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.
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