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Bouligand–Landweber iteration for a non-smooth ill-posed problem

  • Christian ClasonEmail author
  • Vu Huu Nhu
Article
  • 27 Downloads

Abstract

This work is concerned with the iterative regularization of a non-smooth nonlinear ill-posed problem where the forward mapping is merely directionally but not Gâteaux differentiable. Using a Bouligand subderivative of the forward mapping, a modified Landweber method can be applied; however, the standard analysis is not applicable since the Bouligand subderivative mapping is not continuous unless the forward mapping is Gâteaux differentiable. We therefore provide a novel convergence analysis of the modified Landweber method that is based on the concept of asymptotic stability and merely requires a generalized tangential cone condition. These conditions are verified for an inverse source problem for an elliptic PDE with a non-smooth Lipschitz continuous nonlinearity, showing that the corresponding Bouligand–Landweber iteration converges strongly for exact data as well as in the limit of vanishing data if the iteration is stopped according to the discrepancy principle. This is illustrated with a numerical example.

Mathematics Subject Classification

49K20 49K40 90C31 

Notes

Acknowledgements

The authors would like to thank the anonymous reviewers for their detailed and constructive comments that helped to significantly improve the presentation. This work was supported by the DFG under the Grants CL 487/2-1 and RO 2462/6-1, both within the priority programme SPP 1962 “Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization”.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of Duisburg-EssenEssenGermany

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