Lipschitz Stable Determination of Polyhedral Conductivity Inclusions from Local Boundary Measurements

Aspri, Andrea; Beretta, Elena; Francini, Elisa; Vessella, Sergio

doi:10.1137/22m1480550

Cited by 6 publications

(2 citation statements)

References 36 publications

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“…Notice that, in general, a single couple (ϕ, f) is not sufficient to uniquely determine the unknown parameter σ [20]. However, two linearly independent pairs of measurements (ϕ 1 , f 1 ), (ϕ 2 , f 2 ) are sufficient to define the piecewise constant conductivity σ. Lipschitz stability results for the inverse geometrical problem using Dirichlet to Neumann maps have been established in [5] when the interface is polygonal and extended to polyhedral interfaces for 3D problems in [3]. Our adopted approach consists in transforming the inverse problem into an optimization one by constructing a cost function J modeling the energy gap between the solution of the direct problem and the solution of the following Dirichlet problem: v ∈ H 1 (Ω), such that…”

Section: The Inverse Problem and The Kohn-vogelius Functionalmentioning

confidence: 99%

A combination of Kohn-Vogelius and DDM methods for a geometrical inverse problem

Chaabane

Haddar²,

Jerbi

2023

Inverse Problems

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We consider the inverse geometrical problem of identifying the discontinuity curve of an electrical conductivity from boundary measurements. This standard inverse problem is used as a model to introduce and study a combined inversion algorithm coupling a gradient descent on the Kohn-Vogelius cost functional with a domain decomposition method that includes the unknown curve in the domain partitioning. We prove the local convergence of the method in a simpliﬁed case and numerically show its eﬃciency for some two dimensional experiments.

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Section: The Inverse Problem and The Kohn-vogelius Functionalmentioning

confidence: 99%

A combination of Kohn-Vogelius and DDM methods for a geometrical inverse problem

Chaabane

Haddar²,

Jerbi

2023

Inverse Problems

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“…This method is firstly developed in [26] for EIT, and we will use it for our problem (IP). For more about partial data problems, we refer the reader to [27][28][29][30] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Lipschitz stability of recovering the conductivity from internal current densities

Qiu

Zheng

2023

Inverse Problems

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In recent years, coupled physics imaging techniques have been developed to produce clearer images than those produced by electrical impedance tomography. This paper focuses on the inverse problem arising in current density impedance imaging and magneto-acousto-electric tomography.  We consider the electrostatic equation $\nabla\cdot(\sigma\nabla w_b) = 0$ in a bounded domain $\Omega\subset\mathbb{R}^3$ with either the Dirichlet or Neumann boundary condition $b$, where $\sigma$ is a scalar conductivity function. The inverse problem is formulated as recovering $\sigma$ from vector fields $J_b = \sigma\nabla w_b$ with different boundary conditions $b$.  We provide a local Lipschitz stability, stating that near some known $\sigma_0$ and under some regularity assumptions, we can find $b_1$ and $b_2$ by constructing complex geometrical optics (CGO) solutions such that $\|\ln\sigma^{(1)} - \ln\sigma^{(2)}\|_{C^{m,\alpha}(\overline{\Omega})} \le C\sum_{j=1}^2 \|J_{b_j}^{(1)} - J_{b_j}^{(2)}\|_{C^{m,\alpha}(\overline{\Omega})}$.  Furthermore, we modify the CGO solutions using the reflection method to make $b_1$ and $b_2$ vanish on a portion of a plane, and prove a local Lipschitz stability with partial data.

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Lipschitz stability estimate for the simultaneous recovery of two coefficients in the anisotropic Schrödinger type equation via local Cauchy data

Foschiatti

2024

Journal of Mathematical Analysis and Applications

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Lipschitz Stable Determination of Polyhedral Conductivity Inclusions from Local Boundary Measurements

Cited by 6 publications

References 36 publications

A combination of Kohn-Vogelius and DDM methods for a geometrical inverse problem

A combination of Kohn-Vogelius and DDM methods for a geometrical inverse problem

Lipschitz stability of recovering the conductivity from internal current densities

Lipschitz stability estimate for the simultaneous recovery of two coefficients in the anisotropic Schrödinger type equation via local Cauchy data

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