Adaptive Discretizations for the Regularization of Inverse Problems

自然科学、医学和工程领域中的许多复杂过程都是用偏微分方程(PDE)的数学模型来描述的。上述偏微分方程组大多含有未知数据，如空间相关系数函数、源项、初始数据和边界数据，这些未知数据的确定会导致高维逆问题的产生，用偏微分方程组求解逆问题的数值工作量通常比在给定数据集下对底层过程的数值模拟高得多。此外，反问题固有的不稳定性要求使用适当的正则化技术。构造求解这类反问题的有效算法的巨大潜力在于自适应离散化。近年来，在数值模拟中使用自适应概念来选择离散化已经变得很普遍，但反问题背景下的自适应提出了一个新的高度相关的话题。该项目的目标在于找到反问题的自适应离散化的普遍适用和分析合理的方法。在这一过程中，一方面主要关注构造算法的效率，另一方面在正则化方法的背景下进行严格的收敛分析。

Many complex processes in the field of natural sciences, medicine and engineering are described by mathematical models with partial differential equations (PDEs). The mentioned systems of PDEs mostly contain unknown data, e.g. space-dependent coefficient functions, source terms, initial and boundary data, whose determination leads to high-dimensional inverse problems.The numerical effort for solving inverse problems with PDEs is usually much higher than for the numerical simulation of the underlying process with a given data set. Moreover the inherent instability of inverse problems requires the use of appropriate regularization techniques. Great potential for the construction of efficient algorithms for the solution of such inverse problems lies in adaptive discretizations. While the use of adaptive concepts for the choice of the discretization for numerical simulation has become prevalent in the last years, adaptivity in the context of inverse problems presents a new and highly relevant topic.The goal of the project consists in finding generally applicable and analytically justified methods for the adaptive discretization of inverse problems. In this process, the main focus is on the efficiency of the constructed algorithms on the one hand and on the rigorous convergence analysis in the context of regularization methods on the other hand.