TY - JOUR

T1 - Local recovery of a piecewise constant anisotropic conductivity in EIT on domains with exposed corners

AU - de Hoop, Maarten V.

AU - Furuya, Takashi

AU - Lin, Ching Lung

AU - Nakamura, Gen

AU - Vashisth, Manmohan

N1 - Funding Information:
The first author was supported by the Simons Foundation under the MATH + X program, the National Science Foundation under Grant DMS-2108175, and the corporate members of the Geo-Mathematical Imaging Group at Rice University. The second author was supported by Grant-in-Aid for JSPS Fellows (No.21J00119), Japan Society for the Promotion of Science. The third author was partially supported by the Ministry of Science and Technology of Taiwan (Grant No. MOST 108-2115-M-006-018-MY3). The fourth author was supported by JSPS KAKENHI (Grant No. JP19K03554). The fifth author was supported by Start-up Research Grant SRG/2021/001432 from the Science and Engineering Research Board, Government of India.
Publisher Copyright:
© 2023 IOP Publishing Ltd.

PY - 2023/2

Y1 - 2023/2

N2 - We study the local recovery of an unknown piecewise constant anisotropic conductivity in electric impedance tomography on certain bounded Lipschitz domains Ω in R 2 with corners. The measurement is conducted on a connected open subset of the boundary ∂ Ω of Ω containing corners and is given as a localized Neumann-to-Dirichlet map. The above unknown conductivity is defined via a decomposition of Ω into polygonal cells. Specifically, we consider a parallelogram-based decomposition and a trapezoid-based decomposition. We assume that the decomposition is known, but the conductivity on each cell is unknown. We prove that the local recovery is almost surely true near a known piecewise constant anisotropic conductivity γ 0. We do so by proving that the injectivity of the Fréchet derivative F ′ ( γ 0 ) of the forward map F, say, at γ 0 is almost surely true. The proof presented, here, involves defining different classes of decompositions for γ 0 and a perturbation or contrast H in a proper way so that we can find in the interior of a cell for γ 0 exposed single or double corners of a cell of suppH for the former decomposition and latter decomposition, respectively. Then, by adapting the usual proof near such corners, we establish the aforementioned injectivity.

AB - We study the local recovery of an unknown piecewise constant anisotropic conductivity in electric impedance tomography on certain bounded Lipschitz domains Ω in R 2 with corners. The measurement is conducted on a connected open subset of the boundary ∂ Ω of Ω containing corners and is given as a localized Neumann-to-Dirichlet map. The above unknown conductivity is defined via a decomposition of Ω into polygonal cells. Specifically, we consider a parallelogram-based decomposition and a trapezoid-based decomposition. We assume that the decomposition is known, but the conductivity on each cell is unknown. We prove that the local recovery is almost surely true near a known piecewise constant anisotropic conductivity γ 0. We do so by proving that the injectivity of the Fréchet derivative F ′ ( γ 0 ) of the forward map F, say, at γ 0 is almost surely true. The proof presented, here, involves defining different classes of decompositions for γ 0 and a perturbation or contrast H in a proper way so that we can find in the interior of a cell for γ 0 exposed single or double corners of a cell of suppH for the former decomposition and latter decomposition, respectively. Then, by adapting the usual proof near such corners, we establish the aforementioned injectivity.

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U2 - 10.1088/1361-6420/acb008

DO - 10.1088/1361-6420/acb008

M3 - Article

AN - SCOPUS:85146849710

SN - 0266-5611

VL - 39

JO - Inverse Problems

JF - Inverse Problems

IS - 2

M1 - 025005

ER -