Adaptive Discretizations for the Regularization of Inverse Problems

在自然科学、医学和工程领域中，许多复杂的过程都是用偏微分方程（PDE）的数学模型来描述的。偏微分方程组中大多含有未知数据，如空间相关系数函数、源项、初始和边界数据等，这些数据的确定将导致高维反问题的产生，求解偏微分方程反问题的数值工作量通常比给定数据集下的数值模拟高得多。此外，逆问题的固有不稳定性需要使用适当的正则化技术。为解决此类反问题而构造有效算法的巨大潜力在于自适应离散化。虽然在过去的几年中，使用自适应概念来选择数值模拟的离散化方法已经变得普遍，但反问题背景下的自适应性提出了一个新的高度相关的主题。该项目的目标是找到普遍适用的和分析合理的反问题自适应离散化方法。在这个过程中，主要的重点是构造算法的效率，一方面，严格的收敛性分析的正则化方法的背景下，另一方面。

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.