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 & 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.  doi: 10.1364/OL.23.000882.  Google Scholar

[2]

S. R. Arridge and J. C. Schotland, Optical tomography: forward and inverse problems,, Inverse Problems, 25 (2009).  doi: 10.1088/0266-5611/25/12/123010.  Google Scholar

[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.  doi: 10.3934/ipi.2011.5.531.  Google Scholar

[4]

K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane,, Ann. of Math. (2), 163 (2006), 265.  doi: 10.4007/annals.2006.163.265.  Google Scholar

[5]

E. Beretta and E. Francini, Lipschitz stability for the electrical impedance tomography problem: the complex case,, Comm. Partial Differential Equations, 36 (2011), 1723.  doi: 10.1080/03605302.2011.552930.  Google Scholar

[6]

A. Bukhgeim, Recovering a potential from cauchy data in the two-dimensional case,, J. Inverse Ill-Posed Probl., 16 (2008), 19.  doi: 10.1515/jiip.2008.002.  Google Scholar

[7]

A. L. Bukhgeim and G. Uhlmann, Recovering a potential from partial cauchy data,, Comm. Partial Differential Equations, 27 (2002), 653.  doi: 10.1081/PDE-120002868.  Google Scholar

[8]

A. P. Calderón, On an inverse boundary value problem,, in, (1980), 65.   Google Scholar

[9]

A. P. Calderón, On an inverse boundary value problem,, Comput. Appl. Math., 25 (2006), 133.   Google Scholar

[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.  doi: 10.1137/S0036141003422497.  Google Scholar

[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).   Google Scholar

[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.   Google Scholar

[13]

V. Druskin, On the uniqueness of inverse problems from incomplete boundary data,, SIAM J. Appl. Math., 58 (1998), 1591.  doi: 10.1137/S0036139996298292.  Google Scholar

[14]

E. Francini, Recovering a complex coefficient in a planar domain from the Dirichlet-to-Neumann map,, Inverse Problems, 16 (2000), 107.  doi: 10.1088/0266-5611/16/1/309.  Google Scholar

[15]

B. Gebauer, Localized potentials in electrical impedance tomography,, Inverse Probl. Imaging, 2 (2008), 251.  doi: 10.3934/ipi.2008.2.251.  Google Scholar

[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.  doi: 10.3934/ipi.2008.2.355.  Google Scholar

[17]

A. P. Gibson, J. C. Hebden and S. R. Arridge, Recent advances in diffuse optical imaging,, Phys. Med. Biol., 50 (2005).  doi: 10.1088/0031-9155/50/4/R01.  Google Scholar

[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.  doi: 10.1002/cpa.10061.  Google Scholar

[19]

A. Greenleaf, M. Lassas and G. Uhlmann, On nonuniqueness for Calderón's inverse problem,, Math. Res. Lett., 10 (2003), 685.   Google Scholar

[20]

N. I. Grinberg, Local uniqueness for the inverse boundaryproblem for the two-dimensional diffusion equation,, European J. Appl. Math., 11 (2000), 473.  doi: 10.1017/S0956792599004106.  Google Scholar

[21]

B. Harrach, On uniqueness in diffuse optical tomography,, Inverse Problems, 25 (2009).   Google Scholar

[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.  doi: 10.1137/090773970.  Google Scholar

[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.  doi: 10.1090/S0894-0347-10-00656-9.  Google Scholar

[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.  doi: 10.1073/pnas.1011681107.  Google Scholar

[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., ().   Google Scholar

[26]

V. Isakov, "Inverse Problems for Partial Differential Equations," Second edition, Appl. Math. Sci., 127,, Springer, (2006).   Google Scholar

[27]

V. Isakov, On uniqueness in the inverse conductivity problem with local data,, Inverse Probl. Imaging, 1 (2007), 95.  doi: 10.3934/ipi.2007.1.95.  Google Scholar

[28]

C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data,, Ann. of Math. (2), 165 (2007), 567.  doi: 10.4007/annals.2007.165.567.  Google Scholar

[29]

K. Knudsen, The Calderón problem with partial data for less smooth conductivities,, Comm. Partial Differential Equations, 31 (2006), 57.  doi: 10.1080/03605300500361610.  Google Scholar

[30]

R. Kohn, H. Shen, M. Vogelius and M. Weinstein, Cloaking via change of variables in electric impedance tomography,, Inverse Problems, 24 (2008).  doi: 10.1088/0266-5611/24/1/015016.  Google Scholar

[31]

R. Kohn and M. Vogelius, Determining conductivity by boundary measurements,, Comm. Pure Appl. Math., 37 (1984), 289.  doi: 10.1002/cpa.3160370302.  Google Scholar

[32]

R. Kohn and M. Vogelius, Determining conductivity by boundary measurements. II. Interior results,, Comm. Pure Appl. Math., 38 (1985), 643.  doi: 10.1002/cpa.3160380513.  Google Scholar

[33]

J. L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications. I," Grundlehren Math. Wiss., 181,, Springer-Verlag, (1972).   Google Scholar

[34]

C. Miranda, "Partial Differential Equations of Elliptic Type,", Second revised edition, (1970).  doi: 10.1007/978-3-642-87773-5.  Google Scholar

[35]

A. I. Nachman, Reconstructions from boundary measurements,, Ann. of Math. (2), 128 (1988), 531.  doi: 10.2307/1971435.  Google Scholar

[36]

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

[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.  doi: 10.1007/BF01460996.  Google Scholar

[38]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem,, Ann. of Math. (2), 125 (1987), 153.  doi: 10.2307/1971291.  Google Scholar

[39]

G. Uhlmann, Commentary on Calderón's paper (29), on an inverse boundary value problem,, in, (2008), 623.   Google Scholar

show all references

References:
[1]

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

[2]

S. R. Arridge and J. C. Schotland, Optical tomography: forward and inverse problems,, Inverse Problems, 25 (2009).  doi: 10.1088/0266-5611/25/12/123010.  Google Scholar

[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.  doi: 10.3934/ipi.2011.5.531.  Google Scholar

[4]

K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane,, Ann. of Math. (2), 163 (2006), 265.  doi: 10.4007/annals.2006.163.265.  Google Scholar

[5]

E. Beretta and E. Francini, Lipschitz stability for the electrical impedance tomography problem: the complex case,, Comm. Partial Differential Equations, 36 (2011), 1723.  doi: 10.1080/03605302.2011.552930.  Google Scholar

[6]

A. Bukhgeim, Recovering a potential from cauchy data in the two-dimensional case,, J. Inverse Ill-Posed Probl., 16 (2008), 19.  doi: 10.1515/jiip.2008.002.  Google Scholar

[7]

A. L. Bukhgeim and G. Uhlmann, Recovering a potential from partial cauchy data,, Comm. Partial Differential Equations, 27 (2002), 653.  doi: 10.1081/PDE-120002868.  Google Scholar

[8]

A. P. Calderón, On an inverse boundary value problem,, in, (1980), 65.   Google Scholar

[9]

A. P. Calderón, On an inverse boundary value problem,, Comput. Appl. Math., 25 (2006), 133.   Google Scholar

[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.  doi: 10.1137/S0036141003422497.  Google Scholar

[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).   Google Scholar

[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.   Google Scholar

[13]

V. Druskin, On the uniqueness of inverse problems from incomplete boundary data,, SIAM J. Appl. Math., 58 (1998), 1591.  doi: 10.1137/S0036139996298292.  Google Scholar

[14]

E. Francini, Recovering a complex coefficient in a planar domain from the Dirichlet-to-Neumann map,, Inverse Problems, 16 (2000), 107.  doi: 10.1088/0266-5611/16/1/309.  Google Scholar

[15]

B. Gebauer, Localized potentials in electrical impedance tomography,, Inverse Probl. Imaging, 2 (2008), 251.  doi: 10.3934/ipi.2008.2.251.  Google Scholar

[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.  doi: 10.3934/ipi.2008.2.355.  Google Scholar

[17]

A. P. Gibson, J. C. Hebden and S. R. Arridge, Recent advances in diffuse optical imaging,, Phys. Med. Biol., 50 (2005).  doi: 10.1088/0031-9155/50/4/R01.  Google Scholar

[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.  doi: 10.1002/cpa.10061.  Google Scholar

[19]

A. Greenleaf, M. Lassas and G. Uhlmann, On nonuniqueness for Calderón's inverse problem,, Math. Res. Lett., 10 (2003), 685.   Google Scholar

[20]

N. I. Grinberg, Local uniqueness for the inverse boundaryproblem for the two-dimensional diffusion equation,, European J. Appl. Math., 11 (2000), 473.  doi: 10.1017/S0956792599004106.  Google Scholar

[21]

B. Harrach, On uniqueness in diffuse optical tomography,, Inverse Problems, 25 (2009).   Google Scholar

[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.  doi: 10.1137/090773970.  Google Scholar

[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.  doi: 10.1090/S0894-0347-10-00656-9.  Google Scholar

[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.  doi: 10.1073/pnas.1011681107.  Google Scholar

[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., ().   Google Scholar

[26]

V. Isakov, "Inverse Problems for Partial Differential Equations," Second edition, Appl. Math. Sci., 127,, Springer, (2006).   Google Scholar

[27]

V. Isakov, On uniqueness in the inverse conductivity problem with local data,, Inverse Probl. Imaging, 1 (2007), 95.  doi: 10.3934/ipi.2007.1.95.  Google Scholar

[28]

C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data,, Ann. of Math. (2), 165 (2007), 567.  doi: 10.4007/annals.2007.165.567.  Google Scholar

[29]

K. Knudsen, The Calderón problem with partial data for less smooth conductivities,, Comm. Partial Differential Equations, 31 (2006), 57.  doi: 10.1080/03605300500361610.  Google Scholar

[30]

R. Kohn, H. Shen, M. Vogelius and M. Weinstein, Cloaking via change of variables in electric impedance tomography,, Inverse Problems, 24 (2008).  doi: 10.1088/0266-5611/24/1/015016.  Google Scholar

[31]

R. Kohn and M. Vogelius, Determining conductivity by boundary measurements,, Comm. Pure Appl. Math., 37 (1984), 289.  doi: 10.1002/cpa.3160370302.  Google Scholar

[32]

R. Kohn and M. Vogelius, Determining conductivity by boundary measurements. II. Interior results,, Comm. Pure Appl. Math., 38 (1985), 643.  doi: 10.1002/cpa.3160380513.  Google Scholar

[33]

J. L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications. I," Grundlehren Math. Wiss., 181,, Springer-Verlag, (1972).   Google Scholar

[34]

C. Miranda, "Partial Differential Equations of Elliptic Type,", Second revised edition, (1970).  doi: 10.1007/978-3-642-87773-5.  Google Scholar

[35]

A. I. Nachman, Reconstructions from boundary measurements,, Ann. of Math. (2), 128 (1988), 531.  doi: 10.2307/1971435.  Google Scholar

[36]

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

[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.  doi: 10.1007/BF01460996.  Google Scholar

[38]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem,, Ann. of Math. (2), 125 (1987), 153.  doi: 10.2307/1971291.  Google Scholar

[39]

G. Uhlmann, Commentary on Calderón's paper (29), on an inverse boundary value problem,, in, (2008), 623.   Google Scholar

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