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Inverse scattering from an open arc. (English) Zbl 0824.35030
The first part of this paper studies the questions of uniqueness and existence of the direct scattering problem for an open arc with Dirichlet boundary conditions. Then the far field pattern is introduced and a reciprocity principle proved. In Section 4 the author investigates a special Nyström method based on the integral equation approach. Then the inverse problem is studied which is to determine the shape of the arc from the knowledge of the far field pattern. In Section 5 a uniqueness result is shown. Then the author suggests a Newton method for the numerical solution of the inverse scattering problem. He proves that the far field operator is differentiable and gives a characterization of the Fréchet derivative.

MSC:
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35P25 Scattering theory for PDEs
35R30 Inverse problems for PDEs
65Z05 Applications to the sciences
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[1] Atkinson, Math. Comp 56 pp 119– (1991)
[2] ’Die numerische Behandlung von Integralgleichungen erster Art bei Dirichletschen Rand-wertproblemen zur Helmholtzgleichung’, Diplomarbeit, Göttingen, 1994.
[3] and , ’On a quadrature method for a logarithmic integral equation of the first kind’, in: World Scientific Series in Applicable Analysis– Vol. 2. Contributions in Numerical Mathematics (ed.), pp. 127-140, World Scientific, Singapore, 1993.
[4] Colton, SIAM J. Math. Anal. 15 pp 996– (1984)
[5] and , Integral Equation Methods in Scattering Theory, Wiley-Interscience, New York, 1983.
[6] and , Integral Equation Methods in Scattering Theory, Mir, Moscow, 1987 (Russian translation of [5]).
[7] and , Inverse Acoustic and Electromagnetic Scattering Theory, Springer, Berlin, 1992. · Zbl 0760.35053
[8] Colton, SIAM J. Appl. Math. 45 pp 1039– (1985)
[9] Gieseke, Math. Z. 68 pp 54– (1964)
[10] Hayashi, J. Math. Anal. Appl. 44 pp 489– (1973)
[11] Applied and Computational Complex Analysis, Vol 3, Wiley-Interscience, New York, 1986.
[12] Isakov, Comm. Part. Diff. Eqs. 15 pp 1565– (1990)
[13] Kirsch, Inverse Problems 9 pp 81– (1993)
[14] ’Numerical algorithms in inverse scattering theory’, in: Ordinary and Partial Differential Equations ( and , eds), Pitman Research Notes in Mathematics 289, pp. 93-111, Longman, London, 1993.
[15] and , ’On an integral equation of the first kind in inverse acoustic scattering’, in: Inverse Problems ( and , eds), pp. 93-102, ISNM 77, 1986.
[16] Kirsch, Inverse Problems 9 pp 285– (1993)
[17] Linear Integral Equations. Springer, Berlin, Heidelberg, New York, 1989.
[18] Kress, J. Comp. Appl. Math. 42 pp 49– (1992)
[19] and , Scattering Theory, Academic Press, New York, 1967.
[20] Murch, Inverse Problems 4 pp 1117– (1988)
[21] Potthast, Inverse Problems 10 pp 431– (1994)
[22] and , Projektionsverfahren und die näherungsweise Lösung singulärer Gleichungen, Teubner, Leipzig, 1977. · Zbl 0364.65044
[23] Roger, IEEE Trans. Ant. Prop. AP-29 pp 232– (1981)
[24] Tobocman, Inverse Problems 5 pp 1131– (1989) · Zbl 0805.65130
[25] Wang, Wave Motion 13 pp 387– (1991)
[26] Yan, J. Integral Equations Appl. 1 pp 549– (1988)
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