American Institute of Mathematical Sciences

Advanced
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.  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

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

[1]

Masahiro Ikeda, Takahisa Inui, Mamoru Okamoto, Yuta Wakasugi. $ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1967-2008. doi: 10.3934/cpaa.2019090
[2]

Y. Efendiev, Alexander Pankov. Meyers type estimates for approximate solutions of nonlinear elliptic equations and their applications. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 481-492. doi: 10.3934/dcdsb.2006.6.481
[3]

Karen Yagdjian, Anahit Galstian. Fundamental solutions for wave equation in Robertson-Walker model of universe and $L^p-L^q$ -decay estimates. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 483-502. doi: 10.3934/dcdss.2009.2.483
[4]

Selim Esedoḡlu, Fadil Santosa. Error estimates for a bar code reconstruction method. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1889-1902. doi: 10.3934/dcdsb.2012.17.1889
[5]

Samer Dweik. $ L^{p, q} $ estimates on the transport density. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3001-3009. doi: 10.3934/cpaa.2019134
[6]

Ting Zhou. Reconstructing electromagnetic obstacles by the enclosure method. Inverse Problems & Imaging, 2010, 4 (3) : 547-569. doi: 10.3934/ipi.2010.4.547
[7]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. $L^p$ Estimates for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 427-442. doi: 10.3934/dcds.2003.9.427
[8]

Seung-Yeal Ha, Mitsuru Yamazaki. $L^p$-stability estimates for the spatially inhomogeneous discrete velocity Boltzmann model. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 353-364. doi: 10.3934/dcdsb.2009.11.353
[9]

N. V. Krylov. Some $L_{p}$-estimates for elliptic and parabolic operators with measurable coefficients. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2073-2090. doi: 10.3934/dcdsb.2012.17.2073
[10]

Antonio Vitolo. $H^{1,p}$-eigenvalues and $L^\infty$-estimates in quasicylindrical domains. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1315-1329. doi: 10.3934/cpaa.2011.10.1315
[11]

Tadeusz Iwaniec, Gaven Martin, Carlo Sbordone. $L^p$-integrability & weak type $L^{2}$-estimates for the gradient of harmonic mappings of $\mathbb D$. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 145-152. doi: 10.3934/dcdsb.2009.11.145
[12]

Jian Lu, Huaiyu Jian. Topological degree method for the rotationally symmetric $L_p$-Minkowski problem. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 971-980. doi: 10.3934/dcds.2016.36.971
[13]

Kim Knudsen, Jennifer L. Mueller. The born approximation and Calderón's method for reconstruction of conductivities in 3-D. Conference Publications, 2011, 2011 (Special) : 844-853. doi: 10.3934/proc.2011.2011.844
[14]

Xinghong Pan, Jiang Xu. Global existence and optimal decay estimates of the compressible viscoelastic flows in $ L^p $ critical spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2021-2057. doi: 10.3934/dcds.2019085
[15]

Jaeho Choi, Nitin Krishna, Nicole Magill, Alejandro Sarria. On the $ L^p $ regularity of solutions to the generalized Hunter-Saxton system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6349-6365. doi: 10.3934/dcdsb.2019142
[16]

Tuan Anh Dao, Michael Reissig. $ L^1 $ estimates for oscillating integrals and their applications to semi-linear models with $ \sigma $-evolution like structural damping. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5431-5463. doi: 10.3934/dcds.2019222
[17]

Masaru Ikehata. On finding an obstacle with the Leontovich boundary condition via the time domain enclosure method. Inverse Problems & Imaging, 2017, 11 (1) : 99-123. doi: 10.3934/ipi.2017006
[18]

Masaru Ikehata, Mishio Kawashita. On finding a buried obstacle in a layered medium via the time domain enclosure method. Inverse Problems & Imaging, 2018, 12 (5) : 1173-1198. doi: 10.3934/ipi.2018049
[19]

Masaru Ikehata. On finding the surface admittance of an obstacle via the time domain enclosure method. Inverse Problems & Imaging, 2019, 13 (2) : 263-284. doi: 10.3934/ipi.2019014
[20]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences