Regularization and discretization of inverse problems for PDEs in Banach spaces

本项目的目的是对Banach空间中不适定问题的正则化和离散化进行综合分析，特别是在偏微分方程的背景下。这些问题在许多应用中发挥着至关重要的作用，从医学成像到无损检测，再到地球物理勘探，巴拿赫空间设置是由所寻找系数的固有规律性以及稀疏性等结构特征所决定的。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空间中具有点向约束的良定优化问题的自适应离散化之间的空白，以获得明确的源条件和实际参数选择规则，并开发基于功能和目标导向的误差估计的自适应离散化方法，考虑正则化参数的相互依赖性。数据噪声水平和离散误差。这将为巴拿赫空间中参数辨识问题的稳定、高效的数值求解方法提供一个综合的途径。

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.