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 and 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), 820-850. 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), 1327-1341. 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), 237-255. 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), 112-114. 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, Springer-Verlag, Berlin, 1992.

[6]

L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.

[7]

G. Fibich and B. Ilan, Vectorial and random effects in self-focusing and in multiple filamentation, Physica D, 157 (2001), 112-146. 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), 064521. doi: 10.1103/PhysRevB.63.064521.

[9]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Second edition, Grundlehren der Mathematischen Wissenschaften, 224, Springer-Verlag, Berlin, 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), 171-185.

[11]

M. Hanke and A. Kirsch, Sampling methods, in "Handbook of Mathematical Methods in Imaging" (ed. O. Scherzer), Springer, 2011.

[12]

V. Isakov and J. Sylvester, Global uniqueness for a semilinear elliptic inverse problem, Comm. Pure. Appl. Math., 47 (1994), 1403-1410. 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), 3375-3390. doi: 10.2307/2155015.

[14]

E. Jalade, Inverse problem for a nonlinear Helmholtz equation, Ann. I. H. Poincaré Anal. Non Linéaire, 21 (2004), 517-531.

[15]

H. Kang and G. Nakamura, Identification of nonlinearity in a conductivity equation via the Dirichlet-to-Neumann map, Inverse Problems, 18 (2002), 1079-1088. 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), 1489-1512. doi: 10.1088/0266-5611/14/6/009.

[17]

A. Kirsch, New characterizations of solutions in inverse scattering theory, Applicable Analysis, 76 (2000), 319-350. 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, Oxford University Press, Oxford, 2008.

[19]

P. Kügler, Identification of a temperature dependent heat conductivity from single boundary measurements, SIAM J. Num. Anal., 41 (2003), 1543-1563. 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), 751-757. doi: 10.1364/JOSAB.17.000751.

[21]

W. Rundell, Recovering an obstacle and a nonlinear conductivity from Cauchy data, Inverse Problems, 24 (2008), 12 pp. 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), 1323-1337. 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), 1259-1270. doi: 10.1088/0266-5611/23/3/024.

[24]

Z. Sun, On a quasilinear inverse boundary value problem, Mathematische Zeitschrift, 221 (1996), 293-305. 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), 3637-3645. 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), 820-850. 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), 1327-1341. 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), 237-255. 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), 112-114. 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, Springer-Verlag, Berlin, 1992.

[6]

L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.

[7]

G. Fibich and B. Ilan, Vectorial and random effects in self-focusing and in multiple filamentation, Physica D, 157 (2001), 112-146. 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), 064521. doi: 10.1103/PhysRevB.63.064521.

[9]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Second edition, Grundlehren der Mathematischen Wissenschaften, 224, Springer-Verlag, Berlin, 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), 171-185.

[11]

M. Hanke and A. Kirsch, Sampling methods, in "Handbook of Mathematical Methods in Imaging" (ed. O. Scherzer), Springer, 2011.

[12]

V. Isakov and J. Sylvester, Global uniqueness for a semilinear elliptic inverse problem, Comm. Pure. Appl. Math., 47 (1994), 1403-1410. 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), 3375-3390. doi: 10.2307/2155015.

[14]

E. Jalade, Inverse problem for a nonlinear Helmholtz equation, Ann. I. H. Poincaré Anal. Non Linéaire, 21 (2004), 517-531.

[15]

H. Kang and G. Nakamura, Identification of nonlinearity in a conductivity equation via the Dirichlet-to-Neumann map, Inverse Problems, 18 (2002), 1079-1088. 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), 1489-1512. doi: 10.1088/0266-5611/14/6/009.

[17]

A. Kirsch, New characterizations of solutions in inverse scattering theory, Applicable Analysis, 76 (2000), 319-350. 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, Oxford University Press, Oxford, 2008.

[19]

P. Kügler, Identification of a temperature dependent heat conductivity from single boundary measurements, SIAM J. Num. Anal., 41 (2003), 1543-1563. 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), 751-757. doi: 10.1364/JOSAB.17.000751.

[21]

W. Rundell, Recovering an obstacle and a nonlinear conductivity from Cauchy data, Inverse Problems, 24 (2008), 12 pp. 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), 1323-1337. 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), 1259-1270. doi: 10.1088/0266-5611/23/3/024.

[24]

Z. Sun, On a quasilinear inverse boundary value problem, Mathematische Zeitschrift, 221 (1996), 293-305. 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), 3637-3645. doi: 10.1090/S0002-9939-01-06016-6.

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