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Sampling type methods for an inverse waveguide problem
Inverse acoustic obstacle scattering problems using multifrequency measurements
1.  Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstrasse 69, A4040 Linz, Austria, Austria 
References:
[1] 
H.D. Alber and A. G. Ramm, Scattering amplitude and algorithm for solving the inverse scattering problem for a class of nonconvex obstacles, J. Math. Anal. Appl., 117 (1986), 570597. 
[2] 
G. Alessandrini and L. Rondi, Determining a soundsoft polyhedral scatterer by a single farfield measurement, Proc. Amer. Math. Soc., 133 (2005), 16851691 (electronic). 
[3] 
H. Ammari, J. Garnier, H. Kang, M. Lim and K. SΦlna, Multistatic imaging of extended targets, SIAM J. Imaging Sci., 5 (2012), 564600. 
[4] 
G. Bao and F. Triki, Error estimates for the recursive linearization of inverse medium problems, Journal of Computational Mathematics, 28 (2010), 725744. 
[5] 
O. Bucci, L. Crocco, T. Isernia and V. Pascazio, Inverse scattering problems with multifrequency data: reconstruction capabilities and solution strategies, IEEE Transactions on Geoscience and Remote Sensing, 38 (2000), 17491756. 
[6] 
F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory. An Introduction," Interaction of Mechanics and Mathematics, SpringerVerlag, Berlin, 2006. 
[7] 
Y. Chen, Inverse scattering via Heisenberg's uncertainty principle, Inverse Problems, 13 (1997), 253282. 
[8] 
J. Cheng and M. Yamamoto, Global uniqueness in the inverse acoustic scattering problem within polygonal obstacles, Chinese Ann. Math. Ser. B, 25 (2004), 16. 
[9] 
W. Chew and J. Lin, A frequencyhopping approach for microwave imaging of large inhomogeneous bodies, IEEE Microwave and Guided Wave Letters, 5 (1995), 439441. 
[10] 
D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," Second edition, Applied Mathematical Sciences, 93, SpringerVerlag, Berlin, 1998. 
[11] 
D. Colton and B. D. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering, IMA J. Appl. Math., 31 (1983), 253259. 
[12] 
G. B. Folland, "Fourier Analysis and its Applications," The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992. 
[13] 
D. Gintides, Local uniqueness for the inverse scattering problem in acoustics via the FaberKrahn inequality, Inverse Problems, 21 (2005), 11951205. 
[14] 
S. Gutman and A. G. Ramm, Support function method for inverse obstacle scattering problems, in "Acoustics, Mechanics, and the Related Topics of Mathematical Analysis," World Sci. Publ., River Edge, NJ, (2002), 179184. 
[15] 
F. Hettlich and W. Rundell, A second degree method for nonlinear inverse problems, SIAM J. Numer. Anal., 37 (2000), 587620 (electronic). 
[16] 
N. Honda, G. Nakamura and M. Sini, Analytic extension and reconstruction of obstacles from few measurements for elliptic second order operators, Mathematische Annalen, appeared online 04 February, 2012. doi: 10.1007/s0020801207860. 
[17] 
L. Hörmander, "The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis," Reprint of the second (1990) edition, Classics in Mathematics, SpringerVerlag, Berlin, 2003. 
[18] 
V. Isakov, "Inverse Problems for Partial Differential Equations," Second edition, Applied Mathematical Sciences, 127, Springer, New York, 2006. 
[19] 
A. Kirsch, The domain derivative and two applications in inverse scattering theory, Inverse Problems, 9 (1993), 8196. 
[20] 
A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems," Oxford Lecture Series in Mathematics and its Applications, 36, Oxford University Press, Oxford, 2008. 
[21] 
R. Kress, Newton's method for inverse obstacle scattering meets the method of least squares. Special section on imaging, Inverse Problems, 19 (2003), S91S104. 
[22] 
W. McLean, "Strongly Elliptic Systems and Boundary Integral Equations," Cambridge University Press, Cambridge, 2000. 
[23] 
R. Potthast, A survey on sampling and probe methods for inverse problems, Inverse Problems, 22 (2006), R1R47. 
[24] 
A. G. Ramm, "Multidimensional Inverse Scattering Problems," Pitman Monographs and Surveys in Pure and Applied Mathematics, 51, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992. 
[25] 
E. Sincich and M. Sini, Local stability for soft obstacles by a single measurement, Inverse Probl. Imaging, 2 (2008), 301315. 
[26] 
P. Stefanov and G. Uhlmann, Local uniqueness for the fixed energy fixed angle inverse problem in obstacle scattering, Proc. Amer. Math. Soc., 132 (2004), 13511354 (electronic). 
show all references
References:
[1] 
H.D. Alber and A. G. Ramm, Scattering amplitude and algorithm for solving the inverse scattering problem for a class of nonconvex obstacles, J. Math. Anal. Appl., 117 (1986), 570597. 
[2] 
G. Alessandrini and L. Rondi, Determining a soundsoft polyhedral scatterer by a single farfield measurement, Proc. Amer. Math. Soc., 133 (2005), 16851691 (electronic). 
[3] 
H. Ammari, J. Garnier, H. Kang, M. Lim and K. SΦlna, Multistatic imaging of extended targets, SIAM J. Imaging Sci., 5 (2012), 564600. 
[4] 
G. Bao and F. Triki, Error estimates for the recursive linearization of inverse medium problems, Journal of Computational Mathematics, 28 (2010), 725744. 
[5] 
O. Bucci, L. Crocco, T. Isernia and V. Pascazio, Inverse scattering problems with multifrequency data: reconstruction capabilities and solution strategies, IEEE Transactions on Geoscience and Remote Sensing, 38 (2000), 17491756. 
[6] 
F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory. An Introduction," Interaction of Mechanics and Mathematics, SpringerVerlag, Berlin, 2006. 
[7] 
Y. Chen, Inverse scattering via Heisenberg's uncertainty principle, Inverse Problems, 13 (1997), 253282. 
[8] 
J. Cheng and M. Yamamoto, Global uniqueness in the inverse acoustic scattering problem within polygonal obstacles, Chinese Ann. Math. Ser. B, 25 (2004), 16. 
[9] 
W. Chew and J. Lin, A frequencyhopping approach for microwave imaging of large inhomogeneous bodies, IEEE Microwave and Guided Wave Letters, 5 (1995), 439441. 
[10] 
D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," Second edition, Applied Mathematical Sciences, 93, SpringerVerlag, Berlin, 1998. 
[11] 
D. Colton and B. D. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering, IMA J. Appl. Math., 31 (1983), 253259. 
[12] 
G. B. Folland, "Fourier Analysis and its Applications," The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992. 
[13] 
D. Gintides, Local uniqueness for the inverse scattering problem in acoustics via the FaberKrahn inequality, Inverse Problems, 21 (2005), 11951205. 
[14] 
S. Gutman and A. G. Ramm, Support function method for inverse obstacle scattering problems, in "Acoustics, Mechanics, and the Related Topics of Mathematical Analysis," World Sci. Publ., River Edge, NJ, (2002), 179184. 
[15] 
F. Hettlich and W. Rundell, A second degree method for nonlinear inverse problems, SIAM J. Numer. Anal., 37 (2000), 587620 (electronic). 
[16] 
N. Honda, G. Nakamura and M. Sini, Analytic extension and reconstruction of obstacles from few measurements for elliptic second order operators, Mathematische Annalen, appeared online 04 February, 2012. doi: 10.1007/s0020801207860. 
[17] 
L. Hörmander, "The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis," Reprint of the second (1990) edition, Classics in Mathematics, SpringerVerlag, Berlin, 2003. 
[18] 
V. Isakov, "Inverse Problems for Partial Differential Equations," Second edition, Applied Mathematical Sciences, 127, Springer, New York, 2006. 
[19] 
A. Kirsch, The domain derivative and two applications in inverse scattering theory, Inverse Problems, 9 (1993), 8196. 
[20] 
A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems," Oxford Lecture Series in Mathematics and its Applications, 36, Oxford University Press, Oxford, 2008. 
[21] 
R. Kress, Newton's method for inverse obstacle scattering meets the method of least squares. Special section on imaging, Inverse Problems, 19 (2003), S91S104. 
[22] 
W. McLean, "Strongly Elliptic Systems and Boundary Integral Equations," Cambridge University Press, Cambridge, 2000. 
[23] 
R. Potthast, A survey on sampling and probe methods for inverse problems, Inverse Problems, 22 (2006), R1R47. 
[24] 
A. G. Ramm, "Multidimensional Inverse Scattering Problems," Pitman Monographs and Surveys in Pure and Applied Mathematics, 51, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992. 
[25] 
E. Sincich and M. Sini, Local stability for soft obstacles by a single measurement, Inverse Probl. Imaging, 2 (2008), 301315. 
[26] 
P. Stefanov and G. Uhlmann, Local uniqueness for the fixed energy fixed angle inverse problem in obstacle scattering, Proc. Amer. Math. Soc., 132 (2004), 13511354 (electronic). 
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