Regularization and discretization of inverse problems for PDEs in Banach spaces

这个项目的目的是结合分析Banach空间中不适定问题的正则化和离散化，特别是在偏微分方程的背景下。这些问题在从医学成像到无损检测再到地球物理勘探的众多应用中扮演着至关重要的角色，Banach空间的设置由所寻求的系数的内在规律性以及稀疏性等结构特征决定。我们的目标是填补一般Banach空间中已有的抽象正则化理论和Hilbert空间中适定优化问题的自适应离散化之间的空白，得到显式的源条件和实用的参数选择规则，并发展基于泛函和面向目标的误差估计的自适应离散化方法，该方法考虑了正则化参数、数据噪声水平和离散化误差之间的相互依赖。这将为Banach空间中参数辨识问题的稳定有效的数值解方法提供一个完整的途径。

The aim of this project is a combined analysis of regularization and discretization of ill-posed problems in Banach spaces specifically in the context of partial differential equations. Such problems play a crucial role in numerous applications ranging from medical imaging via nondestructive testing to geophysical prospecting, with the Banach space setting mandated by the inherent regularity of the sought coefficients as well as structural features such as sparsity. Our goal is to fill the gap between the existing abstract regularization theory in general Banach spaces and the adaptive discretization of well-posed optimization problems in Hilbert spaces with pointwise constraints to derive explicit source conditions and practical parameter choice rules and to develop adaptive discretization methods based on functional and goal-oriented error estimates that take into account the interdependence of regularization parameter, data noise level and discretization error. This will lead to an integrated approach for the stable and efficient numerical solution method of parameter identification problems in Banach spaces.