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

正则化方法的目的是限制反问题中的重建误差。正则化的基本原理是将重建过程中的研究限制在考虑某些先验信息的解决方案上，例如最大和最小幅度，光滑性或某些守恒原理。当前正则化理论关注的问题是先验信息可以表示为凸泛函的边界，然后利用数学领域的凸分析技术证明正则解的理论性质。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，实际上需要新的措施来评估效率。发展这些措施和条件，保证现代正则化方法的效率是本提案的总体主题，其中包括五个工作包。在第一个和基本的工作包，重点是对非凸吉洪诺夫正则化新的收敛速度结果的验证。第二个工作包处理的后果，在一般吉洪诺夫正则化的Hilbert空间或Banach空间设置的平滑处罚。在第三个工作包中，考虑了源条件和水平集收敛之间的交叉联系。第四个工作包，然而，处理变分源条件和条件稳定性估计的相互作用。具有显式和隐式前向算子的Lavrentiev正则化的新方面是最后的第五个工作包的重点。

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.