November  2011, 5(4): 859-877. doi: 10.3934/ipi.2011.5.859

Reconstructions from boundary measurements on admissible manifolds

1. 

Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, IL 60637-1514, United States

2. 

Department of Mathematics and Statistics, University of Helsinki and University of Jyväskylä, PO Box 68, 00014 Helsinki, Finland

3. 

Department of Mathematics, University of Washington and, Department of Mathematics, University of California, Irvine, CA 92697-3875, United States

Received  November 2010 Revised  August 2011 Published  November 2011

We prove that a potential $q$ can be reconstructed from the Dirichlet-to-Neumann map for the Schrödinger operator $-\Delta_g + q$ in a fixed admissible $3$-dimensional Riemannian manifold $(M,g)$. We also show that an admissible metric $g$ in a fixed conformal class can be constructed from the Dirichlet-to-Neumann map for $\Delta_g$. This is a constructive version of earlier uniqueness results by Dos Santos Ferreira et al. [10] on admissible manifolds, and extends the reconstruction procedure of Nachman [31] in Euclidean space. The main points are the derivation of a boundary integral equation characterizing the boundary values of complex geometrical optics solutions, and the development of associated layer potentials adapted to a cylindrical geometry.
Citation: Carlos E. Kenig, Mikko Salo, Gunther Uhlmann. Reconstructions from boundary measurements on admissible manifolds. Inverse Problems & Imaging, 2011, 5 (4) : 859-877. doi: 10.3934/ipi.2011.5.859
References:
[1]

K. Astala, M. Lassas and L. Päivärinta, Calderón's inverse problem for anisotropic conductivity in the plane,, Comm. PDE, 30 (2005), 207. doi: 10.1081/PDE-200044485. Google Scholar

[2]

K. Astala and L. Päivärinta, A boundary integral equation for Calderón's inverse conductivity problem,, Collect. Math., 2006 (): 127. Google Scholar

[3]

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

[4]

T. Aubin, "Some Nonlinear Problems in Riemannian Geometry,", Springer Monographs in Mathematics, (1998). Google Scholar

[5]

M. Belishev and Y. Kurylev, To the reconstruction of a Riemannian manifold via its spectral data (BC-method),, Comm. PDE, 17 (1992), 767. doi: 10.1080/03605309208820863. Google Scholar

[6]

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

[8]

F. Delbary, P. C. Hansen and K. Knudsen, A direct numerical reconstruction algorithm for the 3D Calderón problem,, in, 290 (2011). doi: 10.1088/1742-6596/290/1/012003. Google Scholar

[9]

D. Dos Santos Ferreira, C. E. Kenig and M. Salo, Determining an unbounded potential from Cauchy data in admissible geometries,, preprint, (2011). Google Scholar

[10]

D. Dos Santos Ferreira, C. E. Kenig, M. Salo and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems,, Invent. Math., 178 (2009), 119. doi: 10.1007/s00222-009-0196-4. Google Scholar

[11]

L. D. Faddeev, Increasing solutions of the Schrödinger equation,, Sov. Phys. Dokl., 10 (1966), 1033. Google Scholar

[12]

B. Frigyik, P. Stefanov and G. Uhlmann, The X-ray transform for a generic family of curves and weights,, J. Geom. Anal., 18 (2008), 89. doi: 10.1007/s12220-007-9007-6. Google Scholar

[13]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Reprint of the 1998 edition, (1998). Google Scholar

[14]

C. Guillarmou and A. Sá Barreto, Inverse problems for Einstein manifolds,, Inverse Probl. Imaging, 3 (2009), 1. doi: 10.3934/ipi.2009.3.1. Google Scholar

[15]

C. Guillarmou and L. Tzou, Calderón inverse problem with partial data on Riemann surfaces,, Duke Math. J., 158 (2011), 83. doi: 10.1215/00127094-1276310. Google Scholar

[16]

C. Guillarmou and L. Tzou, Identification of a connection from Cauchy data on a Riemann surface with boundary,, Geom. Funct. Anal., 21 (2011), 393. doi: 10.1007/s00039-011-0110-2. Google Scholar

[17]

G. M. Henkin and V. Michel, On the explicit reconstruction of a Riemann surface from its Dirichlet-Neumann operator,, Geom. Funct. Anal., 17 (2007), 116. doi: 10.1007/s00039-006-0590-7. Google Scholar

[18]

G. M. Henkin and V. Michel, Inverse conductivity problem on Riemann surfaces,, J. Geom. Anal., 18 (2008), 1033. doi: 10.1007/s12220-008-9035-x. Google Scholar

[19]

G. M. Khenkin and R. G. Novikov, The $\overline{\partial}$ -equation in the multidimensional inverse scattering problem,, Uspekhi Mat. Nauk, 42 (1987), 93. Google Scholar

[20]

G. M. Henkin and R. G. Novikov, On the reconstruction of conductivity of bordered two-dimensional surface in $\mathbbR^3$ from electrical currents measurements on its boundary,, J. Geom. Anal., 21 (2011), 543. doi: 10.1007/s12220-010-9158-8. Google Scholar

[21]

G. Henkina and M. Santacesaria, On an inverse problem for anisotropic conductivity in the plane,, Inverse Problems, 26 (2010). Google Scholar

[22]

G. Henkin and M. Santacesaria, Gel'fand-Calderón's inverse problem for anisotropic conductivities on bordered surfaces in $\mathbbR^3$,, IMRN (to appear), (). Google Scholar

[23]

A. Katchalov and Y. Kurylëv, Incomplete spectral data and the reconstruction of a Riemannian manifold,, J. Inverse Ill-Posed Probl., 1 (1993), 141. doi: 10.1515/jiip.1993.1.2.141. Google Scholar

[24]

A. Katchalov and Y. Kurylev, Multidimensional inverse problem with incomplete boundary spectral data,, Comm. PDE, 23 (1998), 55. doi: 10.1080/03605309808821338. Google Scholar

[25]

A. Katchalov, Y. Kurylev and M. Lassas, "Inverse Boundary Spectral Problems,", Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123 (2001). Google Scholar

[26]

C. E. Kenig, M. Salo and G. Uhlmann, Inverse problems for the anisotropic Maxwell equations,, Duke Math. J., 157 (2011), 369. doi: 10.1215/00127094-1272903. Google Scholar

[27]

M. Lassas, M. Taylor and G. Uhlmann, The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary,, Comm. Anal. Geom., 11 (2003), 207. Google Scholar

[28]

M. Lassas and G. Uhlmann, On determining a Riemannian manifold from the Dirichlet-to-Neumann map,, Ann. Sci. École Norm. Sup. (4), 34 (2001), 771. Google Scholar

[29]

J. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements,, Comm. Pure Appl. Math., 42 (1989), 1097. doi: 10.1002/cpa.3160420804. Google Scholar

[30]

W. McLean, "Strongly Elliptic Systems and Boundary Integral Equations,", Cambridge University Press, (2000). Google Scholar

[31]

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

[32]

A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem,, Ann. of Math. (2), 143 (1996), 71. doi: 10.2307/2118653. Google Scholar

[33]

A. Nachman and B. Street, Reconstruction in the Calderón problem with partial data,, Comm. PDE, 35 (2010), 375. doi: 10.1080/03605300903296322. Google Scholar

[34]

R. G. Novikov, Multidimensional inverse spectral problem for the equation $-\Delta \psi + (v(x) - E u(x))\psi = 0$,, Funct. Anal. Appl., 22 (1988), 263. doi: 10.1007/BF01077418. Google Scholar

[35]

R. G. Novikov, An effectivization of the global reconstruction in the Gel'fand-Calderón inverse problem in three dimensions,, in, 494 (2009), 161. Google Scholar

[36]

P. Ola, L. Päivärinta and E. Somersalo, An inverse boundary value problem in electrodynamics,, Duke Math. J., 70 (1993), 617. doi: 10.1215/S0012-7094-93-07014-7. Google Scholar

[37]

M. Salo, Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field,, Comm. PDE, 31 (2006), 1639. doi: 10.1080/03605300500530420. Google Scholar

[38]

M. Salo and G. Uhlmann, The attenuated ray transform on simple surfaces,, J. Diff. Geom., 88 (2011), 161. Google Scholar

[39]

V. Sharafutdinov, "Integral Geometry of Tensor Fields,", Inverse and Ill-Posed Problems Series, (1994). Google Scholar

[40]

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

[41]

M. E. Taylor, "Tools for PDE. Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials,", Mathematical Surveys and Monographs, 81 (2000). Google Scholar

show all references

References:
[1]

K. Astala, M. Lassas and L. Päivärinta, Calderón's inverse problem for anisotropic conductivity in the plane,, Comm. PDE, 30 (2005), 207. doi: 10.1081/PDE-200044485. Google Scholar

[2]

K. Astala and L. Päivärinta, A boundary integral equation for Calderón's inverse conductivity problem,, Collect. Math., 2006 (): 127. Google Scholar

[3]

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

[4]

T. Aubin, "Some Nonlinear Problems in Riemannian Geometry,", Springer Monographs in Mathematics, (1998). Google Scholar

[5]

M. Belishev and Y. Kurylev, To the reconstruction of a Riemannian manifold via its spectral data (BC-method),, Comm. PDE, 17 (1992), 767. doi: 10.1080/03605309208820863. Google Scholar

[6]

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

[8]

F. Delbary, P. C. Hansen and K. Knudsen, A direct numerical reconstruction algorithm for the 3D Calderón problem,, in, 290 (2011). doi: 10.1088/1742-6596/290/1/012003. Google Scholar

[9]

D. Dos Santos Ferreira, C. E. Kenig and M. Salo, Determining an unbounded potential from Cauchy data in admissible geometries,, preprint, (2011). Google Scholar

[10]

D. Dos Santos Ferreira, C. E. Kenig, M. Salo and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems,, Invent. Math., 178 (2009), 119. doi: 10.1007/s00222-009-0196-4. Google Scholar

[11]

L. D. Faddeev, Increasing solutions of the Schrödinger equation,, Sov. Phys. Dokl., 10 (1966), 1033. Google Scholar

[12]

B. Frigyik, P. Stefanov and G. Uhlmann, The X-ray transform for a generic family of curves and weights,, J. Geom. Anal., 18 (2008), 89. doi: 10.1007/s12220-007-9007-6. Google Scholar

[13]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Reprint of the 1998 edition, (1998). Google Scholar

[14]

C. Guillarmou and A. Sá Barreto, Inverse problems for Einstein manifolds,, Inverse Probl. Imaging, 3 (2009), 1. doi: 10.3934/ipi.2009.3.1. Google Scholar

[15]

C. Guillarmou and L. Tzou, Calderón inverse problem with partial data on Riemann surfaces,, Duke Math. J., 158 (2011), 83. doi: 10.1215/00127094-1276310. Google Scholar

[16]

C. Guillarmou and L. Tzou, Identification of a connection from Cauchy data on a Riemann surface with boundary,, Geom. Funct. Anal., 21 (2011), 393. doi: 10.1007/s00039-011-0110-2. Google Scholar

[17]

G. M. Henkin and V. Michel, On the explicit reconstruction of a Riemann surface from its Dirichlet-Neumann operator,, Geom. Funct. Anal., 17 (2007), 116. doi: 10.1007/s00039-006-0590-7. Google Scholar

[18]

G. M. Henkin and V. Michel, Inverse conductivity problem on Riemann surfaces,, J. Geom. Anal., 18 (2008), 1033. doi: 10.1007/s12220-008-9035-x. Google Scholar

[19]

G. M. Khenkin and R. G. Novikov, The $\overline{\partial}$ -equation in the multidimensional inverse scattering problem,, Uspekhi Mat. Nauk, 42 (1987), 93. Google Scholar

[20]

G. M. Henkin and R. G. Novikov, On the reconstruction of conductivity of bordered two-dimensional surface in $\mathbbR^3$ from electrical currents measurements on its boundary,, J. Geom. Anal., 21 (2011), 543. doi: 10.1007/s12220-010-9158-8. Google Scholar

[21]

G. Henkina and M. Santacesaria, On an inverse problem for anisotropic conductivity in the plane,, Inverse Problems, 26 (2010). Google Scholar

[22]

G. Henkin and M. Santacesaria, Gel'fand-Calderón's inverse problem for anisotropic conductivities on bordered surfaces in $\mathbbR^3$,, IMRN (to appear), (). Google Scholar

[23]

A. Katchalov and Y. Kurylëv, Incomplete spectral data and the reconstruction of a Riemannian manifold,, J. Inverse Ill-Posed Probl., 1 (1993), 141. doi: 10.1515/jiip.1993.1.2.141. Google Scholar

[24]

A. Katchalov and Y. Kurylev, Multidimensional inverse problem with incomplete boundary spectral data,, Comm. PDE, 23 (1998), 55. doi: 10.1080/03605309808821338. Google Scholar

[25]

A. Katchalov, Y. Kurylev and M. Lassas, "Inverse Boundary Spectral Problems,", Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123 (2001). Google Scholar

[26]

C. E. Kenig, M. Salo and G. Uhlmann, Inverse problems for the anisotropic Maxwell equations,, Duke Math. J., 157 (2011), 369. doi: 10.1215/00127094-1272903. Google Scholar

[27]

M. Lassas, M. Taylor and G. Uhlmann, The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary,, Comm. Anal. Geom., 11 (2003), 207. Google Scholar

[28]

M. Lassas and G. Uhlmann, On determining a Riemannian manifold from the Dirichlet-to-Neumann map,, Ann. Sci. École Norm. Sup. (4), 34 (2001), 771. Google Scholar

[29]

J. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements,, Comm. Pure Appl. Math., 42 (1989), 1097. doi: 10.1002/cpa.3160420804. Google Scholar

[30]

W. McLean, "Strongly Elliptic Systems and Boundary Integral Equations,", Cambridge University Press, (2000). Google Scholar

[31]

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

[32]

A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem,, Ann. of Math. (2), 143 (1996), 71. doi: 10.2307/2118653. Google Scholar

[33]

A. Nachman and B. Street, Reconstruction in the Calderón problem with partial data,, Comm. PDE, 35 (2010), 375. doi: 10.1080/03605300903296322. Google Scholar

[34]

R. G. Novikov, Multidimensional inverse spectral problem for the equation $-\Delta \psi + (v(x) - E u(x))\psi = 0$,, Funct. Anal. Appl., 22 (1988), 263. doi: 10.1007/BF01077418. Google Scholar

[35]

R. G. Novikov, An effectivization of the global reconstruction in the Gel'fand-Calderón inverse problem in three dimensions,, in, 494 (2009), 161. Google Scholar

[36]

P. Ola, L. Päivärinta and E. Somersalo, An inverse boundary value problem in electrodynamics,, Duke Math. J., 70 (1993), 617. doi: 10.1215/S0012-7094-93-07014-7. Google Scholar

[37]

M. Salo, Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field,, Comm. PDE, 31 (2006), 1639. doi: 10.1080/03605300500530420. Google Scholar

[38]

M. Salo and G. Uhlmann, The attenuated ray transform on simple surfaces,, J. Diff. Geom., 88 (2011), 161. Google Scholar

[39]

V. Sharafutdinov, "Integral Geometry of Tensor Fields,", Inverse and Ill-Posed Problems Series, (1994). Google Scholar

[40]

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

[41]

M. E. Taylor, "Tools for PDE. Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials,", Mathematical Surveys and Monographs, 81 (2000). Google Scholar

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