Uniqueness of the solution to inverse obstacle scattering problem

Ramm, Alexander G.

doi:10.1016/j.physleta.2005.08.088

Cited by 10 publications

(10 citation statements)

References 10 publications

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“…The columns in Table 6 correspond to the real and the imaginary parts of the scattered fields, and the rows correspond to different values of the angle α . Thus, one has to conclude that, as expected, a coincidence of the radiating solutions at the far field does not imply that the near fields are also coincidental, see [30].…”

Section: Inverse Scattering Methods Based On the Mrcmentioning

confidence: 84%

“…Suppose that an approximate location of the obstacle D is obtained a numerical inversion method, such as the Support Function Method (SFM), see [15,30]. Then one can try to use the MRC method to improve the location of the boundary, see [27].…”

Section: Inverse Scattering Methods Based On the Mrcmentioning

confidence: 99%

“…Various algorithms for direct and inverse scattering problems require global minimization of functions of many variables, see [30]. Since most objective functions contain many local minima, this is a highly nontrivial task.…”

Section: Stability Index Methodsmentioning

confidence: 99%

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Modified Rayleigh conjecture method and its applications

Ramm

Gutman

2008

Nonlinear Analysis: Theory, Methods & Applications

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The Rayleigh conjecture about convergence up to the boundary of the series representing the scattered field in the exterior of an obstacle D is widely used by engineers in applications. However this conjecture is false for some obstacles. AGR introduced the Modified Rayleigh Conjecture (MRC), which is an exact mathematical result. In this paper we present the theoretical basis for the MRC method for 2D and 3D obstacle scattering problems, for static problems, and for scattering by periodic structures. We also present successful numerical algorithms based on the MRC for various scattering problems. The MRC method is easy to implement for both simple and complex geometries. It is shown to be a viable alternative for other obstacle scattering methods. Various direct and inverse scattering problems require finding global minima of functions of several variables. The Stability Index Method (SIM) combines stochastic and deterministic method to accomplish such a minimization.

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Section: Inverse Scattering Methods Based On the Mrcmentioning

confidence: 84%

Section: Inverse Scattering Methods Based On the Mrcmentioning

confidence: 99%

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Modified Rayleigh conjecture method and its applications

Ramm

Gutman

2008

Nonlinear Analysis: Theory, Methods & Applications

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“…It is based on the results in [345,356,395,396,401]. The basic novelty in this theory consists in introducing and using Property C (completeness of the set of products of linear partial differential equations).…”

Section: Scattering By Potentialsmentioning

confidence: 99%

Back Matter

2017

Scattering by Obstacles and Potentials

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“…Only recently the uniqueness of the solution to inverse scattering problem with non-over-determined data A(−β, β, k) and A(β, α 0 , k) was proved, see [7], and [8], [12].…”

Section: Introductionmentioning

confidence: 99%

Inverse Scattering Problem with Underdetermined Data

Ramm

2014

Math. Model. Nat. Phenom.

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Consider the Schrödinger operator −∇ 2 + q with a smooth compactly supported potential q, q = q(x), x ∈ R 3 . Let A(β, α, k) be the corresponding scattering amplitude, k 2 be the energy, α ∈ S 2 be the incident direction, β ∈ S 2 be the direction of scattered wave, S 2 be the unit sphere in R 3 . Assume that k = k 0 > 0 is fixed, and α = α 0 is fixed. Then the scattering data are A(β) = A(β, α 0 , k 0 ) = A q (β) is a function on S 2 . The following inverse scattering problem is studied: IP: Given an arbitrary f ∈ L 2 (S 2 ) and an arbitrary small number > 0, can one find q ∈ C ∞ 0 (D), where D ∈ R 3 is an arbitrary fixed domain, such that ||A q (β) − f (β)|| L 2 (S 2 ) < ? A positive answer to this question is given. A method for constructing such a q is proposed. There are infinitely many such q, not necessarily real-valued.

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Uniqueness of the solution to inverse obstacle scattering problem

Cited by 10 publications

References 10 publications

Modified Rayleigh conjecture method and its applications

Modified Rayleigh conjecture method and its applications

Back Matter

Inverse Scattering Problem with Underdetermined Data

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