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

Reconstruction of penetrable obstacles in the anisotropic acoustic scattering

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.
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.  doi: 10.1515/jiip.1999.7.3.255.  Google Scholar

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

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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.  doi: 10.1002/cpa.3160090407.  Google Scholar

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B. Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution,, In Annales de l'institut Fourier, (1956), 271.   Google Scholar

[7]

V. G. Maz'ja, Sobolev Spaces,, Springer Series in Soviet Mathematics. Springer-Verlag, (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.   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.  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, (2004), 353.   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.  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).  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.  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.  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.  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.  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.  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, (2001), 87.  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.  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, (1956), 271.   Google Scholar

[7]

V. G. Maz'ja, Sobolev Spaces,, Springer Series in Soviet Mathematics. Springer-Verlag, (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.   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.  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, (2004), 353.   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.  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).  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.  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.  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.  doi: 10.1137/060676350.  Google Scholar

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