A class of regularization schemes for linear ill-posed problems in Banach spaces under low order source conditions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916937111437312 |
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| author | Plato, Robert |
| author_facet | Plato, Robert |
| contents | In this work we consider, in a Banach space framework, the regularization of linear ill-posed problems. Our focus is on the recovery of solutions that have a logarithmic source representation. Such cases typically occur in exponentially ill-posed problems like the backwards heat equation. The mathematical framework on logarithms of operators is provided, and convergence rates for a class of parametric regularization schemes are deduced. This class includes the iterated version of Lavrentiev's method and the method of the abstract Cauchy problem. The presentation includes both a priori and a posteriori parameter choice strategies. Finally, we present an example for a logarithmic source representable function with respect to the integration operator. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_05418 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A class of regularization schemes for linear ill-posed problems in Banach spaces under low order source conditions Plato, Robert Numerical Analysis 65J20 65J22 47D06 26A33 In this work we consider, in a Banach space framework, the regularization of linear ill-posed problems. Our focus is on the recovery of solutions that have a logarithmic source representation. Such cases typically occur in exponentially ill-posed problems like the backwards heat equation. The mathematical framework on logarithms of operators is provided, and convergence rates for a class of parametric regularization schemes are deduced. This class includes the iterated version of Lavrentiev's method and the method of the abstract Cauchy problem. The presentation includes both a priori and a posteriori parameter choice strategies. Finally, we present an example for a logarithmic source representable function with respect to the integration operator. |
| title | A class of regularization schemes for linear ill-posed problems in Banach spaces under low order source conditions |
| topic | Numerical Analysis 65J20 65J22 47D06 26A33 |
| url | https://arxiv.org/abs/2509.05418 |