Monotonicity Principle in Tomography of Nonlinear Conducting Materials
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2020
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| _version_ | 1866914965942697984 |
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| author | Esposito, Antonio Corbo Faella, Luisa Piscitelli, Gianpaolo Prakash, Ravi Tamburrino, Antonello |
| author_facet | Esposito, Antonio Corbo Faella, Luisa Piscitelli, Gianpaolo Prakash, Ravi Tamburrino, Antonello |
| contents | We treat an inverse electrical conductivity problem which deals with the reconstruction of nonlinear electrical conductivity starting from boundary measurements in steady currents operations. In this framework, a key role is played by the Monotonicity Principle, which establishes a monotonic relation connecting the unknown material property to the (measured) Dirichlet-to-Neumann operator (DtN). Monotonicity Principles are the foundation for a class of non-iterative and real-time imaging methods and algorithms. In this article, we prove that the Monotonicity Principle for the Dirichlet Energy in nonlinear problems holds under mild assumptions. Then, we show that apart from linear and $p$-Laplacian cases, it is impossible to transfer this Monotonicity result from the Dirichlet Energy to the DtN operator. To overcome this issue, we introduce a new boundary operator, identified as an Average DtN operator. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2007_15431 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Monotonicity Principle in Tomography of Nonlinear Conducting Materials Esposito, Antonio Corbo Faella, Luisa Piscitelli, Gianpaolo Prakash, Ravi Tamburrino, Antonello Analysis of PDEs 35R30, 35J60 We treat an inverse electrical conductivity problem which deals with the reconstruction of nonlinear electrical conductivity starting from boundary measurements in steady currents operations. In this framework, a key role is played by the Monotonicity Principle, which establishes a monotonic relation connecting the unknown material property to the (measured) Dirichlet-to-Neumann operator (DtN). Monotonicity Principles are the foundation for a class of non-iterative and real-time imaging methods and algorithms. In this article, we prove that the Monotonicity Principle for the Dirichlet Energy in nonlinear problems holds under mild assumptions. Then, we show that apart from linear and $p$-Laplacian cases, it is impossible to transfer this Monotonicity result from the Dirichlet Energy to the DtN operator. To overcome this issue, we introduce a new boundary operator, identified as an Average DtN operator. |
| title | Monotonicity Principle in Tomography of Nonlinear Conducting Materials |
| topic | Analysis of PDEs 35R30, 35J60 |
| url | https://arxiv.org/abs/2007.15431 |