., when the penalty takes an infinite value at the true solution gained increasing interest. The considered nonlinearity structure is as in the study B. Hofmann and P. Mathé. Tikhonov regularization with oversmoothing penalty for non-linear ill-posed problems in Hilbert scales. Inverse Problems, 2018. Such analysis can address two fundamental questions. When is it possible to achieve order optimal reconstruction? How to select the regularization parameter? The present study complements previous ones by two main facets. First, an error decomposition into a smoothness dependent and a (smoothness independent) noise propagation term is derived, covering a large range of smoothness conditions. Secondly, parameter selection by balancing principles is presented. A detailed discussion, covering some history and variations of the parameter choice by balancing shows under which conditions such balancing principles yield order optimal reconstruction. A numerical case study, based on some exponential growth model, provides additional insights.
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