Regularization methods in Banach spaces for inverse scattering problems

Banach 空间中逆散射问题的正则化方法

• 批准号: 247299886
• 负责人: Professor Dr. Armin Lechleiter, since 11/2013 (†)
• 金额: --
• 依托单位: Zentrum für Technomathematik (ZeTeM)
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2013
• 资助国家: 德国
• 起止时间: 2012-12-31 至 2016-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/247299886?language=en
• 关键词: Regularization; methods; Banach; spaces; inverse

The solution of inverse scattering problems is of fundamental importance for non-destructive testing problems in, e.g., the engineering and the physical sciences. Examples of possible applications include for example ground penetrating radar measurements for geophysical prospection or the characterization of local defaults in optical structures by light scattering measurements. Taking a mathematical point of view, all these problems can be seen as non-linear inverse parameter identification problems. In this project we investigate inverse scattering problems for penetrable structures that are a-priori known to have a sparse representation in a known wavelet basis. These structures are often called sparse inhomogeneous media. To reconstruct such media we propose to use nonlinear wavelet regularization methods in Banach spaces that are sometimes called sparsity regularization methods. The aim of this project is one the one hand to prove regularization and convergence properties of such methods when applied to inverse scattering and on the other hand to demonstrate these properties numerically. Important ingredients for this analysis is a solution theory for time-harmonic acoustic and electromagnetic wave equations with parameters living in spaces of unbounded functions, as well as proper theory and fast algorithms for wavelet regularization methods when applied to inverse scattering problems. If it is a-priori known that the solution of an inverse scattering problem is sparse in some wavelet basis, then the use of sparsity regularization methods is particularly useful if the inverse problem under investigation is not uniquely solvable without additional assumptions on the solution. Indeed, sparsity regularization methods based on wavelets will in this case automatically pick the sparsest of all possible solutions. We want to demonstrate this property numerically for several practically relevant problems for synthetic as well as for measured data, e.g., for inverse electromagnetic problems with phaseless data and for backscattering problems.

逆散射问题的解对于无损检测问题具有根本的重要性，例如，工程学和物理学。可能的应用实例包括用于地球物理勘探的探地雷达测量或通过光散射测量确定光学结构中局部缺陷的特征。从数学的角度来看，这些问题都可以看作是非线性的逆参数辨识问题。在这个项目中，我们调查的逆散射问题的可穿透的结构，先验已知具有稀疏表示在一个已知的小波基。这些结构通常被称为稀疏非均匀介质。为了重建这样的媒体，我们建议使用非线性小波正则化方法在Banach空间，有时被称为稀疏正则化方法。该项目的目的是一方面证明这些方法的正则化和收敛性能时，应用到逆散射，另一方面，以证明这些属性的数值。这种分析的重要组成部分是时间谐波声波和电磁波方程的解决方案理论与参数生活在空间的无界函数，以及适当的理论和快速算法小波正则化方法时，应用于逆散射问题。如果先验已知逆散射问题的解在某些小波基中是稀疏的，则如果在调查中的逆问题不是唯一可解的，则使用稀疏正则化方法是特别有用的，而无需对解进行额外的假设。事实上，基于小波的稀疏正则化方法在这种情况下会自动选择所有可能解的稀疏性。我们想用数值方法证明这一特性，以解决合成数据和测量数据的几个实际相关问题，例如，用于具有无相位数据的逆电磁问题和用于后向散射问题。