On the required number of electrodes for uniqueness and convex reformulation in an inverse coefficient problem
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866915190411362304 |
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| author | Brojatsch, Andrej Harrach, Bastian |
| author_facet | Brojatsch, Andrej Harrach, Bastian |
| contents | We introduce a computer-assisted proof for the required number of electrodes for uniqueness and global reconstruction for the inverse Robin transmission problem, where the corrosion function on the boundary of an interior object is to be determined from electrode current-voltage measurements. We consider the shunt electrode model where, in contrast to the standard Neumann boundary condition, the applied electrical current is only partially known. The aim is to determine the corrosion coefficient with a finite number of measurements.
In this paper, we present a numerically verifiable criterion that ensures unique solvability of the inverse problem, given a desired resolution. This allows us to explicitly determine the required number and position of the electrodes. Furthermore, we will present an error estimate for noisy data. By rewriting the problem as a convex optimization problem, our aim is to develop a globally convergent reconstruction algorithm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_00482 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the required number of electrodes for uniqueness and convex reformulation in an inverse coefficient problem Brojatsch, Andrej Harrach, Bastian Optimization and Control Numerical Analysis Analysis of PDEs 65M32, 35R30 We introduce a computer-assisted proof for the required number of electrodes for uniqueness and global reconstruction for the inverse Robin transmission problem, where the corrosion function on the boundary of an interior object is to be determined from electrode current-voltage measurements. We consider the shunt electrode model where, in contrast to the standard Neumann boundary condition, the applied electrical current is only partially known. The aim is to determine the corrosion coefficient with a finite number of measurements. In this paper, we present a numerically verifiable criterion that ensures unique solvability of the inverse problem, given a desired resolution. This allows us to explicitly determine the required number and position of the electrodes. Furthermore, we will present an error estimate for noisy data. By rewriting the problem as a convex optimization problem, our aim is to develop a globally convergent reconstruction algorithm. |
| title | On the required number of electrodes for uniqueness and convex reformulation in an inverse coefficient problem |
| topic | Optimization and Control Numerical Analysis Analysis of PDEs 65M32, 35R30 |
| url | https://arxiv.org/abs/2411.00482 |