August  2011, 5(3): 675-694. doi: 10.3934/ipi.2011.5.675

Explicit characterization of the support of non-linear inclusions

1. 

INRIA Saclay–Ile-de-France and CMAP, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France

Received  September 2010 Revised  June 2011 Published  August 2011

We study inverse problems for non-linear penetrable media in the context of scattering theory and impedance tomography. Using a general description of the range of the non-linear far-field operator we show an explicit characterization of the support of a weakly non-linear inhomogeneous scattering object. Application of the same technique to the impedance tomography problem for a monotonic non-linear inclusion yields a characterization of the inclusion's support from the non-linear Neumann-to-Dirichlet operator.
Citation: Armin Lechleiter. Explicit characterization of the support of non-linear inclusions. Inverse Problems & Imaging, 2011, 5 (3) : 675-694. doi: 10.3934/ipi.2011.5.675
References:
[1]

G. Baruch, G. Fibich and S. Tsynkov, High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension,, J. Comput. Phys., 227 (2007), 820. doi: 10.1016/j.jcp.2007.08.022.

[2]

M. Brühl, Explicit characterization of inclusions in electrical impedance tomography,, SIAM J. Math. Anal., 32 (2001), 1327. doi: 10.1137/S003614100036656X.

[3]

F. Cakoni, H. Haddar and D. Gintides, The existence of an infinite discrete set of transmission eigenvalues,, SIAM J. Math. Anal., 42 (2010), 237. doi: 10.1137/090769338.

[4]

J. R. Cannon, Determination of the unknown coefficient $k(u)$ in the equation $\nabla k(u) \nabla u = 0$ from overspecified boundary data,, J. Math. Anal. Appl., 18 (1967), 112. doi: 10.1016/0022-247X(67)90185-0.

[5]

D. L. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", Applied Mathematical Sciences, 93 (1992).

[6]

L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (1998).

[7]

G. Fibich and B. Ilan, Vectorial and random effects in self-focusing and in multiple filamentation,, Physica D, 157 (2001), 112. doi: 10.1016/S0167-2789(01)00293-7.

[8]

M. Friesen and A. Gurevich, Nonlinear current flow in superconductors with restricted geometries,, Phys. Rev. B, 63 (2001). doi: 10.1103/PhysRevB.63.064521.

[9]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Second edition, 224 (1983).

[10]

N. Grinberg and A. Kirsch, The linear sampling method in inverse obstacle scattering for impedance boundary conditions,, J. Inverse Ill-Posed Probl., 10 (2002), 171.

[11]

M. Hanke and A. Kirsch, Sampling methods,, in, (2011).

[12]

V. Isakov and J. Sylvester, Global uniqueness for a semilinear elliptic inverse problem,, Comm. Pure. Appl. Math., 47 (1994), 1403. doi: 10.1002/cpa.3160471005.

[13]

V. Isakov and A. I. Nachman, Global uniqueness for a two-dimensional semilinear elliptic inverse problem,, Trans. Amer. Math. Soc., 347 (1995), 3375. doi: 10.2307/2155015.

[14]

E. Jalade, Inverse problem for a nonlinear Helmholtz equation,, Ann. I. H. Poincaré Anal. Non Linéaire, 21 (2004), 517.

[15]

H. Kang and G. Nakamura, Identification of nonlinearity in a conductivity equation via the Dirichlet-to-Neumann map,, Inverse Problems, 18 (2002), 1079. doi: 10.1088/0266-5611/18/4/309.

[16]

A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator,, Inverse Problems, 14 (1998), 1489. doi: 10.1088/0266-5611/14/6/009.

[17]

A. Kirsch, New characterizations of solutions in inverse scattering theory,, Applicable Analysis, 76 (2000), 319. doi: 10.1080/00036810008840888.

[18]

A. Kirsch and N. I. Grinberg, "The Factorization Method for Inverse Problems,", Oxford Lecture Series in Mathematics and its Applications, 36 (2008).

[19]

P. Kügler, Identification of a temperature dependent heat conductivity from single boundary measurements,, SIAM J. Num. Anal., 41 (2003), 1543. doi: 10.1137/S0036142902415900.

[20]

T. A. Laine and A. T. Friberg, Self-guided waves and exact solutions of the nonlinear Helmholtz equation,, J. Opt. Soc. Am. B Opt.Phys., 17 (2000), 751. doi: 10.1364/JOSAB.17.000751.

[21]

W. Rundell, Recovering an obstacle and a nonlinear conductivity from Cauchy data,, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/5/055015.

[22]

V. Serov and M. Harju, A uniqueness theorem and reconstruction of singularities for a two-dimensional nonlinear Schrödinger equation,, Nonlinearity, 21 (2008), 1323. doi: 10.1088/0951-7715/21/6/010.

[23]

V. Serov, Inverse born approximation for the nonlinear two-dimensional Schrödinger operator,, Inverse Problems, 23 (2007), 1259. doi: 10.1088/0266-5611/23/3/024.

[24]

Z. Sun, On a quasilinear inverse boundary value problem,, Mathematische Zeitschrift, 221 (1996), 293. doi: 10.1007/BF02622117.

[25]

R. Weder, Inverse scattering for the nonlinear Schrödinger equation. II. Reconstruction of the potential and the nonlinearity in the multidimensional case,, Proc. Amer. Math. Soc., 129 (2001), 3637. doi: 10.1090/S0002-9939-01-06016-6.

show all references

References:
[1]

G. Baruch, G. Fibich and S. Tsynkov, High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension,, J. Comput. Phys., 227 (2007), 820. doi: 10.1016/j.jcp.2007.08.022.

[2]

M. Brühl, Explicit characterization of inclusions in electrical impedance tomography,, SIAM J. Math. Anal., 32 (2001), 1327. doi: 10.1137/S003614100036656X.

[3]

F. Cakoni, H. Haddar and D. Gintides, The existence of an infinite discrete set of transmission eigenvalues,, SIAM J. Math. Anal., 42 (2010), 237. doi: 10.1137/090769338.

[4]

J. R. Cannon, Determination of the unknown coefficient $k(u)$ in the equation $\nabla k(u) \nabla u = 0$ from overspecified boundary data,, J. Math. Anal. Appl., 18 (1967), 112. doi: 10.1016/0022-247X(67)90185-0.

[5]

D. L. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", Applied Mathematical Sciences, 93 (1992).

[6]

L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (1998).

[7]

G. Fibich and B. Ilan, Vectorial and random effects in self-focusing and in multiple filamentation,, Physica D, 157 (2001), 112. doi: 10.1016/S0167-2789(01)00293-7.

[8]

M. Friesen and A. Gurevich, Nonlinear current flow in superconductors with restricted geometries,, Phys. Rev. B, 63 (2001). doi: 10.1103/PhysRevB.63.064521.

[9]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Second edition, 224 (1983).

[10]

N. Grinberg and A. Kirsch, The linear sampling method in inverse obstacle scattering for impedance boundary conditions,, J. Inverse Ill-Posed Probl., 10 (2002), 171.

[11]

M. Hanke and A. Kirsch, Sampling methods,, in, (2011).

[12]

V. Isakov and J. Sylvester, Global uniqueness for a semilinear elliptic inverse problem,, Comm. Pure. Appl. Math., 47 (1994), 1403. doi: 10.1002/cpa.3160471005.

[13]

V. Isakov and A. I. Nachman, Global uniqueness for a two-dimensional semilinear elliptic inverse problem,, Trans. Amer. Math. Soc., 347 (1995), 3375. doi: 10.2307/2155015.

[14]

E. Jalade, Inverse problem for a nonlinear Helmholtz equation,, Ann. I. H. Poincaré Anal. Non Linéaire, 21 (2004), 517.

[15]

H. Kang and G. Nakamura, Identification of nonlinearity in a conductivity equation via the Dirichlet-to-Neumann map,, Inverse Problems, 18 (2002), 1079. doi: 10.1088/0266-5611/18/4/309.

[16]

A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator,, Inverse Problems, 14 (1998), 1489. doi: 10.1088/0266-5611/14/6/009.

[17]

A. Kirsch, New characterizations of solutions in inverse scattering theory,, Applicable Analysis, 76 (2000), 319. doi: 10.1080/00036810008840888.

[18]

A. Kirsch and N. I. Grinberg, "The Factorization Method for Inverse Problems,", Oxford Lecture Series in Mathematics and its Applications, 36 (2008).

[19]

P. Kügler, Identification of a temperature dependent heat conductivity from single boundary measurements,, SIAM J. Num. Anal., 41 (2003), 1543. doi: 10.1137/S0036142902415900.

[20]

T. A. Laine and A. T. Friberg, Self-guided waves and exact solutions of the nonlinear Helmholtz equation,, J. Opt. Soc. Am. B Opt.Phys., 17 (2000), 751. doi: 10.1364/JOSAB.17.000751.

[21]

W. Rundell, Recovering an obstacle and a nonlinear conductivity from Cauchy data,, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/5/055015.

[22]

V. Serov and M. Harju, A uniqueness theorem and reconstruction of singularities for a two-dimensional nonlinear Schrödinger equation,, Nonlinearity, 21 (2008), 1323. doi: 10.1088/0951-7715/21/6/010.

[23]

V. Serov, Inverse born approximation for the nonlinear two-dimensional Schrödinger operator,, Inverse Problems, 23 (2007), 1259. doi: 10.1088/0266-5611/23/3/024.

[24]

Z. Sun, On a quasilinear inverse boundary value problem,, Mathematische Zeitschrift, 221 (1996), 293. doi: 10.1007/BF02622117.

[25]

R. Weder, Inverse scattering for the nonlinear Schrödinger equation. II. Reconstruction of the potential and the nonlinearity in the multidimensional case,, Proc. Amer. Math. Soc., 129 (2001), 3637. doi: 10.1090/S0002-9939-01-06016-6.

[1]

Laurent Bourgeois, Houssem Haddar. Identification of generalized impedance boundary conditions in inverse scattering problems. Inverse Problems & Imaging, 2010, 4 (1) : 19-38. doi: 10.3934/ipi.2010.4.19

[2]

Hiroshi Isozaki. Inverse boundary value problems in the horosphere - A link between hyperbolic geometry and electrical impedance tomography. Inverse Problems & Imaging, 2007, 1 (1) : 107-134. doi: 10.3934/ipi.2007.1.107

[3]

Masaya Maeda, Hironobu Sasaki, Etsuo Segawa, Akito Suzuki, Kanako Suzuki. Scattering and inverse scattering for nonlinear quantum walks. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3687-3703. doi: 10.3934/dcds.2018159

[4]

Christodoulos E. Athanasiadis, Vassilios Sevroglou, Konstantinos I. Skourogiannis. The inverse electromagnetic scattering problem by a mixed impedance screen in chiral media. Inverse Problems & Imaging, 2015, 9 (4) : 951-970. doi: 10.3934/ipi.2015.9.951

[5]

Johannes Elschner, Guanghui Hu. Uniqueness in inverse transmission scattering problems for multilayered obstacles. Inverse Problems & Imaging, 2011, 5 (4) : 793-813. doi: 10.3934/ipi.2011.5.793

[6]

Simopekka Vänskä. Stationary waves method for inverse scattering problems. Inverse Problems & Imaging, 2008, 2 (4) : 577-586. doi: 10.3934/ipi.2008.2.577

[7]

Gabriel Katz. Causal holography in application to the inverse scattering problems. Inverse Problems & Imaging, 2019, 13 (3) : 597-633. doi: 10.3934/ipi.2019028

[8]

Fabrice Delbary, Rainer Kress. Electrical impedance tomography using a point electrode inverse scheme for complete electrode data. Inverse Problems & Imaging, 2011, 5 (2) : 355-369. doi: 10.3934/ipi.2011.5.355

[9]

Tan Bui-Thanh, Omar Ghattas. Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves in three dimensions. Inverse Problems & Imaging, 2013, 7 (4) : 1139-1155. doi: 10.3934/ipi.2013.7.1139

[10]

Daniela Calvetti, Paul J. Hadwin, Janne M. J. Huttunen, Jari P. Kaipio, Erkki Somersalo. Artificial boundary conditions and domain truncation in electrical impedance tomography. Part II: Stochastic extension of the boundary map. Inverse Problems & Imaging, 2015, 9 (3) : 767-789. doi: 10.3934/ipi.2015.9.767

[11]

Daniela Calvetti, Paul J. Hadwin, Janne M. J. Huttunen, David Isaacson, Jari P. Kaipio, Debra McGivney, Erkki Somersalo, Joseph Volzer. Artificial boundary conditions and domain truncation in electrical impedance tomography. Part I: Theory and preliminary results. Inverse Problems & Imaging, 2015, 9 (3) : 749-766. doi: 10.3934/ipi.2015.9.749

[12]

Johannes Elschner, Guanghui Hu, Masahiro Yamamoto. Uniqueness in inverse elastic scattering from unbounded rigid surfaces of rectangular type. Inverse Problems & Imaging, 2015, 9 (1) : 127-141. doi: 10.3934/ipi.2015.9.127

[13]

Mourad Sini, Nguyen Trung Thành. Inverse acoustic obstacle scattering problems using multifrequency measurements. Inverse Problems & Imaging, 2012, 6 (4) : 749-773. doi: 10.3934/ipi.2012.6.749

[14]

Frederic Weidling, Thorsten Hohage. Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems. Inverse Problems & Imaging, 2017, 11 (1) : 203-220. doi: 10.3934/ipi.2017010

[15]

Leonardo Marazzi. Inverse scattering on conformally compact manifolds. Inverse Problems & Imaging, 2009, 3 (3) : 537-550. doi: 10.3934/ipi.2009.3.537

[16]

Francesco Demontis, Cornelis Van der Mee. Novel formulation of inverse scattering and characterization of scattering data. Conference Publications, 2011, 2011 (Special) : 343-350. doi: 10.3934/proc.2011.2011.343

[17]

Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems & Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355

[18]

Shui-Nee Chow, Ke Yin, Hao-Min Zhou, Ali Behrooz. Solving inverse source problems by the Orthogonal Solution and Kernel Correction Algorithm (OSKCA) with applications in fluorescence tomography. Inverse Problems & Imaging, 2014, 8 (1) : 79-102. doi: 10.3934/ipi.2014.8.79

[19]

Herbert Egger, Manuel Freiberger, Matthias Schlottbom. On forward and inverse models in fluorescence diffuse optical tomography. Inverse Problems & Imaging, 2010, 4 (3) : 411-427. doi: 10.3934/ipi.2010.4.411

[20]

Tan Bui-Thanh, Quoc P. Nguyen. FEM-based discretization-invariant MCMC methods for PDE-constrained Bayesian inverse problems. Inverse Problems & Imaging, 2016, 10 (4) : 943-975. doi: 10.3934/ipi.2016028

2017 Impact Factor: 1.465

Metrics

  • PDF downloads (3)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]