The book collects and contributes new results on the theory and practice of ill-posed inverse
problems. Different notions of ill-posedness in Banach spaces for linear and nonlinear inverse
problems are discussed not only in standard settings but also in situations up to now not
covered by the literature. Especially ill-posedness of linear operators with uncomplemented
null spaces is examined.Tools for convergence rate analysis of regularization methods are
extended to a wider field of applicability. It is shown that the tool known as variational
source condition always yields convergence rate results. A theory for nonlinear inverse
problems with quadratic structure is developed as well as corresponding regularization methods.
The new methods are applied to a difficult inverse problem from laser optics.Sparsity promoting
regularization is examined in detail from a Banach space point of view. Extensive convergence
analysis reveals new insights into the behavior of Tikhonov-type regularization with sparsity
enforcing penalty.
