Regularization methods in Banach spaces for inverse scattering problems

反散射问题的求解对于工程和物理科学等领域的无损检测问题具有重要的基础意义。可能的应用实例包括，例如用于地球物理勘探的探地雷达测量或通过光散射测量来表征光学结构的局部缺陷。从数学的角度来看，这些问题都可以看作是非线性逆参数辨识问题。在这个项目中，我们研究了在已知小波基中先验已知具有稀疏表示的可穿透结构的逆散射问题。这些结构通常被称为稀疏非均匀介质。为了重建这样的介质，我们建议在巴拿赫空间中使用非线性小波正则化方法，有时也称为稀疏正则化方法。本项目的目的是一方面证明这种方法在应用于逆散射时的正则化和收敛性，另一方面在数值上证明这些性质。该分析的重要组成部分是参数存在于无界函数空间的时谐声波和电磁波方程的解理论，以及应用于逆散射问题的小波正则化方法的适当理论和快速算法。如果先验地知道反散射问题的解在某个小波基中是稀疏的，那么如果所研究的反问题在没有对解的附加假设的情况下不是唯一可解的，那么使用稀疏正则化方法就特别有用。事实上，在这种情况下，基于小波的稀疏性正则化方法将自动选择所有可能解中最稀疏的解。我们想用数值方法来证明这一特性，并应用于几个实际相关的合成问题和测量数据，例如，无相数据的逆电磁问题和后向散射问题。

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.