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Novel Error Measures and Source Conditions of Regularization Methods for Inverse Problems (SCIP)

反问题正则化方法的新颖误差测量和来源条件（SCIP）

基本信息

批准号：
391100538
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
财政年份：
2018
资助国家：
德国
起止时间：
2017-12-31 至 2020-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/391100538?language=en
关键词：
Novel Error Measures Source Conditions

项目摘要

Regularization methods are designed to limit the reconstruction errors in inverse problems. The basic principle of regularization is to limit the investigations in the reconstruction process to solutions which respect certain a-priori information, such as a maximal and minimal magnitude, smoothness, or certain conservation principles.Current regularization theory focuses on problems where the a-priori information can be represented as bounds of convex functionals, and then techniques from the mathematical field of Convex Analysis can be used to prove theoretical properties of the regularized solutions. Recently developed and more efficient regularization methods cannot be analyzed with such techniques, and in fact require novel measures for evaluating the efficiency. The development of such measures and conditions which guarantee the efficiency of modern regularization methods is the overall topic of this proposal which consists of five work packages.In the first and fundamental work package, the focus is on the verification of new convergence rates results for non-convex Tikhonov regularization. The second work package deals with the consequences of over smoothing penalties occurring in general Tikhonov regularization for a Hilbert space or Banach space setting. In the third work package the cross connections between source conditions and the convergence of level sets are under consideration. The fourth work package, however, deals with the interplay of variational source conditions and conditional stability estimates. New aspects of the Lavrentiev regularization with explicit and implicit forward operators are in the focus of the final fifth work package.

设计正则化方法是为了限制反问题的重构误差。正则化的基本原则是将重构过程中的研究限制在尊重某些先验信息的解中，例如最大值和最小值、平滑度或某些守恒原则。目前的正则化理论主要关注先验信息可以表示为凸泛函的界的问题，然后利用凸分析数学领域的技术来证明正则化解的理论性质。最近开发的更有效的正则化方法不能用这种技术进行分析，实际上需要新的方法来评估效率。制定保证现代正规化方法效率的这些措施和条件是本建议的总主题，该建议包括五个工作包。在第一个和基本的工作包中，重点是验证非凸Tikhonov正则化的新收敛率结果。第二个工作包处理Hilbert空间或Banach空间设置的一般Tikhonov正则化中出现的过度平滑惩罚的后果。在第三个工作包中，考虑了源条件与水平集收敛之间的交叉联系。然而，第四个工作包处理变源条件和条件稳定性估计的相互作用。Lavrentiev正则化与显式和隐式前向运算符的新方面是最后第五个工作包的重点。