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November  2014, 8(4): 939-957. doi: 10.3934/ipi.2014.8.939

Stability of the Calderón problem in admissible geometries

Pedro Caro 1, and  Mikko Salo 2,

1. 

Instituto de Ciencias Matemáticas - CSIC, Nicolás Cabrera 13-15, Campus de Cantoblanco UAM, 28049 Madrid, Spain
2. 

Department of Mathematics and Statistics, University of Helsinki and University of Jyväskylä, P.O. Box 35 FI-40014 Jyväskylä

Received  May 2014 Revised  September 2014 Published  November 2014

In this paper we prove log log type stability estimates for inverse boundary value problems on admissible Riemannian manifolds of dimension $n \geq 3$. The stability estimates correspond to the uniqueness results in [13]. These inverse problems arise naturally when studying the anisotropic Calderón problem.
Keywords: Inverse boundary value problems, Calderón problem, stability..
Mathematics Subject Classification: Primary: 35R30, 58J32; Secondary: 35J10, 35R0.
Citation: Pedro Caro, Mikko Salo. Stability of the Calderón problem in admissible geometries. Inverse Problems and Imaging, 2014, 8 (4) : 939-957. doi: 10.3934/ipi.2014.8.939
References:
[1]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172. doi: 10.1080/00036818808839730.

[2]

G. Alessandrini, Open issues of stability for the inverse conductivity problem, J. Inv. Ill-Posed Probl., 15 (2007), 451-460. doi: 10.1515/jiip.2007.025.

[3]

G. Alessandrini and S. Vessella, Lipschitz stability for the inverse conductivity problem, Adv. Appl. Math., 35 (2005), 207-241. doi: 10.1016/j.aam.2004.12.002.

[4]

K. Astala, M. Lassas and L. Päivärinta, Calderón's inverse problem for anisotropic conductivity in the plane, Comm. PDE, 30 (2005), 207-224. doi: 10.1081/PDE-200044485.

[5]

K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Ann. of Math., 163 (2006), 265-299. doi: 10.4007/annals.2006.163.265.

[6]

A. P. Calderón, On an Inverse Boundary Value Problem, Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasileira de Matemática, Río de Janeiro, 1980.

[7]

P. Caro, On an inverse problem in electromagnetism with local data: Stability and uniqueness, Inverse Probl. Imaging, 5 (2011), 297-322. doi: 10.3934/ipi.2011.5.297.

[8]

P. Caro, A. García and J. M. Reyes, Stability of the Calderón problem for less regular conductivities, J. Differential Equations, 254 (2013), 469-492. doi: 10.1016/j.jde.2012.08.018.

[9]

P. Caro, D. Dos Santos Ferreira and A. Ruiz, Stability estimates for the Radon transform with restricted data and applications, Adv. Math., 267 (2014), 523-564. doi: 10.1016/j.aim.2014.08.009.

[10]

P. Caro, D. Dos Santos Ferreira and A. Ruiz, Stability estimates for the Calderón problem with partial data, preprint, arXiv:1405.1217, (2014).

[11]

A. Clop, D. Faraco and A. Ruiz, Stability of Calderón's inverse conductivity problem in the plane for discontinuous conductivities, Inverse Probl. Imaging, 4 (2010), 49-91. doi: 10.3934/ipi.2010.4.49.

[12]

N.S. Dairbekov, G.P. Paternain, P. Stefanov and G. Uhlmann, The boundary rigidity problem in the presence of a magnetic field, Adv. Math., 216 (2007), 535-609. doi: 10.1016/j.aim.2007.05.014.

[13]

D. Dos Santos Ferreira, C. E. Kenig, M. Salo and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Invent. Math., 178 (2009), 119-171. doi: 10.1007/s00222-009-0196-4.

[14]

D. Dos Santos Ferreira, C. E. 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.

[15]

D. Dos Santos Ferreira, Y. Kurylev, M. Lassas and M. Salo, The Calderón problem in transversally anisotropic geometries,, J. Eur. Math. Soc, (). 

[16]

B. Frigyik, P. Stefanov and G. Uhlmann, The X-ray transform for a generic family of curves and weights, J. Geom. Anal., 18 (2008), 89-108. doi: 10.1007/s12220-007-9007-6.

[17]

B. Haberman and D. Tataru, Uniqueness in Calderon's problem with Lipschitz conductivities, Duke Math. J., 162 (2013), 497-516. doi: 10.1215/00127094-2019591.

[18]

H. Heck and J.-N. Wang, Stability estimates for the inverse boundary value problem by partial Cauchy data, Inverse Problems, 22 (2006), 1787-1796. doi: 10.1088/0266-5611/22/5/015.

[19]

H. Heck and J.-N. Wang, Optimal stability estimate of the inverse boundary value problem by partial measurements, preprint, 2007, arXiv:0708.3289.

[20]

H. Kang and K. Yun, Boundary determination of conductivities and Riemannian metrics via local Dirichlet-to-Neumann operator, SIAM J. Math. Anal., 34 (2003), 719-735. doi: 10.1137/S0036141001395042.

[21]

C. E. Kenig, M. Salo and G. Uhlmann, Inverse problems for the anisotropic Maxwell equations, Duke Math. J., 157 (2011), 369-419. doi: 10.1215/00127094-1272903.

[22]

C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data, Ann. of Math., 165 (2007), 567-591. doi: 10.4007/annals.2007.165.567.

[23]

K. Knudsen, M. Lassas, J. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem, Inverse Problems and Imaging, 3 (2009), 599-624. doi: 10.3934/ipi.2009.3.599.

[24]

J. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Commun. Pure Appl. Math., 42 (1989), 1097-1112. doi: 10.1002/cpa.3160420804.

[25]

N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems, 17 (2001), 1435-1444. doi: 10.1088/0266-5611/17/5/313.

[26]

A. Nachman, Reconstructions from boundary measurements, Ann. Math., 128 (1988), 531-576. doi: 10.2307/1971435.

[27]

A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), 71-96. doi: 10.2307/2118653.

[28]

M. Salo, The Calderón problem on Riemannian manifolds, Chapter in Inverse Problems and Applications: Inside Out II (ed. G. Uhlmann), Math. Sci. Res. Inst. Publ., 60, Cambridge Univ. Press, Cambridge, 2013, 167-247.

[29]

M. Salo and G. Uhlmann, The attenuated ray transform on simple surfaces, J. Diff. Geom., 88 (2011), 161-187.

[30]

V. A. Sharafutdinov, Ray transform on Riemannian manifolds. Eight lectures on integral geometry,, , (). 

[31]

P. Stefanov and G. Uhlmann, Stability estimates for the X-ray transform of tensor fields and boundary rigidity, Duke Math. J., 123 (2004), 445-467. doi: 10.1215/S0012-7094-04-12332-2.

[32]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169. doi: 10.2307/1971291.

[33]

G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 123011. doi: 10.1088/0266-5611/25/12/123011.

[34]

S. Vessella, A continuous dependence result in the analytic continuation problem, Forum Math., 11 (1999), 695-703. doi: 10.1515/form.1999.020.

show all references

References:
[1]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172. doi: 10.1080/00036818808839730.

[2]

G. Alessandrini, Open issues of stability for the inverse conductivity problem, J. Inv. Ill-Posed Probl., 15 (2007), 451-460. doi: 10.1515/jiip.2007.025.

[3]

G. Alessandrini and S. Vessella, Lipschitz stability for the inverse conductivity problem, Adv. Appl. Math., 35 (2005), 207-241. doi: 10.1016/j.aam.2004.12.002.

[4]

K. Astala, M. Lassas and L. Päivärinta, Calderón's inverse problem for anisotropic conductivity in the plane, Comm. PDE, 30 (2005), 207-224. doi: 10.1081/PDE-200044485.

[5]

K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Ann. of Math., 163 (2006), 265-299. doi: 10.4007/annals.2006.163.265.

[6]

A. P. Calderón, On an Inverse Boundary Value Problem, Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasileira de Matemática, Río de Janeiro, 1980.

[7]

P. Caro, On an inverse problem in electromagnetism with local data: Stability and uniqueness, Inverse Probl. Imaging, 5 (2011), 297-322. doi: 10.3934/ipi.2011.5.297.

[8]

P. Caro, A. García and J. M. Reyes, Stability of the Calderón problem for less regular conductivities, J. Differential Equations, 254 (2013), 469-492. doi: 10.1016/j.jde.2012.08.018.

[9]

P. Caro, D. Dos Santos Ferreira and A. Ruiz, Stability estimates for the Radon transform with restricted data and applications, Adv. Math., 267 (2014), 523-564. doi: 10.1016/j.aim.2014.08.009.

[10]

P. Caro, D. Dos Santos Ferreira and A. Ruiz, Stability estimates for the Calderón problem with partial data, preprint, arXiv:1405.1217, (2014).

[11]

A. Clop, D. Faraco and A. Ruiz, Stability of Calderón's inverse conductivity problem in the plane for discontinuous conductivities, Inverse Probl. Imaging, 4 (2010), 49-91. doi: 10.3934/ipi.2010.4.49.

[12]

N.S. Dairbekov, G.P. Paternain, P. Stefanov and G. Uhlmann, The boundary rigidity problem in the presence of a magnetic field, Adv. Math., 216 (2007), 535-609. doi: 10.1016/j.aim.2007.05.014.

[13]

D. Dos Santos Ferreira, C. E. Kenig, M. Salo and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Invent. Math., 178 (2009), 119-171. doi: 10.1007/s00222-009-0196-4.

[14]

D. Dos Santos Ferreira, C. E. 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.

[15]

D. Dos Santos Ferreira, Y. Kurylev, M. Lassas and M. Salo, The Calderón problem in transversally anisotropic geometries,, J. Eur. Math. Soc, (). 

[16]

B. Frigyik, P. Stefanov and G. Uhlmann, The X-ray transform for a generic family of curves and weights, J. Geom. Anal., 18 (2008), 89-108. doi: 10.1007/s12220-007-9007-6.

[17]

B. Haberman and D. Tataru, Uniqueness in Calderon's problem with Lipschitz conductivities, Duke Math. J., 162 (2013), 497-516. doi: 10.1215/00127094-2019591.

[18]

H. Heck and J.-N. Wang, Stability estimates for the inverse boundary value problem by partial Cauchy data, Inverse Problems, 22 (2006), 1787-1796. doi: 10.1088/0266-5611/22/5/015.

[19]

H. Heck and J.-N. Wang, Optimal stability estimate of the inverse boundary value problem by partial measurements, preprint, 2007, arXiv:0708.3289.

[20]

H. Kang and K. Yun, Boundary determination of conductivities and Riemannian metrics via local Dirichlet-to-Neumann operator, SIAM J. Math. Anal., 34 (2003), 719-735. doi: 10.1137/S0036141001395042.

[21]

C. E. Kenig, M. Salo and G. Uhlmann, Inverse problems for the anisotropic Maxwell equations, Duke Math. J., 157 (2011), 369-419. doi: 10.1215/00127094-1272903.

[22]

C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data, Ann. of Math., 165 (2007), 567-591. doi: 10.4007/annals.2007.165.567.

[23]

K. Knudsen, M. Lassas, J. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem, Inverse Problems and Imaging, 3 (2009), 599-624. doi: 10.3934/ipi.2009.3.599.

[24]

J. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Commun. Pure Appl. Math., 42 (1989), 1097-1112. doi: 10.1002/cpa.3160420804.

[25]

N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems, 17 (2001), 1435-1444. doi: 10.1088/0266-5611/17/5/313.

[26]

A. Nachman, Reconstructions from boundary measurements, Ann. Math., 128 (1988), 531-576. doi: 10.2307/1971435.

[27]

A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), 71-96. doi: 10.2307/2118653.

[28]

M. Salo, The Calderón problem on Riemannian manifolds, Chapter in Inverse Problems and Applications: Inside Out II (ed. G. Uhlmann), Math. Sci. Res. Inst. Publ., 60, Cambridge Univ. Press, Cambridge, 2013, 167-247.

[29]

M. Salo and G. Uhlmann, The attenuated ray transform on simple surfaces, J. Diff. Geom., 88 (2011), 161-187.

[30]

V. A. Sharafutdinov, Ray transform on Riemannian manifolds. Eight lectures on integral geometry,, , (). 

[31]

P. Stefanov and G. Uhlmann, Stability estimates for the X-ray transform of tensor fields and boundary rigidity, Duke Math. J., 123 (2004), 445-467. doi: 10.1215/S0012-7094-04-12332-2.

[32]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169. doi: 10.2307/1971291.

[33]

G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 123011. doi: 10.1088/0266-5611/25/12/123011.

[34]

S. Vessella, A continuous dependence result in the analytic continuation problem, Forum Math., 11 (1999), 695-703. doi: 10.1515/form.1999.020.

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