Multidimensional inverse scattering problems

Ramm, Alexander G.

doi:10.1109/diped.1999.822120

Cited by 110 publications

(159 citation statements)

References 4 publications

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“…This problem has been discussed in the literature for S being a plane, a sphere, etc, [1][2][3][4], [6], [7]. Inversion formulas, which allow one to calculate f given m(y, r) for all y ∈ S and all r > 0, were obtained for the above cases.…”

Section: This Assumption Is Relaxed Below: It Is Sufficient To Assumementioning

confidence: 99%

“…754-756], [4, pp. 170-175]) are not suitable for practical calculations, and some are ( [6], [7], [2]). The goal of this paper is to give a fairly general condition on S sufficient for M to be injective.…”

Section: This Assumption Is Relaxed Below: It Is Sufficient To Assumementioning

confidence: 99%

“…These formulas are obtained by a method different from the one used in [1][2][3][4]. The injectivity result we obtain, is based on the results in [6], see also [7][8][9].…”

Section: Introductionmentioning

confidence: 99%

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Injectivity of the spherical means operator

Ramm

2002

Comptes Rendus. Mathématique

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Abstract. Let S be a surface in R n which divides the space into two connected components D1 and D2. Let f ∈ C0(R n ) be some real-valued compactly supported function with supp f ⊂ D1. Consider M f := m(y, r) := R n f (z)δ(|y − z| − r)dz, where δ is the delta-function, y ∈ S and r > 0 are arbitrary. A general, local at infinity, condition on S is given, under which M is injective, that is, M f = 0 implies f = 0. The injectivity result is extended to the case when the Fourier transform of f is quasianalytic, so that compactness of support of f is not assumed. A sufficient condition on S is given, under which M −1 can be analytically constructed. Two examples of inversion formulas are given: when S is a plane, and when S is a sphere. These formulas can be used in applications. c 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Injectivité de l'operateur de moyen spheriqueRésumé. Soit S une surface de R n qui divise l'espace en deux composantes connectées D1 and D2.

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Section: This Assumption Is Relaxed Below: It Is Sufficient To Assumementioning

confidence: 99%

Section: This Assumption Is Relaxed Below: It Is Sufficient To Assumementioning

confidence: 99%

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Injectivity of the spherical means operator

Ramm

2002

Comptes Rendus. Mathématique

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“…Therefore it is of interest to experiment numerically with other inversion methods. In [7], p.94, (see also [8,11,12]) a method (SFM-support function method) is proposed for recovery of strictly convex obstacles from the scattering amplitude. This method allows one to recover the support function of the obstacle, and the boundary of the obstacle is obtained from this function explicitly.…”

Section: Remarkmentioning

confidence: 99%

Analysis of a Linear Sampling Method for Identification of Obstacles

Ramm

Gutman

2005

Acta Mathematicae Applicatae Sinica, English Series

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AnalysisDuring the last decade there are many papers published, in which linear sampling methods for identification of an obstacle D are proposed. Such methods (see, for example, [2,3,6]) are based on a numerical verification of the inclusion of some function f := f (α, z), z ∈ R 3 , α ∈ S 2 , in the range R(B) of a certain operator B. It is proved in this paper that the methods, proposed in the above papers, have essential difficulties. This will also be demonstrated by numerical experiments. Although it is true that f ∈ R(B) when z ∈ D, it turns out that in any neighborhood of f , however small, there are elements from R(B). Also, although f ∈ R(B) when z ∈ D, there are elements in every neighborhood of f , however small, which do not belong to R(B) even if z ∈ D. Therefore it is not possible to construct a stable numerical method for the identification of D based on checking the inclusions f ∈ R(B) and f ∈ R(B).We prove below that the range R(B) is dense in the space L 2 (S 2 ). Assumption (A). We assume throughout that k 2 is not a Dirichlet eigenvalue of the Laplacian in D.Let us introduce some notations: N (B) and R(B) are, respectively, the null-space and the range of a linear operator B, D ∈ R 3 is a bounded domain (obstacle) with a smooth boundary S, D = R 3 \ D, u 0 = e ikα·x , k =const> 0, α ∈ S 2 is a unit vector, N is the unit normal to S pointing into D , g = g(x, y, k) := g(|x − y|) := e ik|x−y| 4π|x−y| , f := e −ikα ·z , where z ∈ R 3 and α ∈ S 2 , α := xr −1 , r = |x|, u = u (x, α, k) is the scattering solution: Manuscript

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“…In this case the scattering is adequately described by the Born approximation, and there are methods for the solution of the above Inverse Problems. See [8], [9], [23] and [24] for further details. However, when such an assumption is inappropriate, the preferable method is to match the given observations to a set of solutions for the Direct Problem.…”

Section: Best Fit Profilesmentioning

confidence: 99%

Identification of Multilayered Particles from Scattering Data by a Clustering Method

Gutman

2000

Journal of Computational Physics

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A multilayered particle is illuminated by plane acoustic or electromagnetic waves of one or several frequencies. We consider the inverse scattering problem for the identification of the layers and of the refraction coefficients of the scatterer in a non-Born region of scattering. Local deterministic and global probabilistic minimization methods are studied. A special Reduction Procedure is introduced to reduce the dimensionality of the minimization space. Deep's and the Multilevel Single-Linkage methods for global minimization are used for the solution of the inverse problem. Their performance is analyzed for various multilayer configurations.

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Multidimensional inverse scattering problems

Cited by 110 publications

References 4 publications

Injectivity of the spherical means operator

Injectivity of the spherical means operator

Analysis of a Linear Sampling Method for Identification of Obstacles

Identification of Multilayered Particles from Scattering Data by a Clustering Method

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