Advances in Regularization Theory for Inverse Problems in Banach Spaces

• Applicant: Professor Dr. Irwin Yousept
• Subject Area: Mathematics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 513566305

The goal of this project is to advance regularization theory for inverse problems in the Banach setting with the main focus on (i) variational source conditions, (ii) order optimality, (iii) sufficient conditions for optimal convergence rates, and (iv) converse results. These four research subjects are of paramount importance in the study of linear and nonlinear regularization methods for ill-posed inverse problems. While in the Hilbert setting they have been extensively studied and seem to have reached an adequate stage of development, their investigations in Banach spaces are still wide open and pose great challenges due to the lack of an orthonormal system, particularly for those in the non-Banach-lattice setting. Based on the concepts of unconditionality, $R-$ and $\gamma$-boundedness, serving as a crucial extension of the Hilbertian orthogonality and boundedness, the project shall develop new techniques and theoretical fundaments for (i)-(iv) in Banach spaces through the use of a generalized Littlewood-Paley decomposition. The final objective (v) comprises the application of the abstract theory to the inverse source problem of Maxwell variational inequalities in the Banach setting arising from current detection and measurement in high-technological superconducting cables.

• DFG Programme: Research Grants
• Co-Investigator: Professor Dr. Bernd Hofmann