Reconstruction of obstacle from boundary measurements

Ikehata, Masaru

doi:10.1016/s0165-2125(99)00006-2

Cited by 78 publications

(83 citation statements)

References 10 publications

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“…The proof of Lemma 2.7 is almost the same as that given in [12]. But for the reader's convenience we give the proof in Section 5.…”

Section: Preliminary Resultsmentioning

confidence: 95%

“…In [16], one numerical method is proposed to determine both ∂D and the impedance σ(x). In [1], the authors gave a uniqueness and reconstruction formula for identifying ∂D and the impedance for a Robin-type obstacle from the far-field pattern, by applying the probe method introduced by M. Ikehata (see [10], [11], [12], [13] and [14] for example). Moreover, it has also been noticed that the probe method, the singular sources method given in [23], as well as the factorization method [15], can be applied to determine the boundaries of multiple obstacles if their boundary types are the same (sound-soft or sound-hard).…”

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confidence: 99%

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Recovery of multiple obstacles by probe method

et al. 2009

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Abstract. We consider an inverse scattering problem for multiple obstacles D = N j=1 D j ⊂ R 3 with different types of boundary for D j . By constructing an indicator function from the far-field pattern of the scattered wave, we can firstly reconstruct the shape of all obstacles, then identify the type of boundary for each obstacle, as well as the boundary impedance in the case that obstacles have the Robin-type boundary condition. The novelty of our probe method compared with the existing probe method is that we succeeded in identifying the type of boundary condition for multiple obstacles by analyzing the behavior of both the imaginary part and the real part of the indicator function. The numerical realizations are given to show the performance of this inversion method.

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“…The proof of Lemma 2.7 is almost the same as that given in [12]. But for the reader's convenience we give the proof in Section 5.…”

Section: Preliminary Resultsmentioning

confidence: 95%

mentioning

confidence: 99%

Recovery of multiple obstacles by probe method

et al. 2009

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“…Let σ ∈ N x . We call the sequence ξ = {v n } of H 1 (Ω) solutions of the Helmholtz equation △v + λ 2 v = 0 a needle sequence for (x, σ) for the Helmholtz equation if it satisfies, for any compact set K of The existence of the needle sequence for the Helmholtz equation has been ensured under the additional condition on λ (see Appendixes of [11,18] for the proof):…”

Section: Needle Sequence For Helmholtz Equation and Vekua Transformmentioning

confidence: 99%

Probe method and a Carleman function

Ikehata¹

2007

Inverse Problems

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A Carleman function is a special fundamental solution with a large parameter for the Laplace operator and gives a formula to calculate the value of the solution of the Cauchy problem in a domain for the Laplace equation. The probe method applied to an inverse boundary value problem for the Laplace equation in a bounded domain is based on the existence of a special sequence of harmonic functions which is called a needle sequence. The needle sequence blows up on a special curve which connects a given point inside the domain with a point on the boundary of the domain and is convergent locally outside the curve. The sequence yields a reconstruction formula of unknown discontinuity, such as cavity, inclusion in a given medium from the Dirichlet-to-Neumann map. In this paper an explicit needle sequence in three dimensions is given in a closed-form. It is an application of a Carleman function introduced by Yarmukhamedov. Furthermore, an explicit needle sequence in the probe method applied to the reduction of inverse obstacle scattering problems with an arbitrary fixed wave number to inverse boundary value problems for the Helmholtz equation is also given.

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“…To determine the coefficients in (8), we replace u in (5) by (8). Before the computations, it is useful to note that…”

Section: Determination Of Coefficientsmentioning

confidence: 99%

“…The problem is the same as Inverse Problem 2 on p. 209 of Reference [8] and Section 5.2 on p. 608 of Reference [13]. We will not repeat the proof again and refer the readers to the above articles.…”

Section: Determination Of Coefficientsmentioning

confidence: 99%

An expansion theorem for two‐dimensional elastic waves and its application

Chen

Lin

2006

Math Methods in App Sciences

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SUMMARYWe prove an Atkinson-Wilcox-type expansion for two-dimensional elastic waves in this paper. The approach developed on the two-dimensional Helmholtz equation will be applied in the proof. When the elastic fields are involved, the situation becomes much harder due to two wave solutions propagating at different phase velocities. In the last section, we give an application about the reconstruction of an obstacle from the scattering amplitude.

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Reconstruction of obstacle from boundary measurements

Cited by 78 publications

References 10 publications

Recovery of multiple obstacles by probe method

Recovery of multiple obstacles by probe method

Probe method and a Carleman function

An expansion theorem for two‐dimensional elastic waves and its application

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