Recent progress in the boundary control method

Belishev, M. I.

doi:10.1088/0266-5611/23/5/r01

Cited by 176 publications

(202 citation statements)

References 86 publications

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“…We wish to prove that 6) showing that the cross terms vanish in the limit. To show (3.6), let ε > 0, and decompose ψ = ψ 1 + ψ 2 where…”

Section: Gaussian Beam Quasimodesmentioning

confidence: 99%

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The Calderón problem in transversally anisotropic geometries

Ferreira¹,

Kurylev²,

Lassas³

et al. 2016

J. Eur. Math. Soc.

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Abstract. We consider the anisotropic Calderón problem of recovering a conductivity matrix or a Riemannian metric from electrical boundary measurements in three and higher dimensions. In the earlier work [13], it was shown that a metric in a fixed conformal class is uniquely determined by boundary measurements under two conditions: (1) the metric is conformally transversally anisotropic (CTA), and (2) the transversal manifold is simple. In this paper we will consider geometries satisfying (1) but not (2). The first main result states that the boundary measurements uniquely determine a mixed Fourier transform / attenuated geodesic ray transform (or integral against a more general semiclassical limit measure) of an unknown coefficient. In particular, one obtains uniqueness results whenever the geodesic ray transform on the transversal manifold is injective. The second result shows that the boundary measurements in an infinite cylinder uniquely determine the transversal metric. The first result is proved by using complex geometrical optics solutions involving Gaussian beam quasimodes, and the second result follows from a connection between the Calderón problem and Gel'fand's inverse problem for the wave equation and the boundary control method.

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“…We wish to prove that 6) showing that the cross terms vanish in the limit. To show (3.6), let ε > 0, and decompose ψ = ψ 1 + ψ 2 where…”

Section: Gaussian Beam Quasimodesmentioning

confidence: 99%

“…The corresponding two-dimensional result, involving an additional obstruction arising from the conformal invariance of the Laplace-Beltrami operator, is known [31]. See [6], [7] for another interesting approach to this problem.…”

Section: Introductionmentioning

confidence: 99%

The Calderón problem in transversally anisotropic geometries

Ferreira¹,

Kurylev²,

Lassas³

et al. 2016

J. Eur. Math. Soc.

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“…Using ideas of the BC method [13], we are able to extract the spectral data, λ k , a j k , j = 1, . .…”

Section: The Spectral Estimation Problem In Infinite Dimensional Spacesmentioning

confidence: 99%

On some applications of the boundary control method to spectral estimation and inverse problems

Avdonin

Mikhaylov

2015

Nanosist.: fiz. him. mat.

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“…Being originally proposed for solving the boundary IP for the multidimensional wave equation, the BC method has been successfully applied to all main types of linear equations of mathematical physics (see the review papers [29,30], monograph [31] and references therein). In this paper we use this method in a one-dimensional situation, applying it to the IP for the wave equation on the semi-axis.…”

Section: The Boundary Control Approach To Ips On the Semi-axismentioning

confidence: 99%

Inverse problem on the semi-axis: local approach

Avdonin

Belinskiy

Matthews³

2011

Tamkang J. Math.

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Abstract. We consider the problem of reconstruction of the potential for the wave equation on the semi-axis. We use the local versions of the Gelfand-Levitan and Krein equations, and the linear version of Simon's approach. For all methods, we reduce the problem of reconstruction to a second kind Fredholm integral equation, the kernel and the right-hand-side of which arise from an auxiliary second kind Volterra integral equation. A second-order accurate numerical method for the equations is described and implemented. Then several numerical examples verify that the algorithms can be used to reconstruct an unknown potential accurately. The practicality of each approach is briefly discussed. Accurate data preparation is described and implemented.

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Recent progress in the boundary control method

Cited by 176 publications

References 86 publications

The Calderón problem in transversally anisotropic geometries

The Calderón problem in transversally anisotropic geometries

On some applications of the boundary control method to spectral estimation and inverse problems

Inverse problem on the semi-axis: local approach

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