We exploit the similarities between Tikhonov regularization and Bayesian hierarchical models to propose a regularization scheme that acts like a distributed Tikhonov regularization where the amount of regularization varies from component to component. In the standard formulation, Tikhonov regularization compensates for the inherent ill-conditioning of linear inverse problems by augmenting the data fidelity term measuring the mismatch between the data and the model output with a scaled penalty functional. The selection of the scaling is the core problem in Tikhonov regularization. If an estimate of the amount of noise in the data is available, a popular way is to use the Morozov discrepancy principle, stating that the scaling parameter should be chosen so as to guarantee that the norm of the data fitting error is approximately equal to the norm of the noise in the data. A too small value of the regularization parameter would yield a solution that fits to the noise while a too large value would lead to an excessive penalization of the solution. In many applications, it would be preferable to apply distributed regularization, replacing the regularization scalar by a vector valued parameter, allowing different regularization for different components of the unknown, or for groups of them. A distributed Tikhonov-inspired regularization is particularly well suited when the data have significantly different sensitivity to different components, or to promote sparsity of the solution. The numerical scheme that we propose, while exploiting the Bayesian interpretation of the inverse problem and identifying the Tikhonov regularization with the Maximum A Posteriori (MAP) estimation, requires no statistical tools. A combination of numerical linear algebra and optimization tools makes the scheme computationally efficient and suitable for problems where the matrix is not explicitly available.

翻译：我们利用Tikhonov正则化与贝叶斯层次模型之间的相似性，提出一种类似分布式Tikhonov正则化的正则化方案，其中正则化强度随分量不同而变化。在标准公式中，Tikhonov正则化通过增加一个衡量数据与模型输出之间失配的数据保真项（附加缩放惩罚泛函）来补偿线性反问题的固有病态性。缩放参数的选择是Tikhonov正则化的核心问题。若已知数据中噪声量的估计值，常用方法是采用Morozov偏差原则，即选择缩放参数以确保数据拟合误差的范数近似等于数据中噪声的范数。过小的正则化参数会导致解过度拟合噪声，而过大的参数则会导致对解的过度惩罚。在许多应用中，更倾向于应用分布式正则化，将正则化标量替换为向量值参数，允许对未知量的不同分量或它们的分组施加不同的正则化。当数据对不同分量的敏感度存在显著差异，或需要促进解的稀疏性时，基于分布式Tikhonov思想的正则化尤为适用。我们提出的数值方案虽利用了反问题的贝叶斯解释，并将Tikhonov正则化等同于最大后验概率估计，但无需使用任何统计工具。该方案结合数值线性代数与优化工具，计算高效，尤其适用于矩阵无法显式获取的问题。