# American Institute of Mathematical Sciences

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

 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.
Citation: Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems and 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. [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. [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. [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. [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. [6] M. Brühl, Explicit characterization of inclusions in electrical impedance tomography, SIAM J. Math. Anal., 32 (2001), 1327-1341. doi: 10.1137/S003614100036656X. [7] J. Diestel and J. J. Uhl, "Vector Measures,'' vol. 15 of Mathematical Surveys, American Mathematical Society, Providence, Rhode Island, 1977. [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. [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. [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. [11] B. Gebauer, The factorization method for real elliptic problems, Z. Anal. Anwend., 25 (2006), 81-102. doi: 10.4171/ZAA/1279. [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. [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. [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. [15] N. Hyvönen, Characterizing inclusions in optical tomography, Inverse Problems, 20 (2004), 737-751. doi: 10.1088/0266-5611/20/3/006. [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. [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. [18] N. Hyvönen, Locating transparent regions in optical absorption and scattering tomography, SIAM J. Appl. Math., 67 (2007), 1101-1123. doi: 10.1137/06066299X. [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. [20] A. Kirsch, The factorization method for a class of inverse elliptic problems, Math. Nachr., 278 (2005), 258-277. doi: 10.1002/mana.200310239. [21] S. Kusiak and J. Sylvester, The scattering support, Comm. Pure Appl. Math., 56 (2003), 1525-1548. doi: 10.1002/cpa.3038. [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. [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.

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##### References:
 [1] S. R. Arridge, Optical tomography in medical imaging, Inverse Problems, 15 (1999), R41-R93. doi: 10.1088/0266-5611/15/2/022. [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. [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. [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. [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. [6] M. Brühl, Explicit characterization of inclusions in electrical impedance tomography, SIAM J. Math. Anal., 32 (2001), 1327-1341. doi: 10.1137/S003614100036656X. [7] J. Diestel and J. J. Uhl, "Vector Measures,'' vol. 15 of Mathematical Surveys, American Mathematical Society, Providence, Rhode Island, 1977. [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. [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. [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. [11] B. Gebauer, The factorization method for real elliptic problems, Z. Anal. Anwend., 25 (2006), 81-102. doi: 10.4171/ZAA/1279. [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. [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. [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. [15] N. Hyvönen, Characterizing inclusions in optical tomography, Inverse Problems, 20 (2004), 737-751. doi: 10.1088/0266-5611/20/3/006. [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. [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. [18] N. Hyvönen, Locating transparent regions in optical absorption and scattering tomography, SIAM J. Appl. Math., 67 (2007), 1101-1123. doi: 10.1137/06066299X. [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. [20] A. Kirsch, The factorization method for a class of inverse elliptic problems, Math. Nachr., 278 (2005), 258-277. doi: 10.1002/mana.200310239. [21] S. Kusiak and J. Sylvester, The scattering support, Comm. Pure Appl. Math., 56 (2003), 1525-1548. doi: 10.1002/cpa.3038. [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. [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.
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