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Hanke–Raus rule for Landweber iteration in Banach spaces

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Abstract

We consider the Landweber iteration for solving linear as well as nonlinear inverse problems in Banach spaces. Based on the discrepancy principle, we propose a heuristic parameter choice rule for choosing the regularization parameter which does not require the information on the noise level, so it is purely data-driven. According to a famous veto, convergence in the worst-case scenario cannot be expected in general. However, by imposing certain conditions on the noisy data, we establish a new convergence result which, in addition, requires neither the Gâteaux differentiability of the forward operator nor the reflexivity of the image space. Therefore, we also expand the applied range of the Landweber iteration to cover non-smooth ill-posed inverse problems and to handle the situation that the data is contaminated by various types of noise. Numerical simulations are also reported.

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Acknowledgements

The author also would like to thank Dr. Qinian Jin for the guidance in this research.

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  1. Department of Mathematics, Physics, and Computer Science, University of the Philippines Mindanao, Davao City, 8022, Philippines

    Rommel R. Real

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  1. Rommel R. Real
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Correspondence to Rommel R. Real.

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Real, R.R. Hanke–Raus rule for Landweber iteration in Banach spaces. Numer. Math. 156, 345–373 (2024). https://doi.org/10.1007/s00211-023-01389-1

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