Regularization methods in Banach spaces

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Regularization methods for stable approximate solutions are essential for addressing inverse and ill-posed problems, which appear in diverse fields such as medical imaging, non-destructive testing, finance, and systems biology. Many of these issues involve parameter identification in partial differential equations (PDEs), making them computationally intensive and mathematically complex. Thus, there is a critical demand for stable, efficient solvers and rigorous convergence analyses for these methods. This monograph is divided into five parts. Part I emphasizes the significance of developing and analyzing regularization methods in Banach spaces, highlighting four applications that necessitate this framework and briefly discussing sparsity constraints. Part II provides a summary of the mathematical tools required for analysis in Banach spaces. Part III reviews the current advancements in Tikhonov regularization within these spaces. Part IV focuses on iterative regularization methods, addressing linear operator equations and the iterative solutions of nonlinear equations using gradient-type methods and the iteratively regularized Gauß-Newton method. Finally, Part V presents the method of approximate inverse, which relies on the efficient evaluation of measured data with reconstruction kernels.