# American Institute of Mathematical Sciences

June  2017, 11(3): 455-476. doi: 10.3934/ipi.2017021

## Reconstruction in the partial data Calderón problem on admissible manifolds

 Department of Mathematics, Northeastern University, Boston, MA 02115, USA

Received  April 2016 Revised  February 2017 Published  April 2017

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 [7]. Moreover, the global reconstruction extends and improves the constructive proof of Nachman and Street [14] in the Euclidean setting. We derive a certain boundary integral equation which involves the given partial data and describes the traces of complex geometrical optics solutions. For construction of complex geometrical optics solutions, following [14] and improving their arguments, we use a certain family of Green's functions for the Laplace-Beltrami operator and the corresponding single layer potentials. The constructive inversion problem for local or global geodesic ray transforms is one of the major topics of interest in integral geometry.

Citation: Yernat M. Assylbekov. Reconstruction in the partial data Calderón problem on admissible manifolds. Inverse Problems & Imaging, 2017, 11 (3) : 455-476. doi: 10.3934/ipi.2017021
##### 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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] A. Nachman, Reconstructions from boundary measurements, Ann. Math., 128 (1988), 531-576.  doi: 10.2307/1971435.  Google Scholar [14] A. Nachman and B. Street, Reconstruction in the Calderón problem with partial data, Comm. PDE, 35 (2010), 375-390.  doi: 10.1080/03605300903296322.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse and Ill-Posed Problems Series, VSP, Utrecht, 1994. doi: 10.1515/9783110900095.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] G. Uhlmann, Inverse problems: Seeing the unseen, Bull. Math. Sci., 4 (2014), 209-279.  doi: 10.1007/s13373-014-0051-9.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] A. Nachman, Reconstructions from boundary measurements, Ann. Math., 128 (1988), 531-576.  doi: 10.2307/1971435.  Google Scholar [14] A. Nachman and B. Street, Reconstruction in the Calderón problem with partial data, Comm. PDE, 35 (2010), 375-390.  doi: 10.1080/03605300903296322.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse and Ill-Posed Problems Series, VSP, Utrecht, 1994. doi: 10.1515/9783110900095.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] G. Uhlmann, Inverse problems: Seeing the unseen, Bull. Math. Sci., 4 (2014), 209-279.  doi: 10.1007/s13373-014-0051-9.  Google Scholar [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.  Google Scholar
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