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Adaptive Discretizations for the Regularization of Inverse Problems

逆问题正则化的自适应离散化

基本信息

批准号：
119705116
负责人：
Professor Dr. Boris Vexler
金额：
--
依托单位：
Lehrstuhl für Mathematische Optimierung (M1)
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2009
资助国家：
德国
起止时间：
2008-12-31 至 2013-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/119705116?language=en
关键词：
Adaptive Discretizations Regularization Inverse Problems

项目摘要

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.

自然科学、医学和工程领域的许多复杂过程都是通过偏微分方程（PDE）的数学模型来描述的。上述偏微分方程系统大多包含未知数据，例如空间相关系数函数、源项、初始和边界数据，其确定会导致高维反演问题。使用偏微分方程求解反演问题的数值工作量通常远高于使用给定数据集对基础过程进行数值模拟的工作量。此外，逆问题固有的不稳定性需要使用适当的正则化技术。构建解决此类逆问题的有效算法的巨大潜力在于自适应离散化。虽然在过去几年中使用自适应概念来选择数值模拟的离散化已经变得很普遍，但逆问题背景下的自适应性提出了一个新的且高度相关的主题。该项目的目标在于寻找普遍适用且分析合理的方法来逆问题的自适应离散化。在此过程中，主要关注点一方面是所构造算法的效率，另一方面是正则化方法中严格的收敛性分析。