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Recursive reconstruction of piecewise constant signals by minimization of an energy function
Inverse acoustic scattering using high-order small-inclusion expansion of misfit function
POEMS (ENSTA ParisTech, CNRS, INRIA, Université Paris-Saclay), 91120 Palaiseau, France |
This article concerns an extension of the topological derivative concept for 3D inverse acoustic scattering problems involving the identification of penetrable obstacles, whereby the featured data-misfit cost function $\mathbb{J}$ is expanded in powers of the characteristic radius $a$ of a single small inhomogeneity. The $O(a^6)$ approximation $\mathbb{J}_6$ of $\mathbb{J}$ is derived and justified for a single obstacle of given location, shape and material properties embedded in a 3D acoustic medium of arbitrary shape. The generalization of $\mathbb{J}_6$ to multiple small obstacles is outlined. Simpler and more explicit expressions of $\mathbb{J}_6$ are obtained when the scatterer is centrally-symmetric or spherical. An approximate and computationally light global search procedure, where the location and size of the unknown object are estimated by minimizing $\mathbb{J}_6$ over a search grid, is proposed and demonstrated on numerical experiments, where the identification from known acoustic pressure on the surface of a penetrable scatterer embedded in a acoustic semi-infinite medium, and whose shape may differ from that of the trial obstacle assumed in the expansion of $\mathbb{J}$, is considered.
References:
[1] |
H. Ammari, E. Iakovleva and S. Moskow,
Recovery of small inhomogeneities from the scattering amplitude at a fixed frequency, SIAM J. Math. Anal., 34 (2003), 882-890.
doi: 10.1137/S0036141001392785. |
[2] |
H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, Lecture Notes in Mathematics 1846. Springer-Verlag, 2004.
doi: 10.1007/b98245. |
[3] |
H. Ammari and H. Kang, Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences, Vol. 162. Springer-Verlag, 2007. |
[4] |
H. Ammari and A. Khelifi,
Electromagnetic scattering by small dielectric inhomogeneities, J. Maths Pures Appl., 82 (2003), 749-842.
doi: 10.1016/S0021-7824(03)00033-3. |
[5] |
H. Ammari, J. Garnier, V. Jugnon and H. Kang,
Stability and resolution analysis for a topological derivative based imaging functional, SIAM J. Contr. Opt., 50 (2012), 48-76.
doi: 10.1137/100812501. |
[6] |
C. Bellis and M. Bonnet,
A FEM-based topological sensitivity approach for fast qualitative identification of buried cavities from elastodynamic overdetermined boundary data, Int. J. Solids Struct., 47 (2010), 1221-1242.
|
[7] |
C. Bellis, M. Bonnet and F. Cakoni, Acoustic inverse scattering using topological derivative of far-field measurements-based $L^2$ cost functionals, Inverse Probl., 29 (2013), 075012, 30pp.
doi: 10.1088/0266-5611/29/7/075012. |
[8] |
A. Bendali, P. H. Cocquet and S. Tordeux,
Approximation by multipoles of the multiple acoustic scattering by small obstacles in three dimensions and application to the Foldy theory of isotropic scattering, Arch. Ration. Mech. An., 219 (2016), 1017-1059.
doi: 10.1007/s00205-015-0915-5. |
[9] |
M. Bonnet, Inverse acoustic scattering by small-obstacle expansion of misfit function, Inverse Probl., 24 (2008), 035022, 27pp.
doi: 10.1088/0266-5611/24/3/035022. |
[10] |
M. Bonnet,
Fast identification of cracks using higher-order topological sensitivity for 2-D potential problems, Eng. Anal. Bound. Elem., 35 (2011), 223-235.
doi: 10.1016/j.enganabound.2010.08.007. |
[11] |
M. Bonnet,
A modified volume integral equation for anisotropic elastic or conducting inhomogeneities. Unconditional solvability by Neumann series, J. Integral Eq. Appl., 29 (2017), 271-295.
doi: 10.1216/JIE-2017-29-2-271. |
[12] |
M. Bonnet and R. Cornaggia,
Higher order topological derivatives for three-dimensional anisotropic elasticity, ESAIM: Math. Modell. Numer. Anal., 51 (2017), 2069-2092.
doi: 10.1051/m2an/2017015. |
[13] |
M. Bonnet and B. B. Guzina,
Sounding of finite solid bodies by way of topological derivative, Int. J. Num. Meth. Eng., 61 (2004), 2344-2373.
doi: 10.1002/nme.1153. |
[14] |
F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory, Springer, New York, 2014.
doi: 10.1007/978-1-4614-8827-9. |
[15] |
D.J. Cedio-Fengya, S. Moskow and M. Vogelius,
Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction, Inverse Probl., 14 (1998), 553-595.
doi: 10.1088/0266-5611/14/3/011. |
[16] |
P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2013. |
[17] |
G. R. Feijóo,
A new method in inverse scattering based on the topological derivative, Inverse Probl., 20 (2004), 1819-1840.
doi: 10.1088/0266-5611/20/6/008. |
[18] |
A. D. Ferreira and A. A. Novotny, A new non-iterative reconstruction method for the electrical impedance tomography problem, Inverse Probl., 33 (2017), 035005, 27pp.
doi: 10.1088/1361-6420/aa54e4. |
[19] |
N. A. Gumerov and R. Duraiswami,
Fast multipole methods for the Helmholtz equation in three dimensions, J. Comput. Phys., 215 (2006), 363-383.
doi: 10.1016/j.jcp.2005.10.029. |
[20] |
B. B. Guzina and M. Bonnet,
Small-inclusion asymptotic of misfit functionals for inverse problems in acoustics, Inverse Probl., 22 (2006), 1761-1785.
doi: 10.1088/0266-5611/22/5/014. |
[21] |
B. B. Guzina and I. Chikichev,
From imaging to material identification: A generalized concept of topological sensitivity, J. Mech. Phys. Solids, 55 (2007), 245-279.
doi: 10.1016/j.jmps.2006.07.009. |
[22] |
G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, Springer, 2008. |
[23] |
A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, Oxford, 2008. |
[24] |
A. Laurain, M. Hintermüller, M. Freiberger and H. Scharfetter, Topological sensitivity analysis in fluorescence optical tomography Inverse Probl., 29 (2013), 025003, 30pp.
doi: 10.1088/0266-5611/29/2/025003. |
[25] |
P. A. Martin,
Acoustic scattering by inhomogeneous obstacles, SIAM J. Appl. Math., 64 (2003), 297-308.
doi: 10.1137/S0036139902414379. |
[26] |
M. Masmoudi, J. Pommier and B. Samet,
The topological asymptotic expansion for the Maxwell equations and some applications, Inverse Probl., 21 (2005), 547-564.
doi: 10.1088/0266-5611/21/2/008. |
[27] |
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge, 2000. |
[28] |
T. Mura, Micromechanics of Defects in Solids, Martinus Nijhoff, 1987. |
[29] |
R. Potthast,
A survey on sampling and probe methods for inverse problems, Inverse Probl., 22 (2006), R1-R47.
doi: 10.1088/0266-5611/22/2/R01. |
[30] |
B. Samet, S. Amstutz and M. Masmoudi,
The topological asymptotic for the Helmholtz equation, SIAM J. Control Optim., 42 (2004), 1523-1544.
doi: 10.1137/S0363012902406801. |
[31] |
M. Silva, M. Matalon and D. A. Tortorelli,
Higher order topological derivatives in elasticity, Int. J. Solids Struct., 47 (2010), 3053-3066.
|
show all references
References:
[1] |
H. Ammari, E. Iakovleva and S. Moskow,
Recovery of small inhomogeneities from the scattering amplitude at a fixed frequency, SIAM J. Math. Anal., 34 (2003), 882-890.
doi: 10.1137/S0036141001392785. |
[2] |
H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, Lecture Notes in Mathematics 1846. Springer-Verlag, 2004.
doi: 10.1007/b98245. |
[3] |
H. Ammari and H. Kang, Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences, Vol. 162. Springer-Verlag, 2007. |
[4] |
H. Ammari and A. Khelifi,
Electromagnetic scattering by small dielectric inhomogeneities, J. Maths Pures Appl., 82 (2003), 749-842.
doi: 10.1016/S0021-7824(03)00033-3. |
[5] |
H. Ammari, J. Garnier, V. Jugnon and H. Kang,
Stability and resolution analysis for a topological derivative based imaging functional, SIAM J. Contr. Opt., 50 (2012), 48-76.
doi: 10.1137/100812501. |
[6] |
C. Bellis and M. Bonnet,
A FEM-based topological sensitivity approach for fast qualitative identification of buried cavities from elastodynamic overdetermined boundary data, Int. J. Solids Struct., 47 (2010), 1221-1242.
|
[7] |
C. Bellis, M. Bonnet and F. Cakoni, Acoustic inverse scattering using topological derivative of far-field measurements-based $L^2$ cost functionals, Inverse Probl., 29 (2013), 075012, 30pp.
doi: 10.1088/0266-5611/29/7/075012. |
[8] |
A. Bendali, P. H. Cocquet and S. Tordeux,
Approximation by multipoles of the multiple acoustic scattering by small obstacles in three dimensions and application to the Foldy theory of isotropic scattering, Arch. Ration. Mech. An., 219 (2016), 1017-1059.
doi: 10.1007/s00205-015-0915-5. |
[9] |
M. Bonnet, Inverse acoustic scattering by small-obstacle expansion of misfit function, Inverse Probl., 24 (2008), 035022, 27pp.
doi: 10.1088/0266-5611/24/3/035022. |
[10] |
M. Bonnet,
Fast identification of cracks using higher-order topological sensitivity for 2-D potential problems, Eng. Anal. Bound. Elem., 35 (2011), 223-235.
doi: 10.1016/j.enganabound.2010.08.007. |
[11] |
M. Bonnet,
A modified volume integral equation for anisotropic elastic or conducting inhomogeneities. Unconditional solvability by Neumann series, J. Integral Eq. Appl., 29 (2017), 271-295.
doi: 10.1216/JIE-2017-29-2-271. |
[12] |
M. Bonnet and R. Cornaggia,
Higher order topological derivatives for three-dimensional anisotropic elasticity, ESAIM: Math. Modell. Numer. Anal., 51 (2017), 2069-2092.
doi: 10.1051/m2an/2017015. |
[13] |
M. Bonnet and B. B. Guzina,
Sounding of finite solid bodies by way of topological derivative, Int. J. Num. Meth. Eng., 61 (2004), 2344-2373.
doi: 10.1002/nme.1153. |
[14] |
F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory, Springer, New York, 2014.
doi: 10.1007/978-1-4614-8827-9. |
[15] |
D.J. Cedio-Fengya, S. Moskow and M. Vogelius,
Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction, Inverse Probl., 14 (1998), 553-595.
doi: 10.1088/0266-5611/14/3/011. |
[16] |
P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2013. |
[17] |
G. R. Feijóo,
A new method in inverse scattering based on the topological derivative, Inverse Probl., 20 (2004), 1819-1840.
doi: 10.1088/0266-5611/20/6/008. |
[18] |
A. D. Ferreira and A. A. Novotny, A new non-iterative reconstruction method for the electrical impedance tomography problem, Inverse Probl., 33 (2017), 035005, 27pp.
doi: 10.1088/1361-6420/aa54e4. |
[19] |
N. A. Gumerov and R. Duraiswami,
Fast multipole methods for the Helmholtz equation in three dimensions, J. Comput. Phys., 215 (2006), 363-383.
doi: 10.1016/j.jcp.2005.10.029. |
[20] |
B. B. Guzina and M. Bonnet,
Small-inclusion asymptotic of misfit functionals for inverse problems in acoustics, Inverse Probl., 22 (2006), 1761-1785.
doi: 10.1088/0266-5611/22/5/014. |
[21] |
B. B. Guzina and I. Chikichev,
From imaging to material identification: A generalized concept of topological sensitivity, J. Mech. Phys. Solids, 55 (2007), 245-279.
doi: 10.1016/j.jmps.2006.07.009. |
[22] |
G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, Springer, 2008. |
[23] |
A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, Oxford, 2008. |
[24] |
A. Laurain, M. Hintermüller, M. Freiberger and H. Scharfetter, Topological sensitivity analysis in fluorescence optical tomography Inverse Probl., 29 (2013), 025003, 30pp.
doi: 10.1088/0266-5611/29/2/025003. |
[25] |
P. A. Martin,
Acoustic scattering by inhomogeneous obstacles, SIAM J. Appl. Math., 64 (2003), 297-308.
doi: 10.1137/S0036139902414379. |
[26] |
M. Masmoudi, J. Pommier and B. Samet,
The topological asymptotic expansion for the Maxwell equations and some applications, Inverse Probl., 21 (2005), 547-564.
doi: 10.1088/0266-5611/21/2/008. |
[27] |
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge, 2000. |
[28] |
T. Mura, Micromechanics of Defects in Solids, Martinus Nijhoff, 1987. |
[29] |
R. Potthast,
A survey on sampling and probe methods for inverse problems, Inverse Probl., 22 (2006), R1-R47.
doi: 10.1088/0266-5611/22/2/R01. |
[30] |
B. Samet, S. Amstutz and M. Masmoudi,
The topological asymptotic for the Helmholtz equation, SIAM J. Control Optim., 42 (2004), 1523-1544.
doi: 10.1137/S0363012902406801. |
[31] |
M. Silva, M. Matalon and D. A. Tortorelli,
Higher order topological derivatives in elasticity, Int. J. Solids Struct., 47 (2010), 3053-3066.
|





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