Inverse problems arise in a wide range of applications in science and industry, which aim at recovering or designing physical parameters or quantities from observation data or certain desired targets. The fundamental challenge of the inverse problem lies in its ill-posedness in the sense of Hadamard. To overcome this difficulty, various regularization methods, which produce stable approximate solutions, have been proposed.

The goal of this project is to advance regularization theory for inverse problems in the Banach setting with the main focus on (i) convergence rates, (ii) order optimality, and (iii) converse results. These three 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.

This project is supported by the DFG research grants YO159/5-1.​

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De-Han Chen, Jingzhi Li, and Ye Zhang: A posterior contraction for Bayesian inverse problems in Banach spaces [PDF]Inverse Problems 40 (2024) 045011

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De-Han Chen, Daijun Jiang, Irwin Yousept, Jun Zou: Variational source conditions for inverse Robin and flux problems by partial measurements [PDF] Inverse Problems Imaging 16(2), 283-304, 2022

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De-Han Chen and Irwin Yousept: Variational source conditions in Lp-spaces [PDF], SIAM Journal on Mathematical Analysis 53(3), pp. 2863--2889, 2021

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De-Han Chen, Bernd Hofmann, Irwin Yousept: Oversmoothing Tikhonov regularization in Banach spaces [PDF] Inverse Problems  37 (2021) 085007