American Institute of Mathematical Sciences

May  2008, 2(2): 301-315. doi: 10.3934/ipi.2008.2.301

Local stability for soft obstacles by a single measurement

 1 RICAM, Altenbergerstrasse 69, A4040, Linz, Austria, Austria

Received  October 2007 Revised  March 2008 Published  April 2008

We consider an inverse scattering problem arising in target identification. We prove a local stability result of logarithmic type for the determination of a sound soft obstacle from the far field measurements associated to one single incident wave.
Citation: Eva Sincich, Mourad Sini. Local stability for soft obstacles by a single measurement. Inverse Problems & Imaging, 2008, 2 (2) : 301-315. doi: 10.3934/ipi.2008.2.301
References:
 [1] R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65,, Academic Press, (1975).   Google Scholar [2] V. Adolfsson and L. Escauriaza, $C^{1,\a}$ domains and unique continuation at the boundary,, Comm. Pure Appl. Math, 50 (1997), 935.  doi: 10.1002/(SICI)1097-0312(199710)50:10<935::AID-CPA1>3.0.CO;2-H.  Google Scholar [3] G. Alessandrini, E. Beretta, E. Rosset and S. Vessella, "Optimal Stability for Inverse Elliptic Boundary Value Problems with Unknown Boundaries,", Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 29 (2000), 755.   Google Scholar [4] G. Alessandrini and A. Morassi, Strong unique continuation for the Lamè system of elasticity,, Comm. Partial Differential Equations, 26 (2001), 1787.  doi: 10.1081/PDE-100107459.  Google Scholar [5] G. Alessandrini and L. Rondi, Determining a sound-soft polyhedral scatterer by a single far-field measurement,, Proc. Amer. Math. Soc., 133 (2005), 1685.   Google Scholar [6] G. Alessandrini and E. Rosset, The inverse conductivity problem with one measurement: bounds on the size of the unknown object,, Siam J. Appl. Math., 58 (1998), 1060.  doi: 10.1137/S0036139996306468.  Google Scholar [7] I. Bushuyev, Stability of recovering the near-field wave from the scattering amplitude,, Inverse Problems, 12 (1996), 859.  doi: 10.1088/0266-5611/12/6/004.  Google Scholar [8] F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory,", Interaction of Mechanics and Mathematics, (2006).   Google Scholar [9] J. Cheng and M. Yamamoto, Uniqueness in an inverse scattering problem within non-trapping polygonal obstacles with at most two incoming waves,, [Inverse Problems, 19 (2003), 1361.   Google Scholar [10] D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory,", Pure and Applied Mathematics (New York), (1983).   Google Scholar [11] D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", Appl. Math. Sc. 93, (1992).   Google Scholar [12] D. Colton and B. D. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering,, IMA J. Appl. Math., 31 (1983), 253.  doi: 10.1093/imamat/31.3.253.  Google Scholar [13] J. Elschner and M. Yamamoto, Uniqueness in determining polygonal sound-hard obstacles with a single incoming wave,, Inverse Problems, 22 (2006), 355.  doi: 10.1088/0266-5611/22/1/019.  Google Scholar [14] P. R. Garabedian, "Partial Differential Equations,", Second edition, (1986).   Google Scholar [15] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Grundlehren der Mathematischen Wissenschaften, 224 (1977).   Google Scholar [16] D. Gintides, Local uniqueness for the inverse scattering problem in acoustics via the Faber-Krahn inequality,, Inverse Problems, 21 (2005), 1195.  doi: 10.1088/0266-5611/21/4/001.  Google Scholar [17] N. Honda, G. Nakamura and M. Sini, Analytic extention and reconstruction of obstacles from few measurements for elliptic second order operators,, RICAM Preprint series, (2008).   Google Scholar [18] V. Isakov, Stability estimates for obstacles in inverse scattering,, J. Comp. Appl. Math., 42 (1991), 79.  doi: 10.1016/0377-0427(92)90164-S.  Google Scholar [19] V. Isakov, New stability results for soft obstacles in inverse scattering,, Inverse Problems, 9 (1993), 535.  doi: 10.1088/0266-5611/9/5/003.  Google Scholar [20] D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains,, J. Func. Anal, 130 (1995), 161.  doi: 10.1006/jfan.1995.1067.  Google Scholar [21] H. Liu and J. Zou, Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers,, Inverse Problems, 22 (2006), 515.  doi: 10.1088/0266-5611/22/2/008.  Google Scholar [22] A. Morassi and E. Rosset, Stable determination of cavities in elastic bodies,, Inverse Problems, 20 (2004), 453.  doi: 10.1088/0266-5611/20/2/010.  Google Scholar [23] L. Rondi, Stable determination of sound-soft polyhedral scatterers by a single measurement,, to appear on Indiana Univ. Math. J., ().   Google Scholar [24] A. G. Ramm, "Inverse Problems, Mathematical and Analytical Techniques with Applications to Engineering,", Springer, (2004).   Google Scholar [25] E. Sincich, Stable determination of the surface impedance of an obstacle by far field measurements,, SIAM J. Math. Anal., 38 (2006), 434.  doi: 10.1137/050631513.  Google Scholar [26] E. Sincich, "Stability and Reconstruction for the Determination of Boundary Terms by a Single Measurements,", Ph.D. thesis, (2005).   Google Scholar [27] P. Stefanov and G. Uhlmann, Local uniqueness for the fixed energy fixed angle inverse problem in obstacle scattering,, Proc. Amer. Math. Soc., 132 (2004), 1351.  doi: 10.1090/S0002-9939-03-07363-5.  Google Scholar

show all references

References:
 [1] R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65,, Academic Press, (1975).   Google Scholar [2] V. Adolfsson and L. Escauriaza, $C^{1,\a}$ domains and unique continuation at the boundary,, Comm. Pure Appl. Math, 50 (1997), 935.  doi: 10.1002/(SICI)1097-0312(199710)50:10<935::AID-CPA1>3.0.CO;2-H.  Google Scholar [3] G. Alessandrini, E. Beretta, E. Rosset and S. Vessella, "Optimal Stability for Inverse Elliptic Boundary Value Problems with Unknown Boundaries,", Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 29 (2000), 755.   Google Scholar [4] G. Alessandrini and A. Morassi, Strong unique continuation for the Lamè system of elasticity,, Comm. Partial Differential Equations, 26 (2001), 1787.  doi: 10.1081/PDE-100107459.  Google Scholar [5] G. Alessandrini and L. Rondi, Determining a sound-soft polyhedral scatterer by a single far-field measurement,, Proc. Amer. Math. Soc., 133 (2005), 1685.   Google Scholar [6] G. Alessandrini and E. Rosset, The inverse conductivity problem with one measurement: bounds on the size of the unknown object,, Siam J. Appl. Math., 58 (1998), 1060.  doi: 10.1137/S0036139996306468.  Google Scholar [7] I. Bushuyev, Stability of recovering the near-field wave from the scattering amplitude,, Inverse Problems, 12 (1996), 859.  doi: 10.1088/0266-5611/12/6/004.  Google Scholar [8] F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory,", Interaction of Mechanics and Mathematics, (2006).   Google Scholar [9] J. Cheng and M. Yamamoto, Uniqueness in an inverse scattering problem within non-trapping polygonal obstacles with at most two incoming waves,, [Inverse Problems, 19 (2003), 1361.   Google Scholar [10] D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory,", Pure and Applied Mathematics (New York), (1983).   Google Scholar [11] D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", Appl. Math. Sc. 93, (1992).   Google Scholar [12] D. Colton and B. D. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering,, IMA J. Appl. Math., 31 (1983), 253.  doi: 10.1093/imamat/31.3.253.  Google Scholar [13] J. Elschner and M. Yamamoto, Uniqueness in determining polygonal sound-hard obstacles with a single incoming wave,, Inverse Problems, 22 (2006), 355.  doi: 10.1088/0266-5611/22/1/019.  Google Scholar [14] P. R. Garabedian, "Partial Differential Equations,", Second edition, (1986).   Google Scholar [15] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Grundlehren der Mathematischen Wissenschaften, 224 (1977).   Google Scholar [16] D. Gintides, Local uniqueness for the inverse scattering problem in acoustics via the Faber-Krahn inequality,, Inverse Problems, 21 (2005), 1195.  doi: 10.1088/0266-5611/21/4/001.  Google Scholar [17] N. Honda, G. Nakamura and M. Sini, Analytic extention and reconstruction of obstacles from few measurements for elliptic second order operators,, RICAM Preprint series, (2008).   Google Scholar [18] V. Isakov, Stability estimates for obstacles in inverse scattering,, J. Comp. Appl. Math., 42 (1991), 79.  doi: 10.1016/0377-0427(92)90164-S.  Google Scholar [19] V. Isakov, New stability results for soft obstacles in inverse scattering,, Inverse Problems, 9 (1993), 535.  doi: 10.1088/0266-5611/9/5/003.  Google Scholar [20] D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains,, J. Func. Anal, 130 (1995), 161.  doi: 10.1006/jfan.1995.1067.  Google Scholar [21] H. Liu and J. Zou, Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers,, Inverse Problems, 22 (2006), 515.  doi: 10.1088/0266-5611/22/2/008.  Google Scholar [22] A. Morassi and E. Rosset, Stable determination of cavities in elastic bodies,, Inverse Problems, 20 (2004), 453.  doi: 10.1088/0266-5611/20/2/010.  Google Scholar [23] L. Rondi, Stable determination of sound-soft polyhedral scatterers by a single measurement,, to appear on Indiana Univ. Math. J., ().   Google Scholar [24] A. G. Ramm, "Inverse Problems, Mathematical and Analytical Techniques with Applications to Engineering,", Springer, (2004).   Google Scholar [25] E. Sincich, Stable determination of the surface impedance of an obstacle by far field measurements,, SIAM J. Math. Anal., 38 (2006), 434.  doi: 10.1137/050631513.  Google Scholar [26] E. Sincich, "Stability and Reconstruction for the Determination of Boundary Terms by a Single Measurements,", Ph.D. thesis, (2005).   Google Scholar [27] P. Stefanov and G. Uhlmann, Local uniqueness for the fixed energy fixed angle inverse problem in obstacle scattering,, Proc. Amer. Math. Soc., 132 (2004), 1351.  doi: 10.1090/S0002-9939-03-07363-5.  Google Scholar
 [1] Pedro Serranho. A hybrid method for inverse scattering for Sound-soft obstacles in R3. Inverse Problems & Imaging, 2007, 1 (4) : 691-712. doi: 10.3934/ipi.2007.1.691 [2] Michele Di Cristo. Stability estimates in the inverse transmission scattering problem. Inverse Problems & Imaging, 2009, 3 (4) : 551-565. doi: 10.3934/ipi.2009.3.551 [3] Johannes Elschner, Guanghui Hu. Uniqueness in inverse transmission scattering problems for multilayered obstacles. Inverse Problems & Imaging, 2011, 5 (4) : 793-813. doi: 10.3934/ipi.2011.5.793 [4] Peijun Li, Ganghua Yuan. Increasing stability for the inverse source scattering problem with multi-frequencies. Inverse Problems & Imaging, 2017, 11 (4) : 745-759. doi: 10.3934/ipi.2017035 [5] Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems & Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291 [6] Rodica Toader. Scattering in domains with many small obstacles. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 321-338. doi: 10.3934/dcds.1998.4.321 [7] Frederic Weidling, Thorsten Hohage. Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems. Inverse Problems & Imaging, 2017, 11 (1) : 203-220. doi: 10.3934/ipi.2017010 [8] Christodoulos E. Athanasiadis, Vassilios Sevroglou, Konstantinos I. Skourogiannis. The inverse electromagnetic scattering problem by a mixed impedance screen in chiral media. Inverse Problems & Imaging, 2015, 9 (4) : 951-970. doi: 10.3934/ipi.2015.9.951 [9] Brian Sleeman. The inverse acoustic obstacle scattering problem and its interior dual. Inverse Problems & Imaging, 2009, 3 (2) : 211-229. doi: 10.3934/ipi.2009.3.211 [10] Andreas Kirsch, Albert Ruiz. The Factorization Method for an inverse fluid-solid interaction scattering problem. Inverse Problems & Imaging, 2012, 6 (4) : 681-695. doi: 10.3934/ipi.2012.6.681 [11] Michael V. Klibanov. A phaseless inverse scattering problem for the 3-D Helmholtz equation. Inverse Problems & Imaging, 2017, 11 (2) : 263-276. doi: 10.3934/ipi.2017013 [12] John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems & Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181 [13] Teemu Tyni, Valery Serov. Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line. Inverse Problems & Imaging, 2019, 13 (1) : 159-175. doi: 10.3934/ipi.2019009 [14] Ming Li, Ruming Zhang. Near-field imaging of sound-soft obstacles in periodic waveguides. Inverse Problems & Imaging, 2017, 11 (6) : 1091-1105. doi: 10.3934/ipi.2017050 [15] Yi-Hsuan Lin. Reconstruction of penetrable obstacles in the anisotropic acoustic scattering. Inverse Problems & Imaging, 2016, 10 (3) : 765-780. doi: 10.3934/ipi.2016020 [16] Pedro Caro. On an inverse problem in electromagnetism with local data: stability and uniqueness. Inverse Problems & Imaging, 2011, 5 (2) : 297-322. doi: 10.3934/ipi.2011.5.297 [17] Aymen Jbalia. On a logarithmic stability estimate for an inverse heat conduction problem. Mathematical Control & Related Fields, 2019, 9 (2) : 277-287. doi: 10.3934/mcrf.2019014 [18] Giorgio Menegatti, Luca Rondi. Stability for the acoustic scattering problem for sound-hard scatterers. Inverse Problems & Imaging, 2013, 7 (4) : 1307-1329. doi: 10.3934/ipi.2013.7.1307 [19] Masaya Maeda, Hironobu Sasaki, Etsuo Segawa, Akito Suzuki, Kanako Suzuki. Scattering and inverse scattering for nonlinear quantum walks. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3687-3703. doi: 10.3934/dcds.2018159 [20] Francesco Demontis, Cornelis Van der Mee. Novel formulation of inverse scattering and characterization of scattering data. Conference Publications, 2011, 2011 (Special) : 343-350. doi: 10.3934/proc.2011.2011.343

2018 Impact Factor: 1.469