Reconstruction of the support function for inclusion from boundary measurements

Ikehata, Masaru

doi:10.1515/jiip.2000.8.4.367

Cited by 109 publications

(123 citation statements)

References 14 publications

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“…The method for the proof of Theorem 1.1 is similar to the idea discovered in [4] and does not require a cylindrical structure of the domain. Therein we considered two variants of the inverse conductivity problem proposed by Calderón [2].…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of a source domain from the Cauchy data

Ikehata

1999

Inverse Problems

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We consider an inverse source problem for the Helmholtz equation in a bounded domain. The problem is to reconstruct the shape of the support of a source term from the Cauchy data on the boundary of the solution of the governing equation. We prove that if the shape is a polygon, one can calculate its support function from such data. An application to the inverse boundary value problem is also included.

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Section: Introductionmentioning

confidence: 99%

Reconstruction of a source domain from the Cauchy data

Ikehata

1999

Inverse Problems

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“…Here we are interested in the reconstruction question. Since our method shares the same spirit as Ikehata's enclosure method [11], [12], we will briefly describe Ikehata's ideas to motivate our method. Here we take γ ≡ 1, i.e.,γ = 1 + χ D γ 1 .…”

Section: Inverse Problemsmentioning

confidence: 99%

“…(see [11], [12]). To describe our method, we begin with the following integral inequalities given in [19] (also see [10] for a proof).…”

Section: Inverse Problemsmentioning

confidence: 99%

Reconstructing Discontinuities Using Complex Geometrical Optics Solutions

Uhlmann

Wang²

2008

SIAM J. Appl. Math.

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Abstract. In this paper we provide a framework for constructing general complex geometrical optics solutions for several systems of two variables that can be reduced to a system with the Laplacian as the leading order term. We apply these special solutions to the problem of reconstructing inclusions inside a domain filled with known conductivity from local boundary measurements. Computational results demonstrate the versatility of these solutions to determine electrical inclusions. Key words. geometrical optics, inclusions, numerical algorithm AMS subject classifications. 35R30, 65N21DOI. 10.1137/0606763501. Introduction. Inverse boundary value problems are a class of inverse problems where one attempts to determine the internal parameters of body by making measurements only at the surface of the body. A prototypical example that has received a lot of attention is electrical impedance tomography (EIT). In this inverse method one would like to determine the conductivity distribution inside a body by making voltage and current measurements at the boundary.There are many applications of EIT ranging from early breast cancer detection [32] to geophysical sensing for underground objects; see [18], [24], [25], [27]. The article [28] and the ones reviewed in [29] assume that the measurements are made on the whole boundary. However, it is often possible to make the measurements only on part of the boundary; this is the partial data problem. This is the case for the applications in breast cancer detection and geophysical sensing mentioned above.The boundary information is encoded into the Dirichlet-to-Neumann map associated with the conductivity equation. More precisely, let Ω be an open bounded domain with smooth boundary in R d with d = 2 or 3. Assume that γ(x) > 0 in Ω possesses a suitable regularity. The conductivity equation is described by the following elliptic equation:

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“…The enclosure method [2,3,4,5] is a methodology in inverse problems for partial differential equations. The method yields a partial information about the location of unknown discontinuity which appears as discontinuity of the coefficients of a partial differential equation or a part of the boundary of the common domain of definition of solutions of the equation.…”

Section: Introductionmentioning

confidence: 99%

Enclosure method and reconstruction of a linear crack in an elastic body

Ikehata

Itou

2008

J. Phys.: Conf. Ser.

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Abstract. In this paper we report a recent application of the idea of the enclosure method to an inverse problem related to a crack (inverse crack problem) in an elastic body. The problem is to extract information about the location and shape of an unknown crack from a single set of the surface displacement field and traction on the boundary of an arbitrary homogeneous elastic plate. Both anisotropic and isotropic bodies are considered. In states of both plane stress and plane strain, an extraction formula of an unknown crack is given provided: the crack is linear; one of two end points of the crack is known and located on the boundary of the body; a well-controlled surface traction is given on the boundary of the body.

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Reconstruction of the support function for inclusion from boundary measurements

Cited by 109 publications

References 14 publications

Reconstruction of a source domain from the Cauchy data

Reconstruction of a source domain from the Cauchy data

Reconstructing Discontinuities Using Complex Geometrical Optics Solutions

Enclosure method and reconstruction of a linear crack in an elastic body

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