Oversmoothing regularization models in light of local ill-posedness phenomena

根据局部不适定现象的过平滑正则化模型

• 批准号: 453804957
• 负责人: Professor Dr. Bernd Hofmann
• 金额: --
• 依托单位: Department Mathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/453804957?language=en
• 关键词: Oversmoothing; regularization; models; light; local

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.

在过去的15年中，光滑性在正则化理论中的作用，旨在稳定的近似解的不适定算子方程在希尔伯特或Banach空间设置有显着增长。对于这样的算子方程，代表逆问题与应用在自然科学，工程，成像和金融，平滑的概念是双重的：平滑的抽象函数空间中的元素被重建从噪声数据（解决方案的平滑性），一方面和平滑的线性或非线性前向算子的模型（算子平滑）。这两种平滑度之间存在着强烈的相互作用。成功的正则化方法优选地适应于预期的平滑度和非线性情况，但是这样的预期可能失败。一个典型的例子是具有过平滑惩罚的变分（吉洪诺夫型）正则化，当惩罚泛函高估实际平滑度时发生，使得解元素没有获得有限的惩罚值。对于希尔伯特尺度模型中的过平滑惩罚，最近在申请人及其合著者的决定性参与下取得了实质性的收敛和速率结果。这些结果的细化和改进，以及它们的扩展到Banach空间模型和稀疏促进正则化是本项目的挑战性目标，重点是类非线性反问题，其中的字符和不适定性的程度可以局部分布。在这种情况下，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，以及作为深度学习的一个方面的数据驱动正则化等特定方法，其中必须补偿前向算子的缺失分量。总的来说，通过使用分析方法，离散化方法和数值实验，该项目旨在更深入地了解处理过平滑模型的方法，以及它们在发生局部不适定性现象时的机会和局限性，以便从中受益于最佳正则化程序的选择和正则化参数的适当选择。