# American Institute of Mathematical Sciences

November  2012, 6(4): 663-679. doi: 10.3934/ipi.2012.6.663

## Simultaneous determination of the diffusion and absorption coefficient from boundary data

 1 Department of Mathematics, University of Würzburg, Germany

Received  June 2011 Revised  July 2012 Published  November 2012

We consider the inverse problem of determining both an unknown diffusion and an unknown absorption coefficient from knowledge of (partial) Cauchy data in an elliptic boundary value problem. For piecewise analytic coefficients, we prove a complete characterization of the reconstructible information. It is shown to consist of a combination of both coefficients together with the jumps in the leading order diffusion coefficient and its derivative.
Citation: Bastian Harrach. Simultaneous determination of the diffusion and absorption coefficient from boundary data. Inverse Problems and Imaging, 2012, 6 (4) : 663-679. doi: 10.3934/ipi.2012.6.663
##### References:
 [1] 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. [2] S. R. Arridge and J. C. Schotland, Optical tomography: forward and inverse problems, Inverse Problems, 25 (2009), 123010. doi: 10.1088/0266-5611/25/12/123010. [3] K. Astala, J. Mueller, L. Päivärinta, A. Perämäki and S. Siltanen, Direct electrical impedance tomography for nonsmooth conductivities, Inverse Probl. Imaging, 5 (2011), 531-549. doi: 10.3934/ipi.2011.5.531. [4] K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Ann. of Math. (2), 163 (2006), 265-299. doi: 10.4007/annals.2006.163.265. [5] E. Beretta and E. Francini, Lipschitz stability for the electrical impedance tomography problem: the complex case, Comm. Partial Differential Equations, 36 (2011), 1723-1749. doi: 10.1080/03605302.2011.552930. [6] A. Bukhgeim, Recovering a potential from cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl., 16 (2008), 19-33. doi: 10.1515/jiip.2008.002. [7] A. L. Bukhgeim and G. Uhlmann, Recovering a potential from partial cauchy data, Comm. Partial Differential Equations, 27 (2002), 653-668. doi: 10.1081/PDE-120002868. [8] A. P. Calderón, On an inverse boundary value problem, in "Seminar on Numerical Analysis and its Application to Continuum Physics" (Rio de Janeiro, 1980) (eds. W. H. Meyer and M. A. Raupp), Soc. Brasil. Math., Rio de Janeiro, (1980), 65-73. [9] A. P. Calderón, On an inverse boundary value problem, Comput. Appl. Math., 25 (2006), 133-138. [10] J. Cheng and M. Yamamoto, Determination of two convection coefficients from Dirichlet to Neumann map in the two-dimensional case, SIAM J. Math. Anal., 35 (2004), 1371-1393. doi: 10.1137/S0036141003422497. [11] V. Druskin, The unique solution of the inverse problem of electrical surveying and electrical well-logging for piecewise-continuous conductivity, Earth Physics, 18 (1982). [12] V. Druskin, On uniqueness of the determination of the three-dimensional underground structures from surface measurements with variously positioned steady-state or monochromatic field sources, Sov. Phys.-Solid Earth, 21 (1985), 210-214. [13] V. Druskin, On the uniqueness of inverse problems from incomplete boundary data, SIAM J. Appl. Math., 58 (1998), 1591-1603. doi: 10.1137/S0036139996298292. [14] E. Francini, Recovering a complex coefficient in a planar domain from the Dirichlet-to-Neumann map, Inverse Problems, 16 (2000), 107-119. doi: 10.1088/0266-5611/16/1/309. [15] B. Gebauer, Localized potentials in electrical impedance tomography, Inverse Probl. Imaging, 2 (2008), 251-269. doi: 10.3934/ipi.2008.2.251. [16] B. Gebauer and N. Hyvönen, Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem, Inverse Probl. Imaging, 2 (2008), 355-372. doi: 10.3934/ipi.2008.2.355. [17] A. P. Gibson, J. C. Hebden and S. R. Arridge, Recent advances in diffuse optical imaging, Phys. Med. Biol., 50 (2005), R1-R43. doi: 10.1088/0031-9155/50/4/R01. [18] A. Greenleaf, M. Lassas and G. Uhlmann, The Calderón problem for conormal potentials. I. Global uniqueness and reconstruction, Comm. Pure Appl. Math., 56 (2003), 328-352. doi: 10.1002/cpa.10061. [19] A. Greenleaf, M. Lassas and G. Uhlmann, On nonuniqueness for Calderón's inverse problem, Math. Res. Lett., 10 (2003), 685-694. [20] N. I. Grinberg, Local uniqueness for the inverse boundaryproblem for the two-dimensional diffusion equation, European J. Appl. Math., 11 (2000), 473-489. doi: 10.1017/S0956792599004106. [21] B. Harrach, On uniqueness in diffuse optical tomography, Inverse Problems, 25 (2009), 055010, 14 pp. [22] B. Harrach and J. K. Seo, Exact shape-reconstruction by one-step linearization in electrical impedance tomography, SIAM J. Math. Anal., 42 (2010), 1505-1518. doi: 10.1137/090773970. [23] O. Y. Imanuvilov, G. Uhlmann and M. Yamamoto, The Calderón problem with partial data in two dimensions, J. Amer. Math. Soc., 23 (2010), 655-691. doi: 10.1090/S0894-0347-10-00656-9. [24] O. Y. Imanuvilov, G. Uhlmann and M. Yamamoto, On determination of second order elliptic operators from partial Cauchy data, Proceedings National Academy of Sciences, 108 (2011), 467-472. doi: 10.1073/pnas.1011681107. [25] O. Y. Imanuvilov, G. Uhlmann and M. Yamamoto, Partial cauchy data for general second order elliptic operators in two dimensions, Publ. Research Insti. Math. Sci., to appear. [26] V. Isakov, "Inverse Problems for Partial Differential Equations," Second edition, Appl. Math. Sci., 127, Springer, New York, 2006. [27] V. Isakov, On uniqueness in the inverse conductivity problem with local data, Inverse Probl. Imaging, 1 (2007), 95-105. doi: 10.3934/ipi.2007.1.95. [28] C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data, Ann. of Math. (2), 165 (2007), 567-591. doi: 10.4007/annals.2007.165.567. [29] K. Knudsen, The Calderón problem with partial data for less smooth conductivities, Comm. Partial Differential Equations, 31 (2006), 57-71. doi: 10.1080/03605300500361610. [30] R. Kohn, H. Shen, M. Vogelius and M. Weinstein, Cloaking via change of variables in electric impedance tomography, Inverse Problems, 24 (2008), 015016, 21 pp. doi: 10.1088/0266-5611/24/1/015016. [31] R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Comm. Pure Appl. Math., 37 (1984), 289-298. doi: 10.1002/cpa.3160370302. [32] R. Kohn and M. Vogelius, Determining conductivity by boundary measurements. II. Interior results, Comm. Pure Appl. Math., 38 (1985), 643-667. doi: 10.1002/cpa.3160380513. [33] J. L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications. I," Grundlehren Math. Wiss., 181, Springer-Verlag, Berlin Heidelberg New York, 1972. [34] C. Miranda, "Partial Differential Equations of Elliptic Type," Second revised edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 2, Springer-Verlag, New York-Berlin, 1970. doi: 10.1007/978-3-642-87773-5. [35] A. I. Nachman, Reconstructions from boundary measurements, Ann. of Math. (2), 128 (1988), 531-576. doi: 10.2307/1971435. [36] A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), 71-96. doi: 10.2307/2118653. [37] G. Nakamura, Z. Q. Sun and G. Uhlmann, Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field, Math. Ann., 303 (1995), 377-388. doi: 10.1007/BF01460996. [38] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2), 125 (1987), 153-169. doi: 10.2307/1971291. [39] G. Uhlmann, Commentary on Calderón's paper (29), on an inverse boundary value problem, in "Selected papers of Alberto P. Calderón," Amer. Math. Soc., Providence, RI, (2008), 623-636.

show all references

##### References:
 [1] 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. [2] S. R. Arridge and J. C. Schotland, Optical tomography: forward and inverse problems, Inverse Problems, 25 (2009), 123010. doi: 10.1088/0266-5611/25/12/123010. [3] K. Astala, J. Mueller, L. Päivärinta, A. Perämäki and S. Siltanen, Direct electrical impedance tomography for nonsmooth conductivities, Inverse Probl. Imaging, 5 (2011), 531-549. doi: 10.3934/ipi.2011.5.531. [4] K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Ann. of Math. (2), 163 (2006), 265-299. doi: 10.4007/annals.2006.163.265. [5] E. Beretta and E. Francini, Lipschitz stability for the electrical impedance tomography problem: the complex case, Comm. Partial Differential Equations, 36 (2011), 1723-1749. doi: 10.1080/03605302.2011.552930. [6] A. Bukhgeim, Recovering a potential from cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl., 16 (2008), 19-33. doi: 10.1515/jiip.2008.002. [7] A. L. Bukhgeim and G. Uhlmann, Recovering a potential from partial cauchy data, Comm. Partial Differential Equations, 27 (2002), 653-668. doi: 10.1081/PDE-120002868. [8] A. P. Calderón, On an inverse boundary value problem, in "Seminar on Numerical Analysis and its Application to Continuum Physics" (Rio de Janeiro, 1980) (eds. W. H. Meyer and M. A. Raupp), Soc. Brasil. Math., Rio de Janeiro, (1980), 65-73. [9] A. P. Calderón, On an inverse boundary value problem, Comput. Appl. Math., 25 (2006), 133-138. [10] J. Cheng and M. Yamamoto, Determination of two convection coefficients from Dirichlet to Neumann map in the two-dimensional case, SIAM J. Math. Anal., 35 (2004), 1371-1393. doi: 10.1137/S0036141003422497. [11] V. Druskin, The unique solution of the inverse problem of electrical surveying and electrical well-logging for piecewise-continuous conductivity, Earth Physics, 18 (1982). [12] V. Druskin, On uniqueness of the determination of the three-dimensional underground structures from surface measurements with variously positioned steady-state or monochromatic field sources, Sov. Phys.-Solid Earth, 21 (1985), 210-214. [13] V. Druskin, On the uniqueness of inverse problems from incomplete boundary data, SIAM J. Appl. Math., 58 (1998), 1591-1603. doi: 10.1137/S0036139996298292. [14] E. Francini, Recovering a complex coefficient in a planar domain from the Dirichlet-to-Neumann map, Inverse Problems, 16 (2000), 107-119. doi: 10.1088/0266-5611/16/1/309. [15] B. Gebauer, Localized potentials in electrical impedance tomography, Inverse Probl. Imaging, 2 (2008), 251-269. doi: 10.3934/ipi.2008.2.251. [16] B. Gebauer and N. Hyvönen, Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem, Inverse Probl. Imaging, 2 (2008), 355-372. doi: 10.3934/ipi.2008.2.355. [17] A. P. Gibson, J. C. Hebden and S. R. Arridge, Recent advances in diffuse optical imaging, Phys. Med. Biol., 50 (2005), R1-R43. doi: 10.1088/0031-9155/50/4/R01. [18] A. Greenleaf, M. Lassas and G. Uhlmann, The Calderón problem for conormal potentials. I. Global uniqueness and reconstruction, Comm. Pure Appl. Math., 56 (2003), 328-352. doi: 10.1002/cpa.10061. [19] A. Greenleaf, M. Lassas and G. Uhlmann, On nonuniqueness for Calderón's inverse problem, Math. Res. Lett., 10 (2003), 685-694. [20] N. I. Grinberg, Local uniqueness for the inverse boundaryproblem for the two-dimensional diffusion equation, European J. Appl. Math., 11 (2000), 473-489. doi: 10.1017/S0956792599004106. [21] B. Harrach, On uniqueness in diffuse optical tomography, Inverse Problems, 25 (2009), 055010, 14 pp. [22] B. Harrach and J. K. Seo, Exact shape-reconstruction by one-step linearization in electrical impedance tomography, SIAM J. Math. Anal., 42 (2010), 1505-1518. doi: 10.1137/090773970. [23] O. Y. Imanuvilov, G. Uhlmann and M. Yamamoto, The Calderón problem with partial data in two dimensions, J. Amer. Math. Soc., 23 (2010), 655-691. doi: 10.1090/S0894-0347-10-00656-9. [24] O. Y. Imanuvilov, G. Uhlmann and M. Yamamoto, On determination of second order elliptic operators from partial Cauchy data, Proceedings National Academy of Sciences, 108 (2011), 467-472. doi: 10.1073/pnas.1011681107. [25] O. Y. Imanuvilov, G. Uhlmann and M. Yamamoto, Partial cauchy data for general second order elliptic operators in two dimensions, Publ. Research Insti. Math. Sci., to appear. [26] V. Isakov, "Inverse Problems for Partial Differential Equations," Second edition, Appl. Math. Sci., 127, Springer, New York, 2006. [27] V. Isakov, On uniqueness in the inverse conductivity problem with local data, Inverse Probl. Imaging, 1 (2007), 95-105. doi: 10.3934/ipi.2007.1.95. [28] C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data, Ann. of Math. (2), 165 (2007), 567-591. doi: 10.4007/annals.2007.165.567. [29] K. Knudsen, The Calderón problem with partial data for less smooth conductivities, Comm. Partial Differential Equations, 31 (2006), 57-71. doi: 10.1080/03605300500361610. [30] R. Kohn, H. Shen, M. Vogelius and M. Weinstein, Cloaking via change of variables in electric impedance tomography, Inverse Problems, 24 (2008), 015016, 21 pp. doi: 10.1088/0266-5611/24/1/015016. [31] R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Comm. Pure Appl. Math., 37 (1984), 289-298. doi: 10.1002/cpa.3160370302. [32] R. Kohn and M. Vogelius, Determining conductivity by boundary measurements. II. Interior results, Comm. Pure Appl. Math., 38 (1985), 643-667. doi: 10.1002/cpa.3160380513. [33] J. L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications. I," Grundlehren Math. Wiss., 181, Springer-Verlag, Berlin Heidelberg New York, 1972. [34] C. Miranda, "Partial Differential Equations of Elliptic Type," Second revised edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 2, Springer-Verlag, New York-Berlin, 1970. doi: 10.1007/978-3-642-87773-5. [35] A. I. Nachman, Reconstructions from boundary measurements, Ann. of Math. (2), 128 (1988), 531-576. doi: 10.2307/1971435. [36] A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), 71-96. doi: 10.2307/2118653. [37] G. Nakamura, Z. Q. Sun and G. Uhlmann, Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field, Math. Ann., 303 (1995), 377-388. doi: 10.1007/BF01460996. [38] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2), 125 (1987), 153-169. doi: 10.2307/1971291. [39] G. Uhlmann, Commentary on Calderón's paper (29), on an inverse boundary value problem, in "Selected papers of Alberto P. Calderón," Amer. Math. Soc., Providence, RI, (2008), 623-636.
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