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 and 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, New York, San Francisco, London, 1975.

[2]

V. Adolfsson and L. Escauriaza, $C^{1,\a}$ domains and unique continuation at the boundary, Comm. Pure Appl. Math, 50 (1997), 935-969. doi: 10.1002/(SICI)1097-0312(199710)50:10<935::AID-CPA1>3.0.CO;2-H.

[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), 755-806.

[4]

G. Alessandrini and A. Morassi, Strong unique continuation for the Lamè system of elasticity, Comm. Partial Differential Equations, 26 (2001), 1787-1810. doi: 10.1081/PDE-100107459.

[5]

G. Alessandrini and L. Rondi, Determining a sound-soft polyhedral scatterer by a single far-field measurement, Proc. Amer. Math. Soc., 133 (2005), 1685-1691. Corrigendum, preprint, 2006 (down-loadable at http://www.arxiv.org/archive/math/). arXiv:0601406

[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), 1060-1071. doi: 10.1137/S0036139996306468.

[7]

I. Bushuyev, Stability of recovering the near-field wave from the scattering amplitude, Inverse Problems, 12 (1996), 859-867. doi: 10.1088/0266-5611/12/6/004.

[8]

F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory," Interaction of Mechanics and Mathematics, Springer-Verlag, Berlin, 2006.

[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), 1361-1384; MR2036535], Inverse Problems, 21 (2005).

[10]

D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory," Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.

[11]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," Appl. Math. Sc. 93, Springer-Verlag, Berlin, 1992.

[12]

D. Colton and B. D. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering, IMA J. Appl. Math., 31 (1983), 253-259. doi: 10.1093/imamat/31.3.253.

[13]

J. Elschner and M. Yamamoto, Uniqueness in determining polygonal sound-hard obstacles with a single incoming wave, Inverse Problems, 22 (2006), 355-364. doi: 10.1088/0266-5611/22/1/019.

[14]

P. R. Garabedian, "Partial Differential Equations," Second edition, Chelsea Publishing Co., New York, 1986.

[15]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977.

[16]

D. Gintides, Local uniqueness for the inverse scattering problem in acoustics via the Faber-Krahn inequality, Inverse Problems, 21 (2005), 1195-1205. doi: 10.1088/0266-5611/21/4/001.

[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).

[18]

V. Isakov, Stability estimates for obstacles in inverse scattering, J. Comp. Appl. Math., 42 (1991), 79-89. doi: 10.1016/0377-0427(92)90164-S.

[19]

V. Isakov, New stability results for soft obstacles in inverse scattering, Inverse Problems, 9 (1993), 535-543. doi: 10.1088/0266-5611/9/5/003.

[20]

D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Func. Anal, 130 (1995), 161-219. doi: 10.1006/jfan.1995.1067.

[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), 515-524. doi: 10.1088/0266-5611/22/2/008.

[22]

A. Morassi and E. Rosset, Stable determination of cavities in elastic bodies, Inverse Problems, 20 (2004), 453-480. doi: 10.1088/0266-5611/20/2/010.

[23]

L. Rondi, Stable determination of sound-soft polyhedral scatterers by a single measurement, to appear on Indiana Univ. Math. J., (available on line at http://www.iumj.indiana.edu/Preprints/3217.pdf).

[24]

A. G. Ramm, "Inverse Problems, Mathematical and Analytical Techniques with Applications to Engineering," Springer, 2004.

[25]

E. Sincich, Stable determination of the surface impedance of an obstacle by far field measurements, SIAM J. Math. Anal., 38 (2006), 434-451 (electronic). doi: 10.1137/050631513.

[26]

E. Sincich, "Stability and Reconstruction for the Determination of Boundary Terms by a Single Measurements," Ph.D. thesis, S.I.S.S.A./I.S.A.S., Trieste, Italy, 2005; available on line at http://www.sissa.it/library/.

[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), 1351-1354. doi: 10.1090/S0002-9939-03-07363-5.

show all references

References:
[1]

R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press, New York, San Francisco, London, 1975.

[2]

V. Adolfsson and L. Escauriaza, $C^{1,\a}$ domains and unique continuation at the boundary, Comm. Pure Appl. Math, 50 (1997), 935-969. doi: 10.1002/(SICI)1097-0312(199710)50:10<935::AID-CPA1>3.0.CO;2-H.

[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), 755-806.

[4]

G. Alessandrini and A. Morassi, Strong unique continuation for the Lamè system of elasticity, Comm. Partial Differential Equations, 26 (2001), 1787-1810. doi: 10.1081/PDE-100107459.

[5]

G. Alessandrini and L. Rondi, Determining a sound-soft polyhedral scatterer by a single far-field measurement, Proc. Amer. Math. Soc., 133 (2005), 1685-1691. Corrigendum, preprint, 2006 (down-loadable at http://www.arxiv.org/archive/math/). arXiv:0601406

[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), 1060-1071. doi: 10.1137/S0036139996306468.

[7]

I. Bushuyev, Stability of recovering the near-field wave from the scattering amplitude, Inverse Problems, 12 (1996), 859-867. doi: 10.1088/0266-5611/12/6/004.

[8]

F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory," Interaction of Mechanics and Mathematics, Springer-Verlag, Berlin, 2006.

[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), 1361-1384; MR2036535], Inverse Problems, 21 (2005).

[10]

D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory," Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.

[11]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," Appl. Math. Sc. 93, Springer-Verlag, Berlin, 1992.

[12]

D. Colton and B. D. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering, IMA J. Appl. Math., 31 (1983), 253-259. doi: 10.1093/imamat/31.3.253.

[13]

J. Elschner and M. Yamamoto, Uniqueness in determining polygonal sound-hard obstacles with a single incoming wave, Inverse Problems, 22 (2006), 355-364. doi: 10.1088/0266-5611/22/1/019.

[14]

P. R. Garabedian, "Partial Differential Equations," Second edition, Chelsea Publishing Co., New York, 1986.

[15]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977.

[16]

D. Gintides, Local uniqueness for the inverse scattering problem in acoustics via the Faber-Krahn inequality, Inverse Problems, 21 (2005), 1195-1205. doi: 10.1088/0266-5611/21/4/001.

[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).

[18]

V. Isakov, Stability estimates for obstacles in inverse scattering, J. Comp. Appl. Math., 42 (1991), 79-89. doi: 10.1016/0377-0427(92)90164-S.

[19]

V. Isakov, New stability results for soft obstacles in inverse scattering, Inverse Problems, 9 (1993), 535-543. doi: 10.1088/0266-5611/9/5/003.

[20]

D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Func. Anal, 130 (1995), 161-219. doi: 10.1006/jfan.1995.1067.

[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), 515-524. doi: 10.1088/0266-5611/22/2/008.

[22]

A. Morassi and E. Rosset, Stable determination of cavities in elastic bodies, Inverse Problems, 20 (2004), 453-480. doi: 10.1088/0266-5611/20/2/010.

[23]

L. Rondi, Stable determination of sound-soft polyhedral scatterers by a single measurement, to appear on Indiana Univ. Math. J., (available on line at http://www.iumj.indiana.edu/Preprints/3217.pdf).

[24]

A. G. Ramm, "Inverse Problems, Mathematical and Analytical Techniques with Applications to Engineering," Springer, 2004.

[25]

E. Sincich, Stable determination of the surface impedance of an obstacle by far field measurements, SIAM J. Math. Anal., 38 (2006), 434-451 (electronic). doi: 10.1137/050631513.

[26]

E. Sincich, "Stability and Reconstruction for the Determination of Boundary Terms by a Single Measurements," Ph.D. thesis, S.I.S.S.A./I.S.A.S., Trieste, Italy, 2005; available on line at http://www.sissa.it/library/.

[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), 1351-1354. doi: 10.1090/S0002-9939-03-07363-5.

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