Oversmoothing regularization models in light of local ill-posedness phenomena

在过去的十五年里，光滑性在正则化理论中的作用有了很大的增长，该理论的目标是在Hilbert或Banach空间环境中求不适定算子方程的稳定近似解。对于在自然科学、工程、成像和金融中应用的表示逆问题的这种算子方程，光滑的概念是双重的：一方面是从噪声数据重构的抽象函数空间中的元素的光滑性(解的光滑性)，另一方面是模型中的线性或非线性正向算子的光滑性(算子光滑性)。在这两种发生的平滑变化之间存在着强烈的相互作用。成功的正则化方法最好适应预期的平滑和非线性情况，但这样的期望可能会失败。一个典型的例子是带有过平滑惩罚的变分(Tikhonov型)正则化，当惩罚泛函高估了实际的光滑性，使得解元素没有有限的惩罚值时发生。对于Hilbert尺度模型中的过度平滑惩罚，最近在申请者和他们的合著者的果断参与下，取得了实质性的收敛和比率结果。改进和改进这些结果，并将其推广到Banach空间模型和稀疏性促进正则化是本项目的挑战性目标，重点是一类非线性反问题，其中不适定的性质和程度可以局部分布。在此背景下，我们还考虑了特定类别的逆问题，如2D图像的去自卷积问题和特定方法，如数据驱动正则化作为深度学习的一个方面，其中必须补偿前向算子的缺失分量。总体而言，通过使用分析方法、离散化方法和数值实验，该项目旨在根据发生的局部不适定现象，更深入地了解处理过平滑模型的方法及其机会和局限性，以便从中受益于选择最佳正则化程序和适当选择正则化参数。

In the past fifteen years, the role of smoothness in regularization theory aimed at the stable approximate solution of ill-posed operator equations in a Hilbert or Banach space setting has substantially grown. For such operator equations that represent inverse problems with applications in natural sciences, engineering, imaging and finance, the concept of smoothness is twofold: smoothness of elements in abstract function spaces to be reconstructed from noisy data (solution smoothness) on the one hand and smoothness of the linear or nonlinear forward operator in the model (operator smoothness) on the other hand. There is a strong interplay between both occurring varieties of smoothness. Successful regularization approaches are preferably adapted to expected smoothness and nonlinearity situations, but such expectations can fail. A typical example is the variational (Tikhonov-type) regularization with oversmoothing penalties, occurring when the penalty functional overestimates the actual smoothness such that the solution elements attain no finite penalty values. For oversmoothing penalties in Hilbert scale models, substantial convergence and rates results were recently achieved with the decisive participation of both applicants and their coauthors. The refinement and improvement of these results as well their extension to Banach space models and sparsity promoting regularization are challenging goals of this project with focus on classes of nonlinear inverse problems, where the character and degree of ill-posedness can be locally distributed. In this context, we also consider specific classes of inverse problems like the deautoconvolution problem for 2D-images and specific approaches like the data driven regularization as an aspect of deep learning, where missing components of the forward operator have to be compensated. Overall, by using analytical methods, discretization approaches and numerical experiments, the project intends to deliver a deeper understanding of methods for the treatment of oversmoothing models with their opportunities and limitations in light of occurring local ill-posedness phenomena in order to benefit from this for the selection of optimal regularization procedures and appropriate choices of the regularization parameters.