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.

[2]

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

[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.

[4]

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

[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.

[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.

[7]

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

[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.

[9]

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

[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.

[11]

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

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[19]

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

[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.

[21]

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

[22]

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

[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.

[24]

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

[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).

[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.

[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.

[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.

[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.

[30]

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

[31]

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

[32]

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

[33]

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

[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.

[35]

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

[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.

[37]

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

[38]

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

[39]

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

[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.

[41]

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

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.

[2]

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

[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.

[4]

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

[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.

[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.

[7]

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

[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.

[9]

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

[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.

[11]

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

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[19]

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

[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.

[21]

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

[22]

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

[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.

[24]

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

[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).

[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.

[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.

[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.

[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.

[30]

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

[31]

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

[32]

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

[33]

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

[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.

[35]

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

[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.

[37]

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

[38]

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

[39]

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

[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.

[41]

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

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