Generalized Tikhonov regularization and modern convergence rate theory in Banach spaces

Más información sobre el libro

The development of regularization methods for ill-posed inverse problems has been an ongoing topic in applied mathematics for decades, with Tikhonov's method emerging in the 1960s. Initially, research focused on Tikhonov regularization for linear equations in Hilbert spaces until the late 1980s. In the past 25 years, attention has shifted to nonlinear problems and Banach space settings. This evolution, driven by imaging applications, led to new and generalized versions of Tikhonov’s method, necessitating an expansion of the theoretical framework. Various authors have proposed different concepts to explore the behavior of these new approaches, complicating matters for those interested in the theoretical foundations without engaging in active research. This monograph presents and analyzes a comprehensive formulation of Tikhonov’s method that encompasses nearly all Tikhonov-type approaches found in the literature. An extensive example illustrates the application of these findings to a relevant imaging problem. Additionally, we examine the relationships between modern concepts for obtaining convergence rates, which are crucial in regularization theory. Our results demonstrate that many recent ideas can be unified, ultimately distilling them into just two variations of a single concept.