针对Morozov与等式约束正则化的原始对偶方法及其加速
Primal-dual methods and acceleration for Morozov and equality constrained regularization
浏览论文内容
中文总结 AI 辅助
该研究针对含噪声线性逆问题，在希尔伯特与巴拿赫空间中分析非加速及加速原始对偶方法的正则化特性，推导误差估计并通过数值实验验证结果，扩展了正则化分析的适用框架。
中文摘要 AI 辅助
本研究针对含噪声数据的线性逆问题，开展非加速与加速原始对偶方法的正则化分析。我们在希尔伯特空间中研究Condat-Vũ算法与加速原始对偶混合梯度方法，重点量化数据扰动对重构误差的影响。对于非加速方案，我们推导基于Bregman距离的误差估计；对于加速方案，我们建立范数意义下的误差估计。该研究适用于满足合适扰动条件的一般凸数据保真项，且已针对等式约束正则化与Morozov正则化明确验证。对于非加速方法，分析进一步扩展至巴拿赫空间，考虑非欧几里得几何，包含适配非负解重构的特定非自反设置。所得结果复现经典场景下的已知行为，同时将正则化分析扩展至这些更通用框架。针对稀疏性与熵模型等代表性正则化器的数值实验，验证了理论发现并展示了噪声下的实际性能。
英文摘要
This work develops a regularization analysis of non-accelerated and accelerated primal-dual methods for solving linear inverse problems in the presence of noisy data. We investigate a Condat-Vũ algorithm and an accelerated primal-dual hybrid gradient method in Hilbert spaces, with focus on quantifying the effect of data perturbations on the reconstruction error. For the non-accelerated scheme, we derive error estimates in terms of Bregman distances, whereas for the accelerated scheme we establish error estimates in norm. The study accommodates a general class of convex data fidelities satisfying suitable perturbation conditions, which are verified explicitly for equality constrained and Morozov regularization. For the non-accelerated method, the analysis is further extended to Banach spaces, taking into account non-Euclidean geometries and including a particular non-reflexive setting tailored to nonnegative solution reconstruction. The results recover known behavior in classical settings while extending the regularization analysis to these more general frameworks. Numerical experiments with representative regularizers, including sparsity and entropic models, support the theoretical findings and illustrate practical performance under noise.
