American Institute of Mathematical Sciences

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August  2016, 10(3): 765-780. doi: 10.3934/ipi.2016020

Reconstruction of penetrable obstacles in the anisotropic acoustic scattering

Yi-Hsuan Lin 1,

1. 

Department of Mathematics, National Taiwan University, Taipei 106, Taiwan

Received  February 2015 Revised  April 2016 Published  August 2016

We develop an enclosure-type reconstruction scheme to identify penetrable obstacles in acoustic waves with anisotropic medium in $\mathbb{R}^{3}$. The main difficulty of treating this problem lies in the fact that there are no complex geometrical optics solutions available for the acoustic equation with anisotropic medium in $\mathbb{R}^{3}$. Instead, we will use another type of special solutions called oscillating-decaying solutions. Even though that oscillating-decaying solutions are defined only on the half space, we are able to give necessary boundary inputs by the Runge approximation property. Moreover, since we are considering a Helmholtz-type equation, we turn to Meyers' $L^{p}$ estimate to compare the integrals coming from oscillating-decaying solutions and those from reflected solutions.
Keywords: oscillating-decaying solutions, Meyers $L^{p}$ estimates., reconstruction, Enclosure method, Runge approximation property.
Mathematics Subject Classification: Primary: 35R30; Secondary: 35J1.
Citation: 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
References:
[1]

L, Hörmander, The Analysis of Linear Partial Differential Operators III: Pseudo-differential Operators, volume 274. Springer Science & Business Media, 2007.  Google Scholar

[2]

M. Ikehata, How to draw a picture of an unknown inclusion from boundary measurements, Two mathematical inversion algorithms, J. Inverse Ill-Posed Probl., 7 (1999), 255-271. doi: 10.1515/jiip.1999.7.3.255.  Google Scholar

[3]

M. Ikehata, Enclosing a polygonal cavity in a two-dimensional bounded domain from cauchy data, Inverse Problems, 15 (1999), 1231-1241. doi: 10.1088/0266-5611/15/5/308.  Google Scholar

[4]

M. Ikehata, The enclosure method and its applications, In Analytic extension formulas and their applications, pages 87-103. Springer, 2001. doi: 10.1007/978-1-4757-3298-6_7.  Google Scholar

[5]

P. D. Lax, A stability theorem for solutions of abstract differential equations, and its application to the study of the local behavior of solutions of elliptic equations, Communications on Pure and Applied Mathematics, 9 (1956), 747-766. doi: 10.1002/cpa.3160090407.  Google Scholar

[6]

B. Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, In Annales de l'institut Fourier, volume 6, pages 271-355. Institut Fourier, 1956.  Google Scholar

[7]

V. G. Maz'ja, Sobolev Spaces, Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-09922-3.  Google Scholar

[8]

N. G. Meyers, Lp estimate for the gradient of solutions of second order elliptic divergence equations, Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze, 17 (1963), 189-206.  Google Scholar

[9]

S. Nagayasu, G. Uhlmann and J.-N. Wang, Reconstruction of penetrable obstacles in acoustic scattering, SIAM Journal on Mathematical Analysis, 43 (2011), 189-211. doi: 10.1137/09076218X.  Google Scholar

[10]

G. Nakamura, Applications of the oscillating-decaying solutions to inverse problems, In New analytic and geometric methods in inverse problems, pages 353-365. Springer, 2004.  Google Scholar

[11]

G. Nakamura, G. Uhlmann and J.-N. Wang, Oscillating-decaying solutions, runge approximation property for the anisotropic elasticity system and their applications to inverse problems, Journal de mathématiques pures et appliquées, 84 (2005), 21-54. doi: 10.1016/j.matpur.2004.09.002.  Google Scholar

[12]

M. Sini and K. Yoshida, On the reconstruction of interfaces using complex geometrical optics solutions for the acoustic case, Inverse Problems, 28 (2012), 055013. doi: 10.1088/0266-5611/28/5/055013.  Google Scholar

[13]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Annals of Mathematics, 125 (1987), 153-169. doi: 10.2307/1971291.  Google Scholar

[14]

H. Takuwa, G. Uhlmann and J.-N. Wang, Complex geometrical optics solutions for anisotropic equations and applications, Journal of Inverse and Ill-posed Problems, 16 (2008), 791-804. doi: 10.1515/JIIP.2008.049.  Google Scholar

[15]

G. Uhlmann and J.-N. Wang, Reconstructing discontinuities using complex geometrical optics solutions, SIAM Journal on Applied Mathematics, 68 (2008), 1026-1044. doi: 10.1137/060676350.  Google Scholar

show all references

References:
[1]

L, Hörmander, The Analysis of Linear Partial Differential Operators III: Pseudo-differential Operators, volume 274. Springer Science & Business Media, 2007.  Google Scholar

[2]

M. Ikehata, How to draw a picture of an unknown inclusion from boundary measurements, Two mathematical inversion algorithms, J. Inverse Ill-Posed Probl., 7 (1999), 255-271. doi: 10.1515/jiip.1999.7.3.255.  Google Scholar

[3]

M. Ikehata, Enclosing a polygonal cavity in a two-dimensional bounded domain from cauchy data, Inverse Problems, 15 (1999), 1231-1241. doi: 10.1088/0266-5611/15/5/308.  Google Scholar

[4]

M. Ikehata, The enclosure method and its applications, In Analytic extension formulas and their applications, pages 87-103. Springer, 2001. doi: 10.1007/978-1-4757-3298-6_7.  Google Scholar

[5]

P. D. Lax, A stability theorem for solutions of abstract differential equations, and its application to the study of the local behavior of solutions of elliptic equations, Communications on Pure and Applied Mathematics, 9 (1956), 747-766. doi: 10.1002/cpa.3160090407.  Google Scholar

[6]

B. Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, In Annales de l'institut Fourier, volume 6, pages 271-355. Institut Fourier, 1956.  Google Scholar

[7]

V. G. Maz'ja, Sobolev Spaces, Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-09922-3.  Google Scholar

[8]

N. G. Meyers, Lp estimate for the gradient of solutions of second order elliptic divergence equations, Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze, 17 (1963), 189-206.  Google Scholar

[9]

S. Nagayasu, G. Uhlmann and J.-N. Wang, Reconstruction of penetrable obstacles in acoustic scattering, SIAM Journal on Mathematical Analysis, 43 (2011), 189-211. doi: 10.1137/09076218X.  Google Scholar

[10]

G. Nakamura, Applications of the oscillating-decaying solutions to inverse problems, In New analytic and geometric methods in inverse problems, pages 353-365. Springer, 2004.  Google Scholar

[11]

G. Nakamura, G. Uhlmann and J.-N. Wang, Oscillating-decaying solutions, runge approximation property for the anisotropic elasticity system and their applications to inverse problems, Journal de mathématiques pures et appliquées, 84 (2005), 21-54. doi: 10.1016/j.matpur.2004.09.002.  Google Scholar

[12]

M. Sini and K. Yoshida, On the reconstruction of interfaces using complex geometrical optics solutions for the acoustic case, Inverse Problems, 28 (2012), 055013. doi: 10.1088/0266-5611/28/5/055013.  Google Scholar

[13]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Annals of Mathematics, 125 (1987), 153-169. doi: 10.2307/1971291.  Google Scholar

[14]

H. Takuwa, G. Uhlmann and J.-N. Wang, Complex geometrical optics solutions for anisotropic equations and applications, Journal of Inverse and Ill-posed Problems, 16 (2008), 791-804. doi: 10.1515/JIIP.2008.049.  Google Scholar

[15]

G. Uhlmann and J.-N. Wang, Reconstructing discontinuities using complex geometrical optics solutions, SIAM Journal on Applied Mathematics, 68 (2008), 1026-1044. doi: 10.1137/060676350.  Google Scholar

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