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Foreword
Stability of the Calderón problem in admissible geometries
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 |
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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