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On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach
Identification of obstacles using only the scattered P-waves or the scattered S-waves
1. | Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens |
2. | RICAM, Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040, Linz, Austria |
References:
[1] |
C. J. Alves and R. Kress, On the far-field operator in elastic obstacle scattering, IMA Journal of Applied Mathematics, 67 (2002), 1-21.
doi: 10.1093/imamat/67.1.1. |
[2] |
H. Ammari and H. Kang, "Polarization and Moment Tensors. With Applications to Inverse Problems and Effective Medium Theory," Applied Mathematical Sciences, 162, Springer, New York, 2007. |
[3] |
T. Arens, Linear sampling methods for 2D inverse elastic wave scattering, Inverse Problems, 17 (2001), 1445-1464.
doi: 10.1088/0266-5611/17/5/314. |
[4] |
K. Baganas, B. B. Guzina, A. Charalambopoulos and D. George, A linear sampling method for the inverse transmission problem in near-field elastodynamics, Inverse Problems, 22 (2006), 1835-1853.
doi: 10.1088/0266-5611/22/5/018. |
[5] |
M. Bonnet, "Boundary Integral Methods for Solids and Fluids," Wiley & Sons, 1995. |
[6] |
A. Charalambopoulos, D. Gintides and K. Kiriaki, The linear sampling method for the transmission problem in three-dimensional linear elasticity, Inverse Problems, 18 (2002), 547-558.
doi: 10.1088/0266-5611/18/3/303. |
[7] |
A. Charalambopoulos, D. Gintides and K. Kiriaki, The linear sampling method for non-absorbing penetrable elastic bodies, Inverse Problems, 19 (2003), 549-561.
doi: 10.1088/0266-5611/19/3/305. |
[8] |
D. Gintides and K. Kiriaki, The far-field equations in linear elasticity-An inversion scheme, ZAMM Z. Angew. Math. Mech., 81 (2001), 305-316.
doi: 10.1002/1521-4001(200105)81:5<305::AID-ZAMM305>3.0.CO;2-T. |
[9] |
B. B. Guzina and A. I. Madyarov, A linear sampling approach to inverse elastic scattering in piecewise-homogeneous domains, Inverse Problems, 23 (2007), 1467-1493.
doi: 10.1088/0266-5611/23/4/007. |
[10] |
P. Hahner and G. Hsiao, Uniqueness theorems in inverse obstacle scattering of elastic waves, Inverse Problems, 9 (1993), 525-534.
doi: 10.1088/0266-5611/9/5/002. |
[11] |
N. Honda, R. Potthast, G. Nakamura and M. Sini, The no-response approach and its relation to non-iterative methods for the inverse scattering, Ann. Mat. Pura Appl. (4), 187 (2008), 7-37. |
[12] |
M. Ikehata, Reconstruction of the shape of the inclusion by boundary measurements, Comm. Partial Differential Equations, 23 (1998), 1459-1474. |
[13] |
V. Isakov, On uniqueness in the inverse transmission scattering problem, Commun. Part. Diff. Equ., 15 (1990), 1565-1587. |
[14] |
A. Kirsch and R. Kress, Uniqueness in the inverse obstacle scattering problem, Inverse Problems, 9 (1993), 285-299.
doi: 10.1088/0266-5611/9/2/009. |
[15] |
V. D. Kupradze, "Potential Methods in the Theory of Elasticity," Israel Program for Scientific Translations, Jerusalem, Daniel Davey & Co., Inc., New York, 1965. |
[16] |
V. D. Kupradze, T. G. Gegelia, M. O. Basheleĭshvili and T. V. Burchuladze, "Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity," North-Holland Series in Applied Mathematics and Mechanics, 25, North-Holland Publishing Co., Amsterdam-New York, 1979. |
[17] |
J. J. Liu, G. Nakamura and M. Sini, Reconstruction of the shape and surface impedance from acoustic scattering data for an arbitrary cylinder, SIAM J. Appl. Math., 67 (2007), 1124-1146.
doi: 10.1137/060654220. |
[18] |
G. Nakamura and M. Sini, Obstacle and boundary determination from scattering data, SIAM J. Math. Anal., 39 (2007), 819-837.
doi: 10.1137/060658667. |
[19] |
G. Nakamura, R. Potthast and M. Sini, Unification of the probe and singular sources methods for the inverse boundary value problem by the no-response test, Comm. Partial Differential Equations, 31 (2006), 1505-1528. |
[20] |
F. Nintcheu and B. B. Guzina, Elastic scatterer reconstruction via the adjoint sampling method, SIAM J. Appl. Math., 67 (2007), 1330-1352.
doi: 10.1137/060653123. |
[21] |
R. Potthast, "Point Sources and Multipoles in Inverse Scattering Theory," Chapman & Hall/CRC Research Notes in Mathematics, 427, Chapman & Hall/CRC, Boca Raton, FL, 2001. |
[22] |
F. Simonetti, Localization of pointlike scatterers in solids with subwavelength resolution, Applied Physics Letter, 89 (2006), 094105.
doi: 10.1063/1.2338888. |
[23] |
V. A. Solonnikov, On Green's matrices for elliptic boundary value problems. I, Trudy Mat. Inst. Steklov., 110 (1970), 107-145. |
[24] |
V. A. Solonnikov, The Green's matrices for elliptic boundary value problems. II, Boundary Value Problems of Mathematical Physics, 7, Trudy. Math. Inst. Steklov., 116 (1971), 181-216. |
show all references
References:
[1] |
C. J. Alves and R. Kress, On the far-field operator in elastic obstacle scattering, IMA Journal of Applied Mathematics, 67 (2002), 1-21.
doi: 10.1093/imamat/67.1.1. |
[2] |
H. Ammari and H. Kang, "Polarization and Moment Tensors. With Applications to Inverse Problems and Effective Medium Theory," Applied Mathematical Sciences, 162, Springer, New York, 2007. |
[3] |
T. Arens, Linear sampling methods for 2D inverse elastic wave scattering, Inverse Problems, 17 (2001), 1445-1464.
doi: 10.1088/0266-5611/17/5/314. |
[4] |
K. Baganas, B. B. Guzina, A. Charalambopoulos and D. George, A linear sampling method for the inverse transmission problem in near-field elastodynamics, Inverse Problems, 22 (2006), 1835-1853.
doi: 10.1088/0266-5611/22/5/018. |
[5] |
M. Bonnet, "Boundary Integral Methods for Solids and Fluids," Wiley & Sons, 1995. |
[6] |
A. Charalambopoulos, D. Gintides and K. Kiriaki, The linear sampling method for the transmission problem in three-dimensional linear elasticity, Inverse Problems, 18 (2002), 547-558.
doi: 10.1088/0266-5611/18/3/303. |
[7] |
A. Charalambopoulos, D. Gintides and K. Kiriaki, The linear sampling method for non-absorbing penetrable elastic bodies, Inverse Problems, 19 (2003), 549-561.
doi: 10.1088/0266-5611/19/3/305. |
[8] |
D. Gintides and K. Kiriaki, The far-field equations in linear elasticity-An inversion scheme, ZAMM Z. Angew. Math. Mech., 81 (2001), 305-316.
doi: 10.1002/1521-4001(200105)81:5<305::AID-ZAMM305>3.0.CO;2-T. |
[9] |
B. B. Guzina and A. I. Madyarov, A linear sampling approach to inverse elastic scattering in piecewise-homogeneous domains, Inverse Problems, 23 (2007), 1467-1493.
doi: 10.1088/0266-5611/23/4/007. |
[10] |
P. Hahner and G. Hsiao, Uniqueness theorems in inverse obstacle scattering of elastic waves, Inverse Problems, 9 (1993), 525-534.
doi: 10.1088/0266-5611/9/5/002. |
[11] |
N. Honda, R. Potthast, G. Nakamura and M. Sini, The no-response approach and its relation to non-iterative methods for the inverse scattering, Ann. Mat. Pura Appl. (4), 187 (2008), 7-37. |
[12] |
M. Ikehata, Reconstruction of the shape of the inclusion by boundary measurements, Comm. Partial Differential Equations, 23 (1998), 1459-1474. |
[13] |
V. Isakov, On uniqueness in the inverse transmission scattering problem, Commun. Part. Diff. Equ., 15 (1990), 1565-1587. |
[14] |
A. Kirsch and R. Kress, Uniqueness in the inverse obstacle scattering problem, Inverse Problems, 9 (1993), 285-299.
doi: 10.1088/0266-5611/9/2/009. |
[15] |
V. D. Kupradze, "Potential Methods in the Theory of Elasticity," Israel Program for Scientific Translations, Jerusalem, Daniel Davey & Co., Inc., New York, 1965. |
[16] |
V. D. Kupradze, T. G. Gegelia, M. O. Basheleĭshvili and T. V. Burchuladze, "Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity," North-Holland Series in Applied Mathematics and Mechanics, 25, North-Holland Publishing Co., Amsterdam-New York, 1979. |
[17] |
J. J. Liu, G. Nakamura and M. Sini, Reconstruction of the shape and surface impedance from acoustic scattering data for an arbitrary cylinder, SIAM J. Appl. Math., 67 (2007), 1124-1146.
doi: 10.1137/060654220. |
[18] |
G. Nakamura and M. Sini, Obstacle and boundary determination from scattering data, SIAM J. Math. Anal., 39 (2007), 819-837.
doi: 10.1137/060658667. |
[19] |
G. Nakamura, R. Potthast and M. Sini, Unification of the probe and singular sources methods for the inverse boundary value problem by the no-response test, Comm. Partial Differential Equations, 31 (2006), 1505-1528. |
[20] |
F. Nintcheu and B. B. Guzina, Elastic scatterer reconstruction via the adjoint sampling method, SIAM J. Appl. Math., 67 (2007), 1330-1352.
doi: 10.1137/060653123. |
[21] |
R. Potthast, "Point Sources and Multipoles in Inverse Scattering Theory," Chapman & Hall/CRC Research Notes in Mathematics, 427, Chapman & Hall/CRC, Boca Raton, FL, 2001. |
[22] |
F. Simonetti, Localization of pointlike scatterers in solids with subwavelength resolution, Applied Physics Letter, 89 (2006), 094105.
doi: 10.1063/1.2338888. |
[23] |
V. A. Solonnikov, On Green's matrices for elliptic boundary value problems. I, Trudy Mat. Inst. Steklov., 110 (1970), 107-145. |
[24] |
V. A. Solonnikov, The Green's matrices for elliptic boundary value problems. II, Boundary Value Problems of Mathematical Physics, 7, Trudy. Math. Inst. Steklov., 116 (1971), 181-216. |
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