The Enclosure Method for the Anisotropic Maxwell System

Kuan, Rulin; Lin, Yi; Sini, Mourad

doi:10.1137/15100299x

Cited by 10 publications

(7 citation statements)

References 18 publications

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“…Such study together with the idea in Theorems 1.2 and 1.3 shall open a possibility of the application of the enclosure method to the stationary Maxwell system for an obstacle embedded in an absorbing medium. Note that, for a non-absorbing medium, some results realizing the enclosure method exist, see [21] and [20], and [28] which is based on the author's previous argument in [12].…”

Section: Extending Sini-yoshida's Another Results On a Penetrable Obs...mentioning

confidence: 99%

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Revisiting the probe and enclosure methods

Ikehata

2021

Preprint

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This paper is concerned with reconstruction issue of inverse obstacle problems governed by partial differential equations and consists of two parts.(i) The first part considers the foundation of the probe and enclosure methods for an impenetrable obstacle embedded in a medium governed by the stationary Schrödinger equation. Under a general framework, some natural estimates for a quantity computed from a pair of the Dirichlet and Neumann data on the outer surface of the body occupied by the medium are given. The estimates enables us to derive almost immediately the necessary asymptotic behaviour of indicator functions for both methods.(ii) The second one considers the realization of the enclosure method for a penetrable obstacle embedded in an absorbing medium governed by the stationary Schrödinger equation. The unknown obstacle considered here is modeled by a perturbation term added to the background complex-valued L ∞ -potential. Under a jump condition on the term across the boundary of the obstacle and some kind of regularity for the obstacle surface including Lipschitz one as a special case, the enclosure method using infinitely many pairs of the Dirichlet and Neumann data is established.

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Section: Extending Sini-yoshida's Another Results On a Penetrable Obs...mentioning

confidence: 99%

“…In this sense our approach is better to cover a broad class of boundary. Note that the comment also works for the articles [20] and [21] following their paper.…”

Section: This Yields the Expressionmentioning

confidence: 97%

Revisiting the probe and enclosure methods

Ikehata

2021

Preprint

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“…In these two articles ODS were also used to implement the enclosure method. After these two works, there are many papers discussing ODS for other equations with real anisotropic coefficients, for example [12] for anisotropic Maxwell equations, [14] for anisotropic real conductivity equations, etc.…”

Section: Enclosure Methods and Some Main Difficulties Of This Workmentioning

confidence: 99%

The enclosure method for a generalized anisotropic complex conductivity equation

Kuan¹

2021

Inverse Problems

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We study how to apply the enclosure method to reconstruct an unknown inclusion within a medium in a domain in R n which satisfies the conductivity equation ∇ ⋅ ((σ 0 + iɛ 0)∇u) = 0 with σ 0 and ɛ 0 being real matrix functions. Motivated by some real world applications, we assume the unknown inclusion satisfies an equation of the more general form ∇ ⋅ ( ( σ + i ε ) ∇ u + ζ ∇ u ̄ ) = 0 , where σ, ɛ, ζ are also real matrix functions. Due to the anisotropy, it is in general difficult to find complex geometric optics solutions. Therefore, we construct the oscillating decaying solutions, which is used to test whether a given half-space intersects the unknown inclusion or not.

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“…For the isotropic case, see, for example, [24,25,28,29,41,46,47,49]. For the anisotropic case in the three-dimensions, one can use the oscillating-decaying solutions to reconstruct unknown obstacles in a given medium, see [34,35,42,43]. It is worth mentioning that for the anisotropic case in the plane, one can also construct CGOs via the quasi-conformal map, and we refer readers to [48] for more detailed discussions.…”

Section: Introductionmentioning

confidence: 99%

A direct reconstruction algorithm for the anisotropic inverse conductivity problem based on Calderón’s method in the plane

et al. 2020

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A direct reconstruction algorithm based on Calderón's linearization method for the reconstruction of isotropic conductivities is proposed for anisotropic conductivities in two-dimensions. To overcome the non-uniqueness of the anisotropic inverse conductivity problem, the entries of the unperturbed anisotropic tensors are assumed known a priori, and it remains to reconstruct the multiplicative scalar field. The quasi-conformal map in the plane facilitates the Calderón-based approach for anisotropic conductivities. The method is demonstrated on discontinuous radially symmetric conductivities of high and low contrast.

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The Enclosure Method for the Anisotropic Maxwell System

Cited by 10 publications

References 18 publications

Revisiting the probe and enclosure methods

Revisiting the probe and enclosure methods

The enclosure method for a generalized anisotropic complex conductivity equation

A direct reconstruction algorithm for the anisotropic inverse conductivity problem based on Calderón’s method in the plane

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