Hanke–Raus rule for Landweber iteration in Banach spaces

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 i...

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Bibliographic Details
Published in:Numerische Mathematik Vol. 156; no. 1; pp. 345 - 373
Main Author: Real, Rommel R.
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2024
Springer Nature B.V
Subjects:
Banach spaces
Convergence
Inverse problems
Iterative methods
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Noise levels
Numerical Analysis
Numerical and Computational Physics
Parameters
Regularization
Simulation
Theoretical
65J20 Numerical solutions of ill-posed problems in abstract spaces
regularization
ISSN:0029-599X, 0945-3245
Online Access:Get full text
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Summary: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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ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-023-01389-1