Regularization and discretization of inverse problems for PDEs in Banach spaces

这个项目的目的是在Banach空间中，特别是在偏微分方程的背景下，正则化和不适定问题的离散化的组合分析。此类问题在从医学成像、无损检测到地球物理勘探的众多应用中发挥着至关重要的作用，其中Banach空间设置由所寻求系数的固有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空间中现有的抽象正则化理论与Hilbert空间中具有逐点约束的适定优化问题的自适应离散化之间的差距，以导出显式源条件和实用的参数选择规则，并发展基于泛函和目标导向误差估计的自适应离散化方法，该方法考虑了正则化参数的相互依赖性，数据噪声水平和离散化误差。这将导致一个完整的方法，稳定和有效的数值求解方法的参数识别问题在Banach空间。

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.