30 Years of Calderón’s Problem

Gunther Uhlmann[1]

  • [1] Department of Mathematics University of Washington Seattle, WA 98195 USA Fondation de Sciences Mathématiques de Paris

Séminaire Laurent Schwartz — EDP et applications (2012-2013)

  • Volume: 2012-2013, page 1-25
  • ISSN: 2266-0607

Abstract

top
In this article we survey some of the most important developments since the 1980 paper of A.P. Calderón in which he proposed the problem of determining the conductivity of a medium by making voltage and current measurements at the boundary.

How to cite

top

Uhlmann, Gunther. "30 Years of Calderón’s Problem." Séminaire Laurent Schwartz — EDP et applications 2012-2013 (2012-2013): 1-25. <http://eudml.org/doc/275764>.

@article{Uhlmann2012-2013,
abstract = {In this article we survey some of the most important developments since the 1980 paper of A.P. Calderón in which he proposed the problem of determining the conductivity of a medium by making voltage and current measurements at the boundary.},
affiliation = {Department of Mathematics University of Washington Seattle, WA 98195 USA Fondation de Sciences Mathématiques de Paris},
author = {Uhlmann, Gunther},
journal = {Séminaire Laurent Schwartz — EDP et applications},
keywords = {inverse problem; Dirichlet-to-Neumann map; complex geometrical optics},
language = {eng},
pages = {1-25},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {30 Years of Calderón’s Problem},
url = {http://eudml.org/doc/275764},
volume = {2012-2013},
year = {2012-2013},
}

TY - JOUR
AU - Uhlmann, Gunther
TI - 30 Years of Calderón’s Problem
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2012-2013
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2012-2013
SP - 1
EP - 25
AB - In this article we survey some of the most important developments since the 1980 paper of A.P. Calderón in which he proposed the problem of determining the conductivity of a medium by making voltage and current measurements at the boundary.
LA - eng
KW - inverse problem; Dirichlet-to-Neumann map; complex geometrical optics
UR - http://eudml.org/doc/275764
ER -

References

top
  1. Albin, P, Guillarmou, C., Tzou, L. and Uhlmann, G., Inverse boundary problems for systems in two dimensions, to appear Annales Institut Henri Poincaré. Zbl1310.81052
  2. Alessandrini, G., Stable determination of conductivity by boundary measurements, App. Anal., 27 (1988), 153–172. Zbl0616.35082MR922775
  3. Alessandrini, G., Singular solutions of elliptic equations and the determination of conductivity by boundary measurements, J. Diff. Equations, 84 (1990), 252-272. Zbl0778.35109MR1047569
  4. Alessandrini, G. and Vessella, S., Lipschitz stability for the inverse conductivity problem, Adv. in Appl. Math., 35 (2005), 207–241. Zbl1095.35058MR2152888
  5. Ammari, H. and Uhlmann, G., Reconstruction of the potential from partial Cauchy data for the Schrödinger equation, Indiana Univ. Math. J., 53 (2004), 169-183. Zbl1051.35103MR2048188
  6. Astala, K. and Päivärinta, L., Calderón’s inverse conductivity problem in the plane. Annals of Math., 163 (2006), 265-299. Zbl1111.35004MR2195135
  7. Astala, K., Lassas, M. and Päiväirinta, L., Calderón’s inverse problem for anisotropic conductivity in the plane, Comm. Partial Diff. Eqns., 30 (2005), 207–224. Zbl1129.35483MR2131051
  8. Bal, G., Ren, K., Uhlmann, G, and Zhou, T., Quantitative thermo-acoustics and related problems, Inverse Problems, 27 (2011), 055007. Zbl1217.35207MR2793826
  9. Bal, G. and Uhlmann, G., Inverse diffusion theory of photoacoustics, Inverse Problems, 26 (2010), 085010. Zbl1197.35311MR2658827
  10. Bal, G. and Uhlmann, G., Reconstructions for some coupled-physics inverse problems, Applied Mathematics Letters, 25 (2012), 1030-1033. Zbl1252.65185MR2915119
  11. Bal, G. and Uhlmann, G., Reconstructions of coefficients in scalar second-order elliptic equations from knowledge of their solutions, to appear Comm. Pure Appl. Math. Zbl1273.35308
  12. Barceló, B., Barceló, J.A., and Ruiz, A., Stability of the inverse conductivity problem in the plane for less regular conductivities, J. Differential Equations, 173 (2001), 231-270. Zbl0986.35126MR1834116
  13. Barceló, J.A., Faraco, D. and Ruiz, A., Stability of Calderón’s inverse problem in the plane, Journal des Mathématiques Pures et Appliquées, 88 (2007), 522-556. Zbl1133.35104MR2373740
  14. Belishev, M. I., The Calderón problem for two-dimensional manifolds by the BC-method, SIAM J. Math. Anal., 35 (2003), 172–182. Zbl1048.58019MR2001471
  15. Blaasten, E, Stability and uniqueness for the inverse problem of the Schrödinger equation with potentials in W p , ϵ , arXiv:1106.0632. 
  16. Brown, R., Recovering the conductivity at the boundary from the Dirichlet to Neumann map: a pointwise result, J. Inverse Ill-Posed Probl., 9 (2001), 567–574. Zbl0991.35104MR1881563
  17. Brown, R. and Torres, R., Uniqueness in the inverse conductivity problem for conductivities with 3 / 2 derivatives in L p , p &gt; 2 n , J. Fourier Analysis Appl., 9 (2003), 1049-1056. Zbl1051.35105
  18. Brown, R. and Uhlmann, G., Uniqueness in the inverse conductivity problem with less regular conductivities in two dimensions, Comm. PDE, 22 (1997), 1009-10027. Zbl0884.35167MR1452176
  19. Bukhgeim, A., Recovering the potential from Cauchy data in two dimensions, J. Inverse Ill-Posed Probl., 16 (2008), 19-34. Zbl1142.30018MR2387648
  20. Bukhgeim, A. and Uhlmann, G., Recovering a potential from partial Cauchy data, Comm. PDE, 27 (2002), 653-668. Zbl0998.35063MR1900557
  21. Calderón, A. P., On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), pp. 65–73, Soc. Brasil. Mat., Rio de Janeiro, 1980. MR590275
  22. Calderón, A. P., Boundary value problems for elliptic equations. Outlines of the joint Soviet-American symposium on partial differential equations, 303-304, Novisibirsk (1963). MR203254
  23. Caro, P., Dos Santos Ferreira, D. and Ruiz, A., Stability estimates for the Radon transform with restricted data and applications, arXiv:1211.1887 (2012). 
  24. Caro, P., Garcia, A. and Reyes, J.M., Stability of the Calderón problem for less regular conductivities, J. Differential Equations254 (2013), 469–492. Zbl1273.35311MR2990039
  25. Caro, P., Ola, P. and Salo, M., Inverse boundary value problem for Maxwell equations with local data, Comm. PDE, 34 (2009), 1425-1464. Zbl1185.35321MR2581979
  26. Caro, P. and Zhou, T., On global uniqueness for an IBVP for the time-harmonic Maxwell equations, to appear Anal & PDE, arXiv:1210.7602. Zbl1293.35361
  27. Chanillo S., A problem in electrical prospection and a n -dimensional Borg-Levinson theorem, Proc. AMS, 108 (1990), 761–767. Zbl0702.35035MR998731
  28. Chen, J. and Yang, Y., Quantitative photo-acoustic tomography with partial data, Inverse Problems, 28 (2012), 115014. Zbl1252.35278MR2997231
  29. Chung, F., A partial data result for the magnetic Schrödinger operator, preprint, arXiv:1111.6658. 
  30. Dos Santos Ferreira, D., Kenig, C.E., Sjöstrand, J. and Uhlmann, G., Determining a magnetic Schrödinger operator from partial Cauchy data, Comm. Math. Phys., 271 (2007), 467–488. Zbl1148.35096MR2287913
  31. Dos Santos Ferreira, D., Kenig, C.E., Salo, M., and Uhlmann, G., Limiting Carleman weights and anisotropic inverse problems, Inventiones Math., 178 (2009), 119-171. Zbl1181.35327MR2534094
  32. Eskin, G., Ralston, J., On the inverse boundary value problem for linear isotropic elasticity, Inverse Problems, 18 (2002), 907–921. Zbl1080.35175MR1910209
  33. Francini, E., Recovering a complex coefficient in a planar domain from the Dirichlet-to-Neumann map, Inverse Problems, 16 (2000), 107–119. Zbl0968.35125MR1741230
  34. Garcia, A. and Zhang, G., Reconstruction from boundary measurements for less regular conductivities, preprint, arXiv:1212.0727. 
  35. Greenleaf, A., Lassas, M. and Uhlmann, G., The Calderón problem for conormal potentials, I: Global uniqueness and reconstruction, Comm. Pure Appl. Math, 56 (2003), 328–352. Zbl1061.35165MR1941812
  36. Greenleaf, A., Lassas, M. and Uhlmann, G., Anisotropic conductivities that cannot be detected in EIT, Physiolog. Meas. (special issue on Impedance Tomography), 24 (2003), 413-420. 
  37. Greenleaf, A., Lassas, M. and Uhlmann, G., On nonuniqueness for Calderón’s inverse problem, Math. Res. Lett., 10 (2003), 685-693. Zbl1054.35127MR2024725
  38. Greenleaf, A. and Uhlmann, G., Local uniqueness for the Dirichlet-to-Neumann map via the two-plane transform, Duke Math. J., 108 (2001), 599-617. Zbl1013.35085MR1838663
  39. Guillarmou, C. and Sá Barreto, A., Inverse problems for Einstein manifolds, Inverse Problems and Imaging, 3 (2009), 1-15. Zbl1229.58025MR2558301
  40. Guillarmou, C. and Tzou, L., Calderón inverse problem on Riemann surfaces, Proceedings of CMA, 44 (2009), 129-142. Volume for the AMSI/ANU workshop on Spectral Theory and Harmonic Analysis. Zbl1231.35302
  41. Guillarmou, C. and Tzou, L., Calderón inverse problem with partial data on Riemann surfaces, Duke Math. J., 158 (2011), 83-120. Zbl1222.35212MR2794369
  42. Guillarmou, C. and Tzou, L, Identification of a connection from Cauchy data space on a Riemann surface with boundary, Geometric and Functional Analysis (GAFA), 21 (2011), 393-418. Zbl1260.58011MR2795512
  43. Hähner, P., A periodic Faddeev-type solution operator, J. Differential Equations, 128 (1996), 300–308. Zbl0849.35022MR1392403
  44. Haberman, B. and Tataru, D., Uniqueness in Calderón’s problem with Lipschitz conductivities, to appear Duke Math. J. Zbl1260.35251
  45. Heck, H. and Wang, J.-N., Stability estimates for the inverse boundary value problem by partial Cauchy data, Inverse Problems, 22 (2006), 1787–1796. Zbl1106.35133MR2261266
  46. Henkin, G. and Michel, V., Inverse conductivity problem on Riemann surfaces, J. Geom. Anal., 18 (2008), 1033–1052. Zbl1151.35101MR2438910
  47. Ide, T., Isozaki, H., Nakata S., Siltanen, S. and Uhlmann, G., Probing for electrical inclusions with complex spherical waves, Comm. Pure and Applied Math., 60 (2007), 1415-1442. Zbl1142.35104MR2342953
  48. Ikehata, M., The enclosure method and its applications, Chapter 7 in “Analytic extension formulas and their applications" (Fukuoka, 1999/Kyoto, 2000), Int. Soc. Anal. Appl. Comput., Kluwer Acad. Pub., 9 (2001), 87-103. Zbl0988.35168MR1830379
  49. Imanuvilov, O., Uhlmann, G. and Yamamoto, M., The Calderón problem with partial data in two dimensions, Journal AMS, 23 (2010), 655-691. Zbl1201.35183MR2629983
  50. Imanuvilov, O., Uhlmann, G. and Yamamoto, M., On determination of second order operators from partial Cauchy data, Proceedings National Academy of Sciences., 108 (2011), 467-472. Zbl1256.35203MR2770947
  51. Imanuvilov, O., Uhlmann, G. and Yamamoto, M., Partial data for general second order elliptic operators in two dimensions, Publ. Research Insti. Math. Sci., 48 (2012), 971-1055. Zbl1260.35253MR2999548
  52. Imanuvilov, O., Uhlmann, G. and Yamamoto, M., Inverse boundary problem with Cauchy data on disjoint sets, Inverse Problems, 27 (2011), 085007. Zbl1222.35213MR2819949
  53. Imanuvilov, O., Uhlmann, G. and Yamamoto, M., On reconstruction of Lamé coefficients from partial Cauchy data in three dimensions, Inverse Problems, 28 (2012), 125002. Zbl1264.35281MR2997011
  54. Imanuvilov, O., Uhlmann, G. and Yamamoto, M., Inverse boundary value problem by partial data for the Neumann-to-Dirichlet map in two dimensions, preprint, arXiv:1210.1255. Zbl1321.35024
  55. Imanuvilov, O. and Yamamoto, M., Inverse boundary value for Schrödinger equation in two dimensions, arXiv:1211.1419v1. Zbl1273.35316
  56. Imanuvilov, O. and Yamamoto, M., Uniqueness for inverse boundary problems by Dirichlet-to-Neumann map on arbitrary subboundaries, preprint, arXiv:1303.2159. Zbl1291.35443MR2110612
  57. Isaacson, D., Newell, J. C., Goble, J. C. and Cheney M., Thoracic impedance images during ventilation, Annual Conference of the IEEE Engineering in Medicine and Biology Society, 12 (1990), 106–107. 
  58. Isakov, V., On uniqueness in the inverse conductivity problem with local data, Inverse Problems and Imaging, 1 (2007), 95-105. Zbl1125.35113MR2262748
  59. Isakov, V., Nakamura, G., Uhlmann, G. and Wang, J.-N., Increasing stability of the inverse boundary problem for the Schröedinger equation, to appear Contemp. Math., arXiv:1302.0940. 
  60. Isozaki, H., Inverse spectral problems on hyperbolic manifolds and their applications to inverse boundary value problems in Euclidean space, Amer. J. Math., 126 (2004), 1261–1313. Zbl1069.35092MR2102396
  61. Isozaki, H. and Uhlmann, G., Hyperbolic geometric and the local Dirichlet-to-Neumann map, Advances in Math.188 (2004), 294-314. Zbl1062.35172MR2087229
  62. Jordana, J., Gasulla, J. M. and Paola’s-Areny, R., Electrical resistance tomography to detect leaks from buried pipes, Meas. Sci. Technol., 12 (2001), 1061-1068. 
  63. Jossinet, J., The impedivity of freshly excised human breast tissue, Physiol. Meas., 19 (1998), 61-75. 
  64. Kenig, C. and Salo, M., The Calderón problem with partial data on manifolds and applications, preprint, arXiv:1211.1054. Zbl1335.35301
  65. Kenig, C. and Salo, M., Recent progress in the Calderón problem with partial data, preprint, arXiv:1302.4218. Zbl1330.35534
  66. Kenig, C., Salo, M. and Uhlmann, G., Inverse problems for the anisotropic Maxwell equations", Duke Math. J., 157 (2011), 369-419. Zbl1226.35086MR2783934
  67. Kenig, C., Sjöstrand, J. and Uhlmann, G., The Calderón problem with partial data, Annals of Math., 165 (2007), 567-591. Zbl1127.35079MR2299741
  68. Knudsen, K., The Calderón problem with partial data for less smooth conductivities, Comm. Partial Differential Equations, 31 (2006), 57–71. Zbl1091.35116MR2209749
  69. Knudsen, K. and Salo, M., Determining nonsmooth first order terms from partial boundary measurements, Inverse Problems and Imaging, 1 (2007), 349-369. Zbl1122.35152MR2282273
  70. Kocyigit, I., Acoustic-electric tomography and CGO solutions with internal data, Inverse Problems, 28 (2012), 125004. Zbl1266.78015MR2997013
  71. Kohn, R., Shen, H., Vogelius, M. and Weinstein, M., Cloaking via change of variables in Electrical Impedance Tomography, Inverse Problems24 (2008), 015016 (21pp). Zbl1153.35406MR2384775
  72. Kohn, R. and Vogelius, M., Identification of an unknown conductivity by means of measurements at the boundary, in Inverse Problems, SIAM-AMS Proc., 14 (1984). Zbl0573.35084MR773707
  73. Kohn, R. and Vogelius, M., Determining conductivity by boundary measurements, Comm. Pure Appl. Math., 37 (1984), 289–298. Zbl0586.35089MR739921
  74. Kohn, R. and Vogelius, M., Determining conductivity by boundary measurements II. Interior results, Comm. Pure Appl. Math., 38 (1985), 643–667. Zbl0595.35092MR803253
  75. Krupchyk, K., Lassas, M. and Uhlmann, G., Inverse problems for differential forms on Riemannian manifolds with boundary, Comm. PDE., 36 (2011), 1475-1509. Zbl1227.35245MR2825599
  76. Krupchyk, K., Lassas, M. and Uhlmann, G., Inverse problems with partial data for the magnetic Schrödinger operator in an infinite slab and on a bounded domain Comm. Math. Phys., 312 (2012), 87-126. Zbl1238.35188MR2914058
  77. Krupchyk, K., Lassas, M. and Uhlmann, G., Inverse boundary value problems for the polyharmonic operator, Journal Functional Analysis, 262 (2012), 1781-1801. Zbl1239.35184MR2873860
  78. Krupchyk, K., Lassas, M. and Uhlmann, G, Determining a first order perturbation of the biharmonic operator by partial boundary measurements, to appear Transactions AMS. Zbl1239.35184MR2873860
  79. Krupchyk, K., Uhlmann, G, Determining a magnetic Schrödinger operator with a bounded magnetic potential from boundary measurements, preprint, arXiv:1206.4727. 
  80. Lassas, M. and Uhlmann, G., Determining a Riemannian manifold from boundary measurements, Ann. Sci. École Norm. Sup., 34 (2001), 771–787. Zbl0992.35120MR1862026
  81. Lassas, M., Taylor, M. and Uhlmann, G., The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary, Comm. Geom. Anal., 11 (2003), 207-222. Zbl1077.58012MR2014876
  82. Lee, J. and Uhlmann, G., Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math., 42 (1989), 1097–1112. Zbl0702.35036MR1029119
  83. Li, X. and Uhlmann, G., Inverse problems on a slab, Inverse Problems and Imaging, 4 (2010), 449-462. Zbl1200.35331MR2671106
  84. Mandache, N., Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems, 17 (2001), 1435–1444. Zbl0985.35110MR1862200
  85. Nachman, A., Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), 71-96. Zbl0857.35135MR1370758
  86. Nachman, A., Reconstructions from boundary measurements, Ann. of Math., 128 (1988), 531–576. Zbl0675.35084MR970610
  87. Nachman, A. and Street, B., Reconstruction in the Calderón problem with partial data, Comm. PDE, 35 (2010), 375-390. Zbl1186.35242MR2748629
  88. Nagayasu, S., Uhlmann, G. and Wang, J.-N., Depth dependent stability estimate in electrical impedance tomography, Inverse Problems, 25 (2009), 075001. Zbl1172.35514MR2519853
  89. Nagayasu, S., Uhlmann, G. and Wang, J.-N., Reconstruction of penetrable obstacles in acoustics, SIAM J. Math. Anal., 43 (2011), 189-211. Zbl1234.35315MR2765688
  90. Nagayasu, S, Uhlmann, G. and Wang, J.-N., Increasing stability for the acoustic equation, Inverse Problems, 29 (2013), 020012. Zbl1302.65243
  91. Nakamura, G. and Tanuma, K., Local determination of conductivity at the boundary from the Dirichlet-to-Neumann map, Inverse Problems, 17 (2001), 405–419. Zbl0981.35100MR1843272
  92. Nakamura G. and Uhlmann, G., Global uniqueness for an inverse boundary value problem arising in elasticity, Invent. Math., 118 (1994), 457–474. Erratum: Invent. Math., 152 (2003), 205–207. Zbl0814.35147
  93. Nakamura, G., Sun, Z. and Uhlmann, G., Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field, Math. Annalen, 303 (1995), 377–388. Zbl0843.35134MR1354996
  94. Novikov R. G., Multidimensional inverse spectral problems for the equation - Δ ψ + ( v ( x ) - E u ( x ) ) ψ = 0 , Funktsionalny Analizi Ego Prilozheniya, 22 (1988), 11-12, Translation in Functional Analysis and its Applications, 22 (1988) 263–272. Zbl0689.35098MR976992
  95. Ola, P., Päivärinta, L. and Somersalo, E., An inverse boundary value problem in electrodynamics, Duke Math. J., 70 (1993), 617–653. Zbl0804.35152MR1224101
  96. Ola, P. and Somersalo, E. , Electromagnetic inverse problems and generalized Sommerfeld potentials, SIAM J. Appl. Math., 56 (1996), 1129-1145 Zbl0858.35138MR1398411
  97. Päivärinta, L., Panchenko, A. and Uhlmann, G., Complex geometrical optics for Lipschitz conductivities, Revista Matematica Iberoamericana, 19 (2003), 57-72. Zbl1055.35144MR1993415
  98. Pestov, L. and Uhlmann, G., Two dimensional simple Riemannian manifolds with boundary are boundary distance rigid,Annals of Math., 161 (2005), 1089-1106. Zbl1076.53044MR2153407
  99. Rondi, L., A remark on a paper by G. Alessandrini and S. Vessella: “Lipschitz stability for the inverse conductivity problem" [Adv. in Appl. Math. 35 (2005), 207–241], Adv. in Appl. Math., 36 (2006), 67–69. Zbl1158.35105MR2198854
  100. Salo, M., Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field, Comm. PDE, 31 (2006), 1639-1666. Zbl1119.35119MR2273968
  101. Salo, M., Inverse problems for nonsmooth first order perturbations of the Laplacian, Ann. Acad. Sci. Fenn. Math. Diss., 139 (2004), 67 pp. Zbl1059.35175MR2105191
  102. Salo, M. and Tzou, L., Inverse problems with partial data for a Dirac system: a Carleman estimate approach, Advances in Math., 225 (2010), 487-513. Zbl1197.35329MR2669360
  103. Salo, M. and Wang, J.-N. , Complex spherical waves and inverse problems in unbounded domains, Inverse Problems22 (2006), 2299–2309. Zbl1106.35140MR2277543
  104. Siltanen, S., Müller, J. L. and Isaacson, D., A direct reconstruction algorithm for electrical impedance tomography, IEEE Transactions on Medical Imaging, 21 (2002), 555-559. 
  105. Somersalo, E., Isaacson, D. and Cheney, M., A linearized inverse boundary value problem for Maxwell’s equations, Journal of Comp. and Appl. Math., 42 (1992),123-136. Zbl0757.65128MR1181585
  106. Sun, Z. and Uhlmann, G., Anisotropic inverse problems in two dimensions, Inverse Problems, 19 (2003), 1001-1010. Zbl1054.35139MR2024685
  107. Sun, Z. and Uhlmann, G., Generic uniqueness for an inverse boundary value problem, Duke Math. Journal, 62 (1991), 131–155. Zbl0728.35132MR1104326
  108. Sylvester, J., An anisotropic inverse boundary value problem, Comm. Pure Appl. Math., 43 (1990), 201–232. Zbl0709.35102MR1038142
  109. Sylvester, J. and Uhlmann, G., A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153–169. Zbl0625.35078MR873380
  110. Sylvester, J. and Uhlmann, G., A uniqueness theorem for an inverse boundary value problem in electrical prospection, Comm. Pure Appl. Math., 39 (1986), 92–112. Zbl0611.35088MR820341
  111. Sylvester, J. and Uhlmann, G., Inverse boundary value problems at the boundary – continuous dependence, Comm. Pure Appl. Math., 41 (1988), 197–221. Zbl0632.35074MR924684
  112. Sylvester, J. and Uhlmann, G., Inverse problems in anisotropic media, Contemp. Math., 122 (1991), 105–117. Zbl0748.35057MR1135861
  113. Takuwa, H., Uhlmann, G. and Wang, J.-N., Complex geometrical optics solutions for anisotropic equations and applications, Journal of Inverse and Ill Posed Problems, 16 (2008), 791-804. 29 (1998), 116–133. Zbl1152.35519MR2484149
  114. Tzou, L., Stability estimates for coefficients of magnetic Schrödinger equation from full and partial measurements, Comm. PDE, 33 (2008), 161-184. Zbl1157.35108MR2475324
  115. Uhlmann, G., Calderón’s problem and electrical impedance tomography, Inverse Problems, 25th Anniversary Volume, 25 (2009), 123011 (39pp.) Zbl1181.35339
  116. Uhlmann, G., Editor of Inside Out II: Inverse Problems and Applications, MSRI Publications 60, Cambridge University Press (2012). Zbl1277.65002
  117. Uhlmann, G., Developments in inverse problems since Calderón’s foundational paper, Chapter 19 in “Harmonic Analysis and Partial Differential Equations", University of Chicago Press (1999), 295-345, edited by M. Christ, C. Kenig and C. Sadosky. Zbl0963.35203MR1743870
  118. Uhlmann, G. and Wang, J.-N., Complex spherical waves for the elasticity system and probing of inclusions, SIAM J. Math. Anal., 38 (2007), 1967–1980. Zbl1131.35088MR2299437
  119. Uhlmann, G. and Wang, J.-N., Reconstruction of discontinuities in systems, SIAM J. Appl. Math., 28 (2008), 1026-1044. Zbl1146.35097MR2390978
  120. Uhlmann, G., Wang, J.-N and Wu, C. T., Reconstruction of inclusions in an elastic body, Journal de Mathématiques Pures et Appliquées, 91 (2009), 569-582. Zbl1173.35123MR2531555
  121. Zhdanov, M. S. Keller, G. V., The geoelectrical methods in geophysical exploration, Methods in Geochemistry and Geophysics, 31 (1994), Elsevier. 
  122. Zhou, T., Reconstructing electromagnetic obstacles by the enclosure method, Inverse Problems and Imaging, 4 (2010), 547-569. Zbl1206.35262MR2671111
  123. Zou, Y. and Guo, Z, A review of electrical impedance techniques for breast cancer detection, Med. Eng. Phys., 25 (2003), 79-90. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.