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August  2008, 2(3): 355-372. doi: 10.3934/ipi.2008.2.355

Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem

Bastian Gebauer 1, and  Nuutti Hyvönen 2,

1. 

Institut für Mathematik, Johannes Gutenberg-Universität Maint, 55099 Mainz
2. 

Institute of Mathematics, Helsinki University of Technology, FI-02015 HUT

Received  December 2007 Revised  June 2008 Published  July 2008

In various imaging problems the task is to use the Cauchy data of the solutions to an elliptic boundary value problem to reconstruct the coefficients of the corresponding partial differential equation. Often the examined object has known background properties but is contaminated by inhomogeneities that cause perturbations of the coefficient functions. The factorization method of Kirsch provides a tool for locating such inclusions. In this paper, the factorization technique is studied in the framework of coercive elliptic partial differential equations of the divergence type: Earlier it has been demonstrated that the factorization algorithm can reconstruct the support of a strictly positive (or negative) definite perturbation of the leading order coefficient, or if that remains unperturbed, the support of a strictly positive (or negative) perturbation of the zeroth order coefficient. In this work we show that these two types of inhomogeneities can, in fact, be located simultaneously. Unlike in the earlier articles on the factorization method, our inclusions may have disconnected complements and we also weaken some other a priori assumptions of the method. Our theoretical findings are complemented by two-dimensional numerical experiments that are presented in the framework of the diffusion approximation of optical tomography.
Keywords: inclusions., inverse elliptic boundary value problems, Factorization method.
Mathematics Subject Classification: Primary: 35R30; Secondary: 65N2.
Citation: 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
References:
[1]

S. R. Arridge, Optical tomography in medical imaging, Inverse Problems, 15 (1999), R41-R93. doi: 10.1088/0266-5611/15/2/022.  Google Scholar

[2]

S. R. Arridge and W. R. B. Lionheart, Nonuniqueness in diffusion-based optical tomography, Opt. Lett., 23 (1998), 882-884. doi: 10.1364/OL.23.000882.  Google Scholar

[3]

K. Astala and L. Päivärinta, Calderon's inverse conductivity problem in the plane, Ann. of Math., 163 (2006), 265-299. doi: 10.4007/annals.2006.163.265.  Google Scholar

[4]

G. Bal, Reconstructions in impedance and optical tomography with singular interfaces, Inverse Problems, 21 (2005), 113-131. doi: 10.1088/0266-5611/21/1/008.  Google Scholar

[5]

M. Brühl and M. Hanke, Numerical implementation of two noniterative methods for locating inclusions by impedance tomography, Inverse Problems, 16 (2000), 1029-1042. doi: 10.1088/0266-5611/16/4/310.  Google Scholar

[6]

M. Brühl, Explicit characterization of inclusions in electrical impedance tomography, SIAM J. Math. Anal., 32 (2001), 1327-1341. doi: 10.1137/S003614100036656X.  Google Scholar

[7]

J. Diestel and J. J. Uhl, "Vector Measures,'' vol. 15 of Mathematical Surveys, American Mathematical Society, Providence, Rhode Island, 1977. Google Scholar

[8]

H. W. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems,'' vol. 375 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, 1996. Google Scholar

[9]

F. Frühauf, B. Gebauer and O. Scherzer, Detecting interfaces in a parabolic-elliptic problem from surface measurements, SIAM J. Numer. Anal., 45 (2007), 810-836. doi: 10.1137/050641545.  Google Scholar

[10]

B. Gebauer, M. Hanke, A. Kirsch, W. Muniz and C. Schneider, A sampling method for detecting buried objects using electromagnetic scattering, Inverse Problems, 21 (2005), 2035-2050. doi: 10.1088/0266-5611/21/6/015.  Google Scholar

[11]

B. Gebauer, The factorization method for real elliptic problems, Z. Anal. Anwend., 25 (2006), 81-102. doi: 10.4171/ZAA/1279.  Google Scholar

[12]

B. Gebauer and N. Hyvönen, Factorization method and irregular inclusions in electrical impedance tomography, Inverse Problems, 23 (2007), 2159-2170. doi: 10.1088/0266-5611/23/5/020.  Google Scholar

[13]

M. Hanke and M. Brühl, Recent progress in electrical impedance tomography, Inverse Problems, 19 (2003), S65-S90. doi: 10.1088/0266-5611/19/6/055.  Google Scholar

[14]

J. Heino and E. Somersalo, Estimation of optical absorption in anisotropic background, Inverse Problems, 18 (2002), 559-573. doi: 10.1088/0266-5611/18/3/304.  Google Scholar

[15]

N. Hyvönen, Characterizing inclusions in optical tomography, Inverse Problems, 20 (2004), 737-751. doi: 10.1088/0266-5611/20/3/006.  Google Scholar

[16]

N. Hyvönen, Application of a weaker formulation of the factorization method to the characterization of absorbing inclusions in optical tomography, Inverse Problems, 21 (2005), 1331-1343. doi: 10.1088/0266-5611/21/4/009.  Google Scholar

[17]

N. Hyvönen, Application of the factorization method to the characterization of weak inclusions in electrical impedance tomography, Adv. in Appl. Math., 39 (2007), 197-221. doi: 10.1016/j.aam.2006.12.004.  Google Scholar

[18]

N. Hyvönen, Locating transparent regions in optical absorption and scattering tomography, SIAM J. Appl. Math., 67 (2007), 1101-1123. doi: 10.1137/06066299X.  Google Scholar

[19]

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.  Google Scholar

[20]

A. Kirsch, The factorization method for a class of inverse elliptic problems, Math. Nachr., 278 (2005), 258-277. doi: 10.1002/mana.200310239.  Google Scholar

[21]

S. Kusiak and J. Sylvester, The scattering support, Comm. Pure Appl. Math., 56 (2003), 1525-1548. doi: 10.1002/cpa.3038.  Google Scholar

[22]

A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), 71-96. doi: 10.2307/2118653.  Google Scholar

[23]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169. doi: 10.2307/1971291.  Google Scholar

show all references

References:
[1]

S. R. Arridge, Optical tomography in medical imaging, Inverse Problems, 15 (1999), R41-R93. doi: 10.1088/0266-5611/15/2/022.  Google Scholar

[2]

S. R. Arridge and W. R. B. Lionheart, Nonuniqueness in diffusion-based optical tomography, Opt. Lett., 23 (1998), 882-884. doi: 10.1364/OL.23.000882.  Google Scholar

[3]

K. Astala and L. Päivärinta, Calderon's inverse conductivity problem in the plane, Ann. of Math., 163 (2006), 265-299. doi: 10.4007/annals.2006.163.265.  Google Scholar

[4]

G. Bal, Reconstructions in impedance and optical tomography with singular interfaces, Inverse Problems, 21 (2005), 113-131. doi: 10.1088/0266-5611/21/1/008.  Google Scholar

[5]

M. Brühl and M. Hanke, Numerical implementation of two noniterative methods for locating inclusions by impedance tomography, Inverse Problems, 16 (2000), 1029-1042. doi: 10.1088/0266-5611/16/4/310.  Google Scholar

[6]

M. Brühl, Explicit characterization of inclusions in electrical impedance tomography, SIAM J. Math. Anal., 32 (2001), 1327-1341. doi: 10.1137/S003614100036656X.  Google Scholar

[7]

J. Diestel and J. J. Uhl, "Vector Measures,'' vol. 15 of Mathematical Surveys, American Mathematical Society, Providence, Rhode Island, 1977. Google Scholar

[8]

H. W. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems,'' vol. 375 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, 1996. Google Scholar

[9]

F. Frühauf, B. Gebauer and O. Scherzer, Detecting interfaces in a parabolic-elliptic problem from surface measurements, SIAM J. Numer. Anal., 45 (2007), 810-836. doi: 10.1137/050641545.  Google Scholar

[10]

B. Gebauer, M. Hanke, A. Kirsch, W. Muniz and C. Schneider, A sampling method for detecting buried objects using electromagnetic scattering, Inverse Problems, 21 (2005), 2035-2050. doi: 10.1088/0266-5611/21/6/015.  Google Scholar

[11]

B. Gebauer, The factorization method for real elliptic problems, Z. Anal. Anwend., 25 (2006), 81-102. doi: 10.4171/ZAA/1279.  Google Scholar

[12]

B. Gebauer and N. Hyvönen, Factorization method and irregular inclusions in electrical impedance tomography, Inverse Problems, 23 (2007), 2159-2170. doi: 10.1088/0266-5611/23/5/020.  Google Scholar

[13]

M. Hanke and M. Brühl, Recent progress in electrical impedance tomography, Inverse Problems, 19 (2003), S65-S90. doi: 10.1088/0266-5611/19/6/055.  Google Scholar

[14]

J. Heino and E. Somersalo, Estimation of optical absorption in anisotropic background, Inverse Problems, 18 (2002), 559-573. doi: 10.1088/0266-5611/18/3/304.  Google Scholar

[15]

N. Hyvönen, Characterizing inclusions in optical tomography, Inverse Problems, 20 (2004), 737-751. doi: 10.1088/0266-5611/20/3/006.  Google Scholar

[16]

N. Hyvönen, Application of a weaker formulation of the factorization method to the characterization of absorbing inclusions in optical tomography, Inverse Problems, 21 (2005), 1331-1343. doi: 10.1088/0266-5611/21/4/009.  Google Scholar

[17]

N. Hyvönen, Application of the factorization method to the characterization of weak inclusions in electrical impedance tomography, Adv. in Appl. Math., 39 (2007), 197-221. doi: 10.1016/j.aam.2006.12.004.  Google Scholar

[18]

N. Hyvönen, Locating transparent regions in optical absorption and scattering tomography, SIAM J. Appl. Math., 67 (2007), 1101-1123. doi: 10.1137/06066299X.  Google Scholar

[19]

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.  Google Scholar

[20]

A. Kirsch, The factorization method for a class of inverse elliptic problems, Math. Nachr., 278 (2005), 258-277. doi: 10.1002/mana.200310239.  Google Scholar

[21]

S. Kusiak and J. Sylvester, The scattering support, Comm. Pure Appl. Math., 56 (2003), 1525-1548. doi: 10.1002/cpa.3038.  Google Scholar

[22]

A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), 71-96. doi: 10.2307/2118653.  Google Scholar

[23]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169. doi: 10.2307/1971291.  Google Scholar

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