August  2018, 12(4): 921-953. doi: 10.3934/ipi.2018039

Inverse acoustic scattering using high-order small-inclusion expansion of misfit function

POEMS (ENSTA ParisTech, CNRS, INRIA, Université Paris-Saclay), 91120 Palaiseau, France

Received  May 2017 Revised  March 2018 Published  June 2018

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.

Citation: Marc Bonnet. Inverse acoustic scattering using high-order small-inclusion expansion of misfit function. Inverse Problems & Imaging, 2018, 12 (4) : 921-953. doi: 10.3934/ipi.2018039
References:
[1]

H. AmmariE. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

H. AmmariJ. GarnierV. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

A. BendaliP. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

D.J. Cedio-FengyaS. 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. Google Scholar

[16]

P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2013. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, Springer, 2008.Google Scholar

[23]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, Oxford, 2008. Google Scholar

[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. Google Scholar

[25]

P. A. Martin, Acoustic scattering by inhomogeneous obstacles, SIAM J. Appl. Math., 64 (2003), 297-308. doi: 10.1137/S0036139902414379. Google Scholar

[26]

M. MasmoudiJ. 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. Google Scholar

[27]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge, 2000. Google Scholar

[28]

T. Mura, Micromechanics of Defects in Solids, Martinus Nijhoff, 1987.Google Scholar

[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. Google Scholar

[30]

B. SametS. Amstutz and M. Masmoudi, The topological asymptotic for the Helmholtz equation, SIAM J. Control Optim., 42 (2004), 1523-1544. doi: 10.1137/S0363012902406801. Google Scholar

[31]

M. SilvaM. Matalon and D. A. Tortorelli, Higher order topological derivatives in elasticity, Int. J. Solids Struct., 47 (2010), 3053-3066. Google Scholar

show all references

References:
[1]

H. AmmariE. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

H. AmmariJ. GarnierV. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

A. BendaliP. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

D.J. Cedio-FengyaS. 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. Google Scholar

[16]

P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2013. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, Springer, 2008.Google Scholar

[23]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, Oxford, 2008. Google Scholar

[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. Google Scholar

[25]

P. A. Martin, Acoustic scattering by inhomogeneous obstacles, SIAM J. Appl. Math., 64 (2003), 297-308. doi: 10.1137/S0036139902414379. Google Scholar

[26]

M. MasmoudiJ. 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. Google Scholar

[27]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge, 2000. Google Scholar

[28]

T. Mura, Micromechanics of Defects in Solids, Martinus Nijhoff, 1987.Google Scholar

[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. Google Scholar

[30]

B. SametS. Amstutz and M. Masmoudi, The topological asymptotic for the Helmholtz equation, SIAM J. Control Optim., 42 (2004), 1523-1544. doi: 10.1137/S0363012902406801. Google Scholar

[31]

M. SilvaM. Matalon and D. A. Tortorelli, Higher order topological derivatives in elasticity, Int. J. Solids Struct., 47 (2010), 3053-3066. Google Scholar

Figure 1.  Identification of a penetrable scatterer $(\mathring{B}, \mathring{\rho}, \mathring{c})$ in a acoustic half-space: geometry and notation (the dark shaded part is the search region $\mathcal{S}$)
Figure 2.  Iso-surfaces of $\hat{J}_{6}(\mathit{\boldsymbol{z}})$ for $\hat{J}_{6}=\zeta J_{6}^{\text{min}}$, with $\zeta=0.6$ (top left), $\zeta=0.7$ (top right), $\zeta=0.8$ (bottom left) and $\zeta=0.9$ (bottom right): obstacle (E), testing configuration $5\times5$ and noise-free data. The iso-surfaces and true obstacle location are emphasized by projections on coordinate planes
Figure 3.  Contour plots of $\mathit{\boldsymbol{z}}\mapsto{\mathcal{T}}_3(\mathit{\boldsymbol{z}})$ (left column) and $\mathit{\boldsymbol{z}}\mapsto\hat{J}_{6}(\mathit{\boldsymbol{z}})$ (right column) in the horizontal plane containing the true obstacle center $\hat {\mathit{\boldsymbol{x}}}$, for obstacle (B), testing configuration $5\times5$ and noise-free data. Testing frequencies are $kd=1$ (top row), $kd=2$ (middle row) and $kd=5$ (bottom row)
Figure 4.  Contour plots of $\mathit{\boldsymbol{z}}\mapsto{\mathcal{T}}_3(\mathit{\boldsymbol{z}})$ (left column) and $\mathit{\boldsymbol{z}}\mapsto\hat{J}_{6}(\mathit{\boldsymbol{z}})$ (right column) in a vertical plane containing the true obstacle center $\hat {\mathit{\boldsymbol{x}}}$, for obstacle (B), testing configuration $5\times5$ and noise-free data. Testing frequencies are $kd=1$ (top row), $kd=2$ (middle row) and $kd=5$ (bottom row)
Figure 5.  Sensitivity of search procedure to trial physical parameters: contour map of $(\beta, \eta)\mapsto \hat{J}_6(\hat {\mathit{\boldsymbol{x}}};\beta, \eta)$ (actual obstacle parameters are $\hat{\beta}=1, \, \hat{\eta}=-0.5$)
Table 1.  Relative error $R(\hat {\mathit{\boldsymbol{x}}})/\hat{R}\, -1$ on obstacle radius, for obstacles (S), (E) and (B) of known location, testing configurations $2\times2$, $5\times5$ and $10\times10$, and noise-free synthetic data.
$kd=1$ $ kd=2$ $ kd=5$
(S) $ 2\times 2$ $-1.1$e$-02$ $-4.7$e$-02$ $-2.1$e$-01$
$ 5\times 5$$-1.1$e$-02$$-4.9$e$-02$$-2.2$e$-01$
$10\times10$$-1.1$e$-02$$-5.0$e$-02$$-2.3$e$-01$
(E) $ 2\times 2$$-1.8$e$-03$$-3.7$e$-02$$-2.1$e$-01$
$ 5\times 5$$-5.0$e$-03$$-4.1$e$-02$$-2.2$e$-01$
$10\times10$$-5.8$e$-03$$-4.2$e$-02$$-2.2$e$-01$
(B) $ 2\times 2$$-4.4$e$-03$$-4.0$e$-02$$-2.1$e$-01$
$ 5\times 5$$-8.2$e$-03$$-4.4$e$-02$$-2.2$e$-01$
$10\times10$$-9.2$e$-03$$-4.5$e$-02$$-2.3$e$-01$
$kd=1$ $ kd=2$ $ kd=5$
(S) $ 2\times 2$ $-1.1$e$-02$ $-4.7$e$-02$ $-2.1$e$-01$
$ 5\times 5$$-1.1$e$-02$$-4.9$e$-02$$-2.2$e$-01$
$10\times10$$-1.1$e$-02$$-5.0$e$-02$$-2.3$e$-01$
(E) $ 2\times 2$$-1.8$e$-03$$-3.7$e$-02$$-2.1$e$-01$
$ 5\times 5$$-5.0$e$-03$$-4.1$e$-02$$-2.2$e$-01$
$10\times10$$-5.8$e$-03$$-4.2$e$-02$$-2.2$e$-01$
(B) $ 2\times 2$$-4.4$e$-03$$-4.0$e$-02$$-2.1$e$-01$
$ 5\times 5$$-8.2$e$-03$$-4.4$e$-02$$-2.2$e$-01$
$10\times10$$-9.2$e$-03$$-4.5$e$-02$$-2.3$e$-01$
Table 2.  Relative error $R(\hat {\mathit{\boldsymbol{x}}})/\hat{R}\, -1$ on obstacle radius for obstacles (S), (E) and (B) of unknown location: testing configurations $5\times5$, $10\times10$ and $20\times20$, noise-free synthetic data. A distance $|{\mathit{\boldsymbol{x}}}_{\rm{est}}-\hat {\mathit{\boldsymbol{x}}}|= d\sqrt{5}/20$ is found for all cases.
$kd=1$$ kd=2$$ kd=5$
(S) $ 2\times 2$$-1.9$e$-02$$-5.6$e$-02$$-2.3$e$-01$
$ 5\times 5$$-2.1$e$-02$$-6.0$e$-02$$-2.4$e$-01$
$10\times10$$-2.2$e$-02$$-6.2$e$-02$$-2.5$e$-01$
(E)$ 2\times 2$$-4.7$e$-03$$-4.6$e$-02$$-2.3$e$-01$
$ 5\times 5$$-3.5$e$-03$$-5.2$e$-02$$-2.4$e$-01$
$10\times10$$-3.4$e$-03$$-3.5$e$-02$$-2.4$e$-01$
(B)$ 2\times 2$$-2.2$e$-03$$-3.5$e$-02$$-2.1$e$-01$
$ 5\times 5$$-1.2$e$-03$$-3.7$e$-02$$-2.2$e$-01$
$10\times10$$-1.1$e$-03$$-3.9$e$-02$$-2.2$e$-01$
$kd=1$$ kd=2$$ kd=5$
(S) $ 2\times 2$$-1.9$e$-02$$-5.6$e$-02$$-2.3$e$-01$
$ 5\times 5$$-2.1$e$-02$$-6.0$e$-02$$-2.4$e$-01$
$10\times10$$-2.2$e$-02$$-6.2$e$-02$$-2.5$e$-01$
(E)$ 2\times 2$$-4.7$e$-03$$-4.6$e$-02$$-2.3$e$-01$
$ 5\times 5$$-3.5$e$-03$$-5.2$e$-02$$-2.4$e$-01$
$10\times10$$-3.4$e$-03$$-3.5$e$-02$$-2.4$e$-01$
(B)$ 2\times 2$$-2.2$e$-03$$-3.5$e$-02$$-2.1$e$-01$
$ 5\times 5$$-1.2$e$-03$$-3.7$e$-02$$-2.2$e$-01$
$10\times10$$-1.1$e$-03$$-3.9$e$-02$$-2.2$e$-01$
Table 3.  Offset $|{\mathit{\boldsymbol{x}}}_{\rm{est}}-\hat {\mathit{\boldsymbol{x}}}|$ and relative error $\varepsilon_R : = R_{\rm{est}}/\hat{R}\, -1$ on obstacle radius for obstacles (S), (E) and (B): testing configurations $2\times2$, $5\times5$ and $10\times10$, synthetic data with 2% relative noise on total field. Where $|{\mathit{\boldsymbol{x}}}_{\rm{est}}-\hat {\mathit{\boldsymbol{x}}}|$ is unacceptably large, $\varepsilon_R$ is deemed irrelevant and not shown.
$ka=1$$ka=2$ m$ka=5$
$|{\mathit{\boldsymbol{x}}}_{\rm{est}}-\hat {\mathit{\boldsymbol{x}}}|$$\varepsilon_R$$|{\mathit{\boldsymbol{x}}}_{\rm{est}}-\hat {\mathit{\boldsymbol{x}}}|$$\varepsilon_R$$|{\mathit{\boldsymbol{x}}}_{\rm{est}}-\hat {\mathit{\boldsymbol{x}}}|$$\varepsilon_R$
(S) $ 2\times 2$$6.5$e$+00$——$4.9$e$+00$——$1.1$e$-01$$ -2.1$e$-01$
$ 5\times 5$$4.3$e$+00$——$ -8.3$e$-02$$5.6$e$-01$$1.1$e$-01$$ -2.3$e$-01$
$10\times10$$4.2$e$+00$——$ -5.2$e$-03$$1.1$e$-01$$1.1$e$-01$$ -2.4$e$-01$
(E) $ 2\times 2$$5.4$e$+00$——$6.0$e$+00$——$1.1$e$-01$$ -1.8$e$-01$
$ 5\times 5$$4.2$e$+00$——$1.1$e$-01$$ -3.5$e$-02$$1.1$e$-01$$ -2.4$e$-01$
$10\times10$$4.8$e$+00$——$2.3$e$-01$$ -6.7$e$-02$$1.1$e$-01$$ -2.5$e$-01$
(B)$ 2\times 2$$2.8$e$+00$——$6.7$e$+00$——$1.1$e$-01$$ -1.8$e$-01$
$ 5\times 5$$4.4$e$+00$——$3.1$e$-01$$ -4.2$e$-02$$1.1$e$-01$$ -2.2$e$-01$
$10\times10$$2.4$e$+00$——$2.3$e$-01$$ -3.7$e$-02$$1.1$e$-01$$ -2.3$e$-01$
$ka=1$$ka=2$ m$ka=5$
$|{\mathit{\boldsymbol{x}}}_{\rm{est}}-\hat {\mathit{\boldsymbol{x}}}|$$\varepsilon_R$$|{\mathit{\boldsymbol{x}}}_{\rm{est}}-\hat {\mathit{\boldsymbol{x}}}|$$\varepsilon_R$$|{\mathit{\boldsymbol{x}}}_{\rm{est}}-\hat {\mathit{\boldsymbol{x}}}|$$\varepsilon_R$
(S) $ 2\times 2$$6.5$e$+00$——$4.9$e$+00$——$1.1$e$-01$$ -2.1$e$-01$
$ 5\times 5$$4.3$e$+00$——$ -8.3$e$-02$$5.6$e$-01$$1.1$e$-01$$ -2.3$e$-01$
$10\times10$$4.2$e$+00$——$ -5.2$e$-03$$1.1$e$-01$$1.1$e$-01$$ -2.4$e$-01$
(E) $ 2\times 2$$5.4$e$+00$——$6.0$e$+00$——$1.1$e$-01$$ -1.8$e$-01$
$ 5\times 5$$4.2$e$+00$——$1.1$e$-01$$ -3.5$e$-02$$1.1$e$-01$$ -2.4$e$-01$
$10\times10$$4.8$e$+00$——$2.3$e$-01$$ -6.7$e$-02$$1.1$e$-01$$ -2.5$e$-01$
(B)$ 2\times 2$$2.8$e$+00$——$6.7$e$+00$——$1.1$e$-01$$ -1.8$e$-01$
$ 5\times 5$$4.4$e$+00$——$3.1$e$-01$$ -4.2$e$-02$$1.1$e$-01$$ -2.2$e$-01$
$10\times10$$2.4$e$+00$——$2.3$e$-01$$ -3.7$e$-02$$1.1$e$-01$$ -2.3$e$-01$
Table 4.  Offset $|{\mathit{\boldsymbol{x}}}_{\rm{est}}-\hat {\mathit{\boldsymbol{x}}}|$ and relative error $\varepsilon_R : = R_{\rm{est}}/\hat{R}\, -1$ on obstacle radius for obstacles (S), (E) and (B): testing configurations $2\times2$, $5\times5$ and $10\times10$, synthetic data with 5% relative noise on total field. Where $\varepsilon_R$ is unacceptably large, $R_{\rm{est}}/\hat{R}\;-1$ is deemed irrelevant and not shown.
$ka=1$$ka=2$$ka=5$
$|{\mathit{\boldsymbol{x}}}_{\rm{est}}-\hat {\mathit{\boldsymbol{x}}}|$$\varepsilon_R$$|{\mathit{\boldsymbol{x}}}_{\rm{est}}-\hat {\mathit{\boldsymbol{x}}}|$$\varepsilon_R$$|{\mathit{\boldsymbol{x}}}_{\rm{est}}-\hat {\mathit{\boldsymbol{x}}}|$$\varepsilon_R$
(S)$ 2\times 2$$3.5$e$+00$——$6.9$e$+00$——$3.9$e$+00$$ -2.4$e$-01$
$ 5\times 5$$3.7$e$+00$——$7.2$e$+00$——$1.1$e$-01$$ -2.3$e$-01$
$10\times10$$5.6$e$+00$——$8.5$e$+00$——$1.1$e$-01$
(E)$ 2\times 2$$5.1$e$+00$——$5.0$e$+00$——$4.2$e$+00$——
$ 5\times 5$$6.4$e$+00$——$3.7$e$+00$——$1.1$e$-01$$ -1.9$e$-01$
$10\times10$$4.9$e$+00$——$5.4$e$+00$——$1.1$e$-01$$ -2.1$e$-01$
(B)$ 2\times 2$$5.9$e$+00$——$3.8$e$+00$——$6.1$e$+00$——
$ 5\times 5$$4.8$e$+00$——$5.1$e$+00$——$1.1$e$-01$$ -2.1$e$-01$
$10\times10$$3.8$e$+00$——$6.0$e$+00$——$1.1$e$-01$$ -2.1$e$-01$
$ka=1$$ka=2$$ka=5$
$|{\mathit{\boldsymbol{x}}}_{\rm{est}}-\hat {\mathit{\boldsymbol{x}}}|$$\varepsilon_R$$|{\mathit{\boldsymbol{x}}}_{\rm{est}}-\hat {\mathit{\boldsymbol{x}}}|$$\varepsilon_R$$|{\mathit{\boldsymbol{x}}}_{\rm{est}}-\hat {\mathit{\boldsymbol{x}}}|$$\varepsilon_R$
(S)$ 2\times 2$$3.5$e$+00$——$6.9$e$+00$——$3.9$e$+00$$ -2.4$e$-01$
$ 5\times 5$$3.7$e$+00$——$7.2$e$+00$——$1.1$e$-01$$ -2.3$e$-01$
$10\times10$$5.6$e$+00$——$8.5$e$+00$——$1.1$e$-01$
(E)$ 2\times 2$$5.1$e$+00$——$5.0$e$+00$——$4.2$e$+00$——
$ 5\times 5$$6.4$e$+00$——$3.7$e$+00$——$1.1$e$-01$$ -1.9$e$-01$
$10\times10$$4.9$e$+00$——$5.4$e$+00$——$1.1$e$-01$$ -2.1$e$-01$
(B)$ 2\times 2$$5.9$e$+00$——$3.8$e$+00$——$6.1$e$+00$——
$ 5\times 5$$4.8$e$+00$——$5.1$e$+00$——$1.1$e$-01$$ -2.1$e$-01$
$10\times10$$3.8$e$+00$——$6.0$e$+00$——$1.1$e$-01$$ -2.1$e$-01$
Table 5.  Offset $|{\mathit{\boldsymbol{x}}}_{\rm{est}}-\hat {\mathit{\boldsymbol{x}}}|$ and relative error $\varepsilon_R : = R_{\rm{est}}/\hat{R}\, -1$ for obstacle (S) of unknown location: testing configurations $2\times2$, $5\times5$ and $10\times10$, synthetic data with 20% relative noise on scattered field.
$ka=1$$ka=2$$ka=5$
$|{\mathit{\boldsymbol{x}}}_{\rm{est}}-\hat {\mathit{\boldsymbol{x}}}|$$\varepsilon_R$$|{\mathit{\boldsymbol{x}}}_{\rm{est}}-\hat {\mathit{\boldsymbol{x}}}|$$\varepsilon_R$$|{\mathit{\boldsymbol{x}}}_{\rm{est}}-\hat {\mathit{\boldsymbol{x}}}|$$\varepsilon_R$
$ 2\times 2$$1.1$e$-01$$6.7$e$-03$$1.1$e$-01$$ -5.3$e$-02$$1.1$e$-01$$ -2.3$e$-01$
$ 5\times 5$$1.1$e$-01$$ -2.1$e$-02$$1.1$e$-01$$ -6.6$e$-02$$1.1$e$-01$$ -2.4$e$-01$
$10\times10$$1.1$e$-01$$ -2.1$e$-02$$1.1$e$-01$$ -6.3$e$-02$$1.1$e$-01$$ -2.5$e$-01$
$ka=1$$ka=2$$ka=5$
$|{\mathit{\boldsymbol{x}}}_{\rm{est}}-\hat {\mathit{\boldsymbol{x}}}|$$\varepsilon_R$$|{\mathit{\boldsymbol{x}}}_{\rm{est}}-\hat {\mathit{\boldsymbol{x}}}|$$\varepsilon_R$$|{\mathit{\boldsymbol{x}}}_{\rm{est}}-\hat {\mathit{\boldsymbol{x}}}|$$\varepsilon_R$
$ 2\times 2$$1.1$e$-01$$6.7$e$-03$$1.1$e$-01$$ -5.3$e$-02$$1.1$e$-01$$ -2.3$e$-01$
$ 5\times 5$$1.1$e$-01$$ -2.1$e$-02$$1.1$e$-01$$ -6.6$e$-02$$1.1$e$-01$$ -2.4$e$-01$
$10\times10$$1.1$e$-01$$ -2.1$e$-02$$1.1$e$-01$$ -6.3$e$-02$$1.1$e$-01$$ -2.5$e$-01$
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