Convergence rates under a range invariance condition with application to electrical impedance tomography
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909152915226624 |
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| author | Kaltenbacher, Barbara |
| author_facet | Kaltenbacher, Barbara |
| contents | This paper is devoted to proving convergence rates of variational and iterative regularization methods under variational source conditions VSCs for inverse problems whose linearization satisfies a range invariance condition. In order to achieve this, often an appropriate relaxation of the problem needs to be found that is usually based on an augmentation of the set of unknowns and leads to a particularly structured reformulation of the inverse problem. We analyze three approaches that make use of this structure, namely a variational and a Newton type scheme, whose convergence without rates has already been established in \cite{rangeinvar}; additionally we propose a split minimization approach that can be show to satisfy the same rates results. \\ The range invariance condition has been verified for several coefficient identification problems for partial differential equations from boundary observations as relevant in a variety of tomographic imaging modalities. Our motivation particularly comes from the by now classical inverse problem of electrical impedance tomography EIT and we study both the original formulation by a diffusion type equation and its reformulation as a Schrödinger equation. For both of them we find relaxations that can be proven to satisfy the range invariance condition. Combining results on VSCs from \cite{Diss-Weidling} with the abstract framework for the three approaches mentioned above, we arrive at convergence rates results for the variational, split minimization and Newton type method in EIT. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_18704 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence rates under a range invariance condition with application to electrical impedance tomography Kaltenbacher, Barbara Numerical Analysis 65J22, 65N21, 35R30 This paper is devoted to proving convergence rates of variational and iterative regularization methods under variational source conditions VSCs for inverse problems whose linearization satisfies a range invariance condition. In order to achieve this, often an appropriate relaxation of the problem needs to be found that is usually based on an augmentation of the set of unknowns and leads to a particularly structured reformulation of the inverse problem. We analyze three approaches that make use of this structure, namely a variational and a Newton type scheme, whose convergence without rates has already been established in \cite{rangeinvar}; additionally we propose a split minimization approach that can be show to satisfy the same rates results. \\ The range invariance condition has been verified for several coefficient identification problems for partial differential equations from boundary observations as relevant in a variety of tomographic imaging modalities. Our motivation particularly comes from the by now classical inverse problem of electrical impedance tomography EIT and we study both the original formulation by a diffusion type equation and its reformulation as a Schrödinger equation. For both of them we find relaxations that can be proven to satisfy the range invariance condition. Combining results on VSCs from \cite{Diss-Weidling} with the abstract framework for the three approaches mentioned above, we arrive at convergence rates results for the variational, split minimization and Newton type method in EIT. |
| title | Convergence rates under a range invariance condition with application to electrical impedance tomography |
| topic | Numerical Analysis 65J22, 65N21, 35R30 |
| url | https://arxiv.org/abs/2403.18704 |