Regularization methods in Banach spaces for inverse scattering problems

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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.