Regularization of nonlinear ill-posed problems in Banach spaces and conditional stability

• Applicant: Professor Dr. Bernd Hofmann
• Subject Area: Mathematics
• Term: from 2011 to 2016
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 190672901

• Final Report Year: 2016

Final Report Abstract

The work on this research project in the original three years period from 2011 to 2014 was a consequence of the trilateral Chinese-Finnish-German call in 2010 entitled ‘Inverse Problems Initiative’ with the aim to bring together relevant and competitive international research teams in the field of inverse problems. Funding within this initiative was available for multilateral collaborative research projects consisting of researchers from at least two of the partner countries. In particular, our project was based on a joint proposal of the TU Chemnitz German research group (PI: Bernd Hofmann) and the Fudan University Shanghai Chinese research group (PI: Jin Cheng). For completion of the project work from German side a fourth year was funded by DFG. Following the original research plan the work could be concluded by the end of 2016. In particular, this additional year allowed us to raise a couple of new and challenging questions which will be part of an Austrian-German (DACH) research project on regularization, in cooperation with Prof. Dr. Otmar Scherzer (Univ. Vienna), under preparation (acronym: SCIP) until spring 2017. The project to be reported has contributed to the regularization theory for nonlinear and locally ill-posed inverse problems. With respect to the full four years period its goals have been threefold. The first goal was directed towards extensions of results for variational (Tikhonov type) regularization with general convex penalties from a Hilbert space setting to forward operators mapping between Banach spaces, including the case of non-reflexive spaces. Published project results prove the capability of variational source conditions as a powerful tool for obtaining substantial knowledge and new convergence rates results. In particular, for the sparsity promoting ℓ1-regularization and elastic-net, improved rates could be derived from variational inequalities expressing the relevant local solution smoothness. As a second goal of the project one has to mention an improved understanding of the role of conditional stability estimates under the auspices of regularization. Specifically, it was not clear why a stability estimate additionally requires regularization for ensuring stable solutions to an ill-posed operator equation. Now it could be emphasized that using regularization under conditional stability is like putting into the hole while playing golf, where here the hole is the mostly unknown domain of stability. In this context, also the analysis of the Lavrentiev regularization appeared as an interesting new aspect and as a third goal.

Publications

• Regularization properties of the sequential discrepancy principle for Tikhonov regularization in Banach spaces. Appl. Anal., 93(7):1382–1400, 2014
  S. W. Anzengruber, B. Hofmann, P. Mathé
  (See online at https://doi.org/10.1080/00036811.2013.833326)
• The index function and Tikhonov regularization for ill-posed problems. J. Comput. Appl. Math., 265:110–119, 2014
  J. Cheng, B. Hofmann, S. Lu
  (See online at https://doi.org/10.1016/j.cam.2013.09.035)
• A unified approach to convergence rates for ℓ1-regularization and lacking sparsity. J. Inverse Ill-Posed Probl., 24(2):139– 148, 2016
  J. Flemming, B. Hofmann, I. Veselić
  (See online at https://doi.org/10.1515/jiip-2015-0058)
• Conditional stability versus ill-posedness for operator equations with monotone operators in Hilbert space. Inverse Problems, 32(12):125003 (23pp), 2016
  R.I. Boţ, B. Hofmann
  (See online at https://doi.org/10.1088/0266-5611/32/12/125003)
• Lavrentiev’s regularization method in Hilbert spaces revisited. Inverse Probl. Imaging, 10(3):741–764, 2016
  B. Hofmann, B. Kaltenbacher, E. Resmerita
  (See online at https://doi.org/10.3934/ipi.2016019)
• Optimal rates for Lavrentiev regularization with adjoint source conditions. Mathematics of Computation
  R. Plato, P. Mathé, B. Hofmann
  (See online at https://doi.org/10.1090/mcom/3237)
• Tikhonov regularization with oversmoothing penalties. Preprint 2016-8, Preprintreihe der Fakultät für Mathematik der Technischen Universität Chemnitz (ISSN 1614-8835), 2016
  D. Gerth
  
• Elastic-net regularization versus ℓ1-regularization for linear inverse problems with quasi-sparse solutions. Inverse Problems, 33(1):015004 (17pp), 2017
  D. Chen, B. Hofmann, J. Zou
  (See online at https://doi.org/10.1088/1361-6420/33/1/015004)