Variational Source Conditions, Quadratic Inverse Problems, Sparsity Promoting Regularization

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