Novel Error Measures and Source Conditions of Regularization Methods for Inverse Problems (SCIP)

• Applicant: Professor Dr. Bernd Hofmann
• Subject Area: Mathematics
• Term: from 2018 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 391100538

• Final Report Year: 2022

Final Report Abstract

This project has aimed at the analytical and numerical treatment of operator equations with linear and preferably nonlinear forward operators in Banach and Hilbert spaces, which serve as models for ill-posed inverse problems. Our focus was on novel contributions to modern regularization theory in the sense of new results on convergence rates for different interesting varieties of regularization methods that were rarely treated in the literature of the past years. One important facet, with numerous new results obtained, was the treatment of variational regularization methods with oversmoothing penalties. To obtain convergence rates in regularization approaches for the stable approximate solution of ill-posed operator equations, additional conditions imposed on the potential solutions are required, which are usually called source conditions. Such source conditions are characterizations of various types of smoothness in a very general sense with respect to and in adaption to properties of the forward operators and to properties of the specific Banach spaces. Besides the long-established range-type source conditions, also approximate source conditions, conditional stability conditions, and since 2007 the variational source conditions play some prominent role. Moreover, for nonlinear inverse problems, structural nonlinearity conditions have to been taken into consideration. As important details of the project results to be emphasized should be mentioned first the comprehensive analytical and numerical studies on the interplay of conditional stability estimates and variational source conditions, and secondly the new assertions on convergence and convergence rates of Tikhonov-regularized solutions with oversmoothing penalties in Hilbert scales that go well beyond the previous state of knowledge. The certainly achieved main objective of the project has been some progress in the deeper understanding of the role of the specific source conditions and of the interplay of solutions smoothness and smoothing properties of the forward operators in the light of nonlinearity conditions whenever the inverse problem is nonlinear. This deeper understanding allows not only mathematicians, but also practitioners in natural science, engineering and finance, to use better designed models and regularization approaches for an efficient treatment of their own inverse problems in an adapted manner.

Publications

• Case studies and a pitfall for nonlinear variational regularization under conditional stability. Chapter 9 (pp. 177–203) in: Inverse Problems and Related Topics: Shanghai, China, October 12–14, 2018 (Eds.: J. Cheng, S. Lu and M. Yamamoto). Book 310 of Springer Proceedings in Mathematics & Statistics. Springer Nature, Singapore 2020
  D. Gerth, B. Hofmann and C. Hofmann
  
• Tikhonov regularization in Hilbert scales under conditional stability assumptions. Inverse Problems, 34(11):115015, 2018
  H. Egger and B. Hofmann
  
• Penalty-based smoothness conditions in convex variational regularization. Journal of Inverse and Ill-Posed Problems, 27(2):283–300, 2019
  B. Hofmann, S. Kindermann and P. Mathé
  
• Simultaneous identification of volatility and interest rate functions a two-parameter regularization approach. Electronic Transactions on Numerical Analysis, 51:99–117, 2019
  C. Hofmann, B. Hofmann and A. Pichler
  
• Tikhonov regularization with l^0-term complementing a convex penalty: l^1-convergence under sparsity constraints. Journal of Inverse and Ill-Posed Problems, 27(4):575–590, 2019
  W. Wang, S. Lu, B. Hofmann and J. Cheng
  
• Contributions to regularization theory and practice of certain nonlinear inverse problems. PhD thesis (Dissertation), TU Chemnitz, Fakultät für Mathematik
  Christopher Hofmann
  
• Convergence results and low-order rates for nonlinear Tikhonov regularization with oversmoothing penalty term. Electronic Transactions on Numerical Analysis, 53:313–328, 2020
  B. Hofmann and R. Plato
  
• The impact of the discrepancy principle on the Tikhonov-regularized solutions with oversmoothing Penalties. Mathematics, 8(3):331, 2020
  B. Hofmann and C. Hofmann
  
• Oversmoothing Tikhonov regularization in Banach spaces. Inverse Problems, 37(8):085007, 2021
  D.-H. Chen, B. Hofmann and I. Yousept
  
• The Hausdorff moment problem in the light of ill-posedness of type I. Eurasian Journal of Mathematical and Computer Applications, 9(2):57–87, 2021
  D. Gerth, B. Hofmann, C. Hofmann and S. Kindermann