Abstract
In this article we propose a novel nonstationary iterated Tikhonov (nIT) type method for obtaining stable approximate solutions to ill-posed operator equations modeled by linear operators acting between Banach spaces. We propose a novel a posteriori strategy for choosing the sequence of regularization parameters (or, equivalently, the Lagrange multipliers) for the nIT iteration, aiming to obtain a fast decay of the residual.
Numerical experiments are presented for a 1D convolution problem (smooth Tikhonov functional and Banach parameter-space), and for a 2D deblurring problem (nonsmooth Tikhonov functional and Hilbert parameter-space).
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Notes
- 1.
Normally, the differentiability and convexity properties of this functional are independent of the particular choice of p > 1.
- 2.
Here (2) is replaced by Ag = y δ.
- 3.
For simplicity, all legends in this figure refers to the space L 1; however, we used p = 1.001 in the computations.
- 4.
For the purpose of comparison, the iteration error is plotted in the in L 2-norm for both choices of the parameter space X = L 2 and X = L 1.001.
- 5.
In this situation we have smooth penalization terms and Banach parameter-spaces.
- 6.
In this situation we have nonsmooth penalization terms and Hilbert parameter-spaces.
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Acknowledgements
F. Margotti is supported by CAPES and IMPA. A. Leitão acknowledges support from CNPq (grant 309767/2013-0), and from the AvH Foundation.
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Machado, M.P., Margotti, F., Leitão, A. (2018). On Nonstationary Iterated Tikhonov Methods for Ill-Posed Equations in Banach Spaces. In: Hofmann, B., Leitão, A., Zubelli, J. (eds) New Trends in Parameter Identification for Mathematical Models. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-70824-9_10
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DOI: https://doi.org/10.1007/978-3-319-70824-9_10
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