Recovery of the singularities of a potential from backscattering data in general dimension

Meroño, Cristóbal J.

doi:10.1016/j.jde.2018.11.003

Cited by 7 publications

(2 citation statements)

References 34 publications

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“…Thus, the Born approximation should recover the leading discontinuities of the potential. This property is well known in backscattering and in the fixed angle scattering problems, see among others [31,32,37,39,40]. The recover of singularities has also been studied in the Calderón problem with d = 2 in [19,25].…”

Section: The Born Approximation: the Radial Casementioning

confidence: 95%

The Born approximation in the three-dimensional Calderón problem

Barceló¹,

Castro²,

Macià³

et al. 2021

Preprint

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Uniqueness and reconstruction in the three-dimensional Calderón inverse conductivity problem can be reduced to the study of the inverse boundary problem for Schrödinger operators −∆ + q. We study the Born approximation of q in the ball, which amounts to studying the linearization of the inverse problem. We first analyze this approximation for real and radial potentials in any dimension. We show that this approximation is well-defined and obtain a closed formula that involves the spectrum of the Dirichlet-to-Neumann map associated to −∆ + q. We then turn to general real and essentially bounded potentials in three dimensions and introduce the notion of averaged Born approximation, which captures the invariance properties of the exact inverse problem. We obtain explicit formulas for the averaged Born approximation in terms of the matrix elements of the Dirichlet to Neumann map in the basis spherical harmonics. Motivated by these formulas we also study the high-energy behaviour of the matrix elements of the Dirichlet to Neumann map.1

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Section: The Born Approximation: the Radial Casementioning

confidence: 95%

The Born approximation in the three-dimensional Calderón problem

Barceló¹,

Castro²,

Macià³

et al. 2021

Preprint

Self Cite

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“…It is well known, for example, that it contains the leading singularities of q, and that one can reconstruct q explicitly from its Born approximation in the case of small potentials. See, respectively, [Mer18,Mer19] and [BCLV18,BCLV19] for recent results on both questions and for further references. This suggest that γ exp could have similar numerical applications.…”

Section: Introductionmentioning

confidence: 99%

The Born approximation in the three-dimensional Calderón problem II: Numerical reconstruction in the radial case

Barceló¹,

Castro²,

Macià³

et al. 2022

Preprint

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In this work we illustrate a number of properties of the Born approximation in the three-dimensional Calderón inverse conductivity problem by numerical experiments. The results are based on an explicit representation formula for the Born approximation recently introduced by the authors. We focus on the particular case of radial conductivities in the ball BR ⊂ R 3 of radius R, in which the linearization of the Calderon problem is equivalent to a Hausdorff moment problem. We give numerical evidences that the Born approximation is well defined for L ∞ conductivities, and present a novel numerical algorithm to reconstruct a radial conductivity from the Born approximation under a suitable smallness assumption. We also show that the Born approximation has depth-dependent uniqueness and approximation capabilities depending on the distance (depth) to the boundary ∂BR. We then investigate how increasing the radius R affects the quality of the Born approximation, and the existence of a scattering limit as R → ∞. Similar properties are also illustrated in the inverse boundary problem for the Schrödinger operator −∆ + q, and strong recovery of singularity results are observed in this case.

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Multidimensional Inverse Scattering for the Schrödinger Equation

Novikov

2022

Springer Proceedings in Mathematics &Amp; Statistics

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Recovery of the singularities of a potential from backscattering data in general dimension

Cited by 7 publications

References 34 publications

The Born approximation in the three-dimensional Calderón problem

The Born approximation in the three-dimensional Calderón problem

The Born approximation in the three-dimensional Calderón problem II: Numerical reconstruction in the radial case

Multidimensional Inverse Scattering for the Schrödinger Equation

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