On existence of a variational regularization parameter under Morozov's discrepancy principle
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915340808617984 |
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| author | Ding, Liang Li, Long Han, Weimin Wang, Wei |
| author_facet | Ding, Liang Li, Long Han, Weimin Wang, Wei |
| contents | Morozov's discrepancy principle is commonly adopted in Tikhonov regularization for choosing the regularization parameter. Nevertheless, for a general non-linear inverse problem, the discrepancy $\|F(x_α^δ)-y^δ\|_Y$ does not depend continuously on $α$ and it is questionable whether there exists a regularization parameter $α$ such that $τ_1δ\leq \|F(x_α^δ)-y^δ\|_Y\leq τ_2 δ$ $(1\le τ_1<τ_2)$. In this paper, we prove the existence of $α$ under Morozov's discrepancy principle if $τ_2\ge (3+2γ)τ_1$, where $γ>0$ is a parameter in a tangential cone condition for the nonlinear operator $F$. Furthermore, we present results on the convergence of the regularized solutions under Morozov's discrepancy principle. Numerical results are reported on the efficiency of the proposed approach. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_11397 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On existence of a variational regularization parameter under Morozov's discrepancy principle Ding, Liang Li, Long Han, Weimin Wang, Wei Numerical Analysis Optimization and Control 47J06 G.1.6 Morozov's discrepancy principle is commonly adopted in Tikhonov regularization for choosing the regularization parameter. Nevertheless, for a general non-linear inverse problem, the discrepancy $\|F(x_α^δ)-y^δ\|_Y$ does not depend continuously on $α$ and it is questionable whether there exists a regularization parameter $α$ such that $τ_1δ\leq \|F(x_α^δ)-y^δ\|_Y\leq τ_2 δ$ $(1\le τ_1<τ_2)$. In this paper, we prove the existence of $α$ under Morozov's discrepancy principle if $τ_2\ge (3+2γ)τ_1$, where $γ>0$ is a parameter in a tangential cone condition for the nonlinear operator $F$. Furthermore, we present results on the convergence of the regularized solutions under Morozov's discrepancy principle. Numerical results are reported on the efficiency of the proposed approach. |
| title | On existence of a variational regularization parameter under Morozov's discrepancy principle |
| topic | Numerical Analysis Optimization and Control 47J06 G.1.6 |
| url | https://arxiv.org/abs/2506.11397 |