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A direct D-bar method for partial boundary data electrical impedance tomography with a priori information
Reconstruction in the partial data Calderón problem on admissible manifolds
Department of Mathematics, Northeastern University, Boston, MA 02115, USA |
We consider the problem of developing a method to reconstruct a potential $q$ from the partial data Dirichlet-to-Neumann map for the Schrödinger equation $(-Δ_g+q)u=0$ on a fixed admissible manifold $(M,g)$. If the part of the boundary that is inaccessible for measurements satisfies a flatness condition in one direction, then we reconstruct the local attenuated geodesic ray transform of the one-dimensional Fourier transform of the potential $q$. This allows us to reconstruct $q$ locally, if the local (unattenuated) geodesic ray transform is constructively invertible. We also reconstruct $q$ globally, if $M$ satisfies certain concavity condition and if the global geodesic ray transform can be inverted constructively. These are reconstruction procedures for the corresponding uniqueness results given by Kenig and Salo [
References:
[1] |
A. Bukhgeim and G. Uhlmann,
Recovering a potential from partial Cauchy data, Comm. Partial Diff. Eq., 27 (2002), 653-668.
doi: 10.1081/PDE-120002868. |
[2] |
A. Calderón,
On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasil. Mat., (Rio de Janeiro), (1980), 65-73.
|
[3] |
D. Dos Santos Ferreira, C. Kenig, M. Salo and G. Uhlmann,
Limiting Carleman weights and anisotropic inverse problems, Invent. Math., 178 (2009), 119-17.
doi: 10.1007/s00222-009-0196-4. |
[4] |
D. Dos Santos Ferreira, C. Kenig and M. Salo,
Determining an unbounded potential from Cauchy data in admissible geometries, Comm. PDE, 38 (2013), 50-68.
doi: 10.1080/03605302.2012.736911. |
[5] |
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. |
[6] |
C. Kenig and M. Salo,
Recent progress in the Calderón problem with partial data, Contemp.
Math., 615 (2014), 193-222.
doi: 10.1090/conm/615/12245. |
[7] |
C. Kenig and M. Salo,
The Calderón problem with partial data on manifolds and applications, Analysis & PDE, 6 (2013), 2003-2048.
doi: 10.2140/apde.2013.6.2003. |
[8] |
C. Kenig, M. Salo and G. Uhlmann,
Reconstructions from boundary measurements on admissible manifolds, Inverse Probl. Imaging, 5 (2011), 859-877.
doi: 10.3934/ipi.2011.5.859. |
[9] |
C. 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. |
[10] |
V. Krishnan,
On the inversion formulas of Pestov and Uhlmann for the geodesic ray transform, J. Inv. Ill-Posed Problems, 18 (2010), 401-408.
doi: 10.1515/JIIP.2010.017. |
[11] |
J. Lee and G. Uhlmann,
Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math., 42 (1989), 1097-1112.
doi: 10.1002/cpa.3160420804. |
[12] |
J. -L. Lions and E. Magenes,
Problémes Aux Limites Non Homogénes et Applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968. |
[13] |
A. Nachman,
Reconstructions from boundary measurements, Ann. Math., 128 (1988), 531-576.
doi: 10.2307/1971435. |
[14] |
A. Nachman and B. Street,
Reconstruction in the Calderón problem with partial data, Comm.
PDE, 35 (2010), 375-390.
doi: 10.1080/03605300903296322. |
[15] |
R. G. Novikov,
Multidimensional inverse spectral problem for the equation $-Δψ+(v(x)-Eu(x))ψ=0$, Funct. Anal. Appl., 22 (1988), 263-272.
doi: 10.1007/BF01077418. |
[16] |
L. Pestov and G. Uhlmann,
On the Characterization of the Range and Inversion Formulas for the Geodesic X-Ray Transform, International Math. Research Notices, 80 (2004), 4331-4347.
doi: 10.1155/S1073792804142116. |
[17] |
M. Salo and G. Uhlmann,
The attenuated ray transform on simple surfaces, J. Diff. Geom., 88 (2011), 161-187.
doi: 10.4310/jdg/1317758872. |
[18] |
V. A. Sharafutdinov,
Integral Geometry of Tensor Fields, Inverse and Ill-Posed Problems Series, VSP, Utrecht, 1994.
doi: 10.1515/9783110900095. |
[19] |
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. |
[20] |
M. E. Taylor,
Partial Differential Equations I. Basic Theory, Applied Mathematical Sciences, 115. Springer, New York, 2011.
doi: 10.1007/978-1-4419-7055-8. |
[21] |
G. Uhlmann,
Inverse problems: Seeing the unseen, Bull. Math. Sci., 4 (2014), 209-279.
doi: 10.1007/s13373-014-0051-9. |
[22] |
G. Uhlmann and A. Vasy,
The inverse problem for the local geodesic ray transform, Inventiones Mathematicae, 205 (2016), 83-120.
doi: 10.1007/s00222-015-0631-7. |
show all references
References:
[1] |
A. Bukhgeim and G. Uhlmann,
Recovering a potential from partial Cauchy data, Comm. Partial Diff. Eq., 27 (2002), 653-668.
doi: 10.1081/PDE-120002868. |
[2] |
A. Calderón,
On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasil. Mat., (Rio de Janeiro), (1980), 65-73.
|
[3] |
D. Dos Santos Ferreira, C. Kenig, M. Salo and G. Uhlmann,
Limiting Carleman weights and anisotropic inverse problems, Invent. Math., 178 (2009), 119-17.
doi: 10.1007/s00222-009-0196-4. |
[4] |
D. Dos Santos Ferreira, C. Kenig and M. Salo,
Determining an unbounded potential from Cauchy data in admissible geometries, Comm. PDE, 38 (2013), 50-68.
doi: 10.1080/03605302.2012.736911. |
[5] |
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. |
[6] |
C. Kenig and M. Salo,
Recent progress in the Calderón problem with partial data, Contemp.
Math., 615 (2014), 193-222.
doi: 10.1090/conm/615/12245. |
[7] |
C. Kenig and M. Salo,
The Calderón problem with partial data on manifolds and applications, Analysis & PDE, 6 (2013), 2003-2048.
doi: 10.2140/apde.2013.6.2003. |
[8] |
C. Kenig, M. Salo and G. Uhlmann,
Reconstructions from boundary measurements on admissible manifolds, Inverse Probl. Imaging, 5 (2011), 859-877.
doi: 10.3934/ipi.2011.5.859. |
[9] |
C. 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. |
[10] |
V. Krishnan,
On the inversion formulas of Pestov and Uhlmann for the geodesic ray transform, J. Inv. Ill-Posed Problems, 18 (2010), 401-408.
doi: 10.1515/JIIP.2010.017. |
[11] |
J. Lee and G. Uhlmann,
Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math., 42 (1989), 1097-1112.
doi: 10.1002/cpa.3160420804. |
[12] |
J. -L. Lions and E. Magenes,
Problémes Aux Limites Non Homogénes et Applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968. |
[13] |
A. Nachman,
Reconstructions from boundary measurements, Ann. Math., 128 (1988), 531-576.
doi: 10.2307/1971435. |
[14] |
A. Nachman and B. Street,
Reconstruction in the Calderón problem with partial data, Comm.
PDE, 35 (2010), 375-390.
doi: 10.1080/03605300903296322. |
[15] |
R. G. Novikov,
Multidimensional inverse spectral problem for the equation $-Δψ+(v(x)-Eu(x))ψ=0$, Funct. Anal. Appl., 22 (1988), 263-272.
doi: 10.1007/BF01077418. |
[16] |
L. Pestov and G. Uhlmann,
On the Characterization of the Range and Inversion Formulas for the Geodesic X-Ray Transform, International Math. Research Notices, 80 (2004), 4331-4347.
doi: 10.1155/S1073792804142116. |
[17] |
M. Salo and G. Uhlmann,
The attenuated ray transform on simple surfaces, J. Diff. Geom., 88 (2011), 161-187.
doi: 10.4310/jdg/1317758872. |
[18] |
V. A. Sharafutdinov,
Integral Geometry of Tensor Fields, Inverse and Ill-Posed Problems Series, VSP, Utrecht, 1994.
doi: 10.1515/9783110900095. |
[19] |
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. |
[20] |
M. E. Taylor,
Partial Differential Equations I. Basic Theory, Applied Mathematical Sciences, 115. Springer, New York, 2011.
doi: 10.1007/978-1-4419-7055-8. |
[21] |
G. Uhlmann,
Inverse problems: Seeing the unseen, Bull. Math. Sci., 4 (2014), 209-279.
doi: 10.1007/s13373-014-0051-9. |
[22] |
G. Uhlmann and A. Vasy,
The inverse problem for the local geodesic ray transform, Inventiones Mathematicae, 205 (2016), 83-120.
doi: 10.1007/s00222-015-0631-7. |
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