Numerical Realization of Algorithm for Exact Solution of Two-Dimensional Monochromatic inverse Problem of Acoustical Scattering

Bogatyrev, A V; Burov, V. A.; Morozov, S. A.; Rumyantseva, O. D.; Sukhov, E. G.

doi:10.1007/0-306-47107-8_8

Cited by 10 publications

(7 citation statements)

References 2 publications

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“…This is an important open problem for the acoustic case. In this connection, in addition to results of [32][33][34] and [4,6,7], we would like to mention also that.…”

Section: Open Problem For the Acoustic Casementioning

confidence: 87%

“…The work [32] was stimulated also by the "one-dimensional" works [25] and [21]. The reconstruction procedure of [18,27,28,30] was implemented numerically in [6][7][8].…”

Section: Remark 14mentioning

confidence: 99%

“…6) In addition, (1.5) is an equation for f (k, ·) on R d for each fixed k ∈ R d , k 2 = E, and the scattering amplitude arises as the restriction f | M E . We consider the following monochromatic inverse scattering problem for equation (1.1): Problem 1.1 Given f on M E at fixed E > 0, find v on R d (at least approximately, but sufficiently stably for numerical implementations).…”

Section: Introductionmentioning

confidence: 99%

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The $\bar{\partial}$ -Approach to Monochromatic Inverse Scattering in Three Dimensions

Novikov

2008

J Geom Anal

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We discuss a method for monochromatic inverse scattering in three dimensions of [Novikov in Int. Math. Res. Papers 2005(6):287-349, 2005] and implemented numerically in [Alekseenko et al. in Acoust. J. 54(3), 2008]. This method is obtained as a development of the∂-approach to inverse scattering at fixed energy in dimension d ≥ 3 of [Beals and Coifman in Proc. Symp. Pure Math. 43:45-70, 1985] and [Henkin and Novikov in Usp. Mat. Nauk 42(3):93-152, 1987] and involves, in particular, some results of [Faddeev in Itogi Nauki Tech. Sovr. Prob. Math. 3:93-180, 1965, 1974] and some ideas of the soliton theory (in particular, some ideas going back to [Manakov in Usp. Mat. Nauk 31(5):245-246, 1976] and [Dubrovin et al. in Dokl. Akad. Nauk SSSR 229:15-18, 1976]). Also, our studies go back, in particular, to [Regge in Nuovo Cimento 14:951-976, 1959]. This article is an extended version of the talk given at International Conference in Mathematics in honor of G. Henkin at the occasion of his 65th birthday.

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“…This is an important open problem for the acoustic case. In this connection, in addition to results of [32][33][34] and [4,6,7], we would like to mention also that.…”

Section: Open Problem For the Acoustic Casementioning

confidence: 87%

“…The work [32] was stimulated also by the "one-dimensional" works [25] and [21]. The reconstruction procedure of [18,27,28,30] was implemented numerically in [6][7][8].…”

Section: Remark 14mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The $\bar{\partial}$ -Approach to Monochromatic Inverse Scattering in Three Dimensions

Novikov

2008

J Geom Anal

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“…for n = 1, see [16,17,18,19]. In addition, as concerns numerical implementations of these algorithms for Problem 2 for n = 1, see [4,6], and references therein.…”

Section: Introductionmentioning

confidence: 99%

Monochromatic Reconstruction Algorithms for Two-dimensional Multi-channel Inverse Problems

Novikov

Santacesaria

2012

International Mathematics Research Notices

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Abstract. We consider two inverse problems for the multi-channel two-dimensional Schrödinger equation at fixed positive energy, i.e. the equation −∆ψ + V (x)ψ = Eψ at fixed positive E, where V is a matrixvalued potential. The first is the Gel'fand inverse problem on a bounded domain D at fixed energy and the second is the inverse fixed-energy scattering problem on the whole plane R 2 . We present in this paper two algorithms which give efficient approximate solutions to these problems: in particular, in both cases we show that the potential V is reconstructed with Lipschitz stability by these algorithms up to O(E −(m−2)/2 ) in the uniform norm as E → +∞, under the assumptions that V is m-times differentiable in L 1 , for m ≥ 3, and has sufficient boundary decay.

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“…In particular, in the present work we give first mathematically justified approach for reconstructing coefficient v from boundary measurements for (1.1) via inverse scattering without the assumption that E is not a Dirichlet eigenvalue for −∆ + v in D. In addition, numerical efficiency of related inverse scattering techniques was shown in [3], [9], [11], [12]; see also [8].…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of a Potential From the Impedance Boundary Map

Isaev¹,

Novikov²

2013

EJMCA

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We consider the inverse boundary value problem for the Schrödinger equation at fixed energy with boundary measurements represented as the impedance boundary map (or Robin-to-Robin map). We give formulas and equations for finding (generalized) scattering data for the aforementioned equation from boundary measurements in this impedance representation. Combining these results with results of the inverse scattering theory we obtain efficient methods for reconstructing potential from the impedance boundary map. To our knowledge, results of the present work are new already for the case of Neumannto-Dirichlet map.

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Numerical Realization of Algorithm for Exact Solution of Two-Dimensional Monochromatic inverse Problem of Acoustical Scattering

Cited by 10 publications

References 2 publications

The $\bar{\partial}$ -Approach to Monochromatic Inverse Scattering in Three Dimensions

The $\bar{\partial}$ -Approach to Monochromatic Inverse Scattering in Three Dimensions

Monochromatic Reconstruction Algorithms for Two-dimensional Multi-channel Inverse Problems

Reconstruction of a Potential From the Impedance Boundary Map

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