30 Years of Calderón’s Problem
Séminaire Laurent Schwartz — EDP et applications (2012-2013), Exposé no. 13, 25 p.

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.

DOI : 10.5802/slsedp.40
Uhlmann, Gunther 1, 2

1 Department of Mathematics University of Washington Seattle, WA 98195 USA
2 Fondation de Sciences Mathématiques de Paris
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Uhlmann, Gunther. 30 Years of Calderón’s Problem. Séminaire Laurent Schwartz — EDP et applications (2012-2013), Exposé no. 13, 25 p. doi : 10.5802/slsedp.40. http://archive.numdam.org/articles/10.5802/slsedp.40/

[1] Albin, P, Guillarmou, C., Tzou, L. and Uhlmann, G., Inverse boundary problems for systems in two dimensions, to appear Annales Institut Henri Poincaré.

[2] Alessandrini, G., Stable determination of conductivity by boundary measurements, App. Anal., 27 (1988), 153–172. | MR | Zbl

[3] Alessandrini, G., Singular solutions of elliptic equations and the determination of conductivity by boundary measurements, J. Diff. Equations, 84 (1990), 252-272. | MR | Zbl

[4] Alessandrini, G. and Vessella, S., Lipschitz stability for the inverse conductivity problem, Adv. in Appl. Math., 35 (2005), 207–241. | MR | Zbl

[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. | MR | Zbl

[6] Astala, K. and Päivärinta, L., Calderón’s inverse conductivity problem in the plane. Annals of Math., 163 (2006), 265-299. | MR | Zbl

[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. | MR | Zbl

[8] Bal, G., Ren, K., Uhlmann, G, and Zhou, T., Quantitative thermo-acoustics and related problems, Inverse Problems, 27 (2011), 055007. | MR | Zbl

[9] Bal, G. and Uhlmann, G., Inverse diffusion theory of photoacoustics, Inverse Problems, 26 (2010), 085010. | MR | Zbl

[10] Bal, G. and Uhlmann, G., Reconstructions for some coupled-physics inverse problems, Applied Mathematics Letters, 25 (2012), 1030-1033. | MR | Zbl

[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.

[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. | MR | Zbl

[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. | MR | Zbl

[14] Belishev, M. I., The Calderón problem for two-dimensional manifolds by the BC-method, SIAM J. Math. Anal., 35 (2003), 172–182. | MR | Zbl

[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. | MR | Zbl

[17] Brown, R. and Torres, R., Uniqueness in the inverse conductivity problem for conductivities with 3/2 derivatives in L p ,p>2n, J. Fourier Analysis Appl., 9 (2003), 1049-1056. | Zbl

[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. | MR

[19] Bukhgeim, A., Recovering the potential from Cauchy data in two dimensions, J. Inverse Ill-Posed Probl., 16 (2008), 19-34. | MR | Zbl

[20] Bukhgeim, A. and Uhlmann, G., Recovering a potential from partial Cauchy data, Comm. PDE, 27 (2002), 653-668. | MR | Zbl

[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. | MR

[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). | MR

[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 Equations 254 (2013), 469–492. | MR

[25] Caro, P., Ola, P. and Salo, M., Inverse boundary value problem for Maxwell equations with local data, Comm. PDE, 34 (2009), 1425-1464. | MR | Zbl

[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.

[27] Chanillo S., A problem in electrical prospection and a n-dimensional Borg-Levinson theorem, Proc. AMS, 108 (1990), 761–767. | MR | Zbl

[28] Chen, J. and Yang, Y., Quantitative photo-acoustic tomography with partial data, Inverse Problems, 28 (2012), 115014. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[32] Eskin, G., Ralston, J., On the inverse boundary value problem for linear isotropic elasticity, Inverse Problems, 18 (2002), 907–921. | MR | Zbl

[33] Francini, E., Recovering a complex coefficient in a planar domain from the Dirichlet-to-Neumann map, Inverse Problems, 16 (2000), 107–119. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[39] Guillarmou, C. and Sá Barreto, A., Inverse problems for Einstein manifolds, Inverse Problems and Imaging, 3 (2009), 1-15. | MR | Zbl

[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. | Zbl

[41] Guillarmou, C. and Tzou, L., Calderón inverse problem with partial data on Riemann surfaces, Duke Math. J., 158 (2011), 83-120. | MR | Zbl

[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. | MR

[43] Hähner, P., A periodic Faddeev-type solution operator, J. Differential Equations, 128 (1996), 300–308. | MR | Zbl

[44] Haberman, B. and Tataru, D., Uniqueness in Calderón’s problem with Lipschitz conductivities, to appear Duke Math. J.

[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. | MR | Zbl

[46] Henkin, G. and Michel, V., Inverse conductivity problem on Riemann surfaces, J. Geom. Anal., 18 (2008), 1033–1052. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[49] Imanuvilov, O., Uhlmann, G. and Yamamoto, M., The Calderón problem with partial data in two dimensions, Journal AMS, 23 (2010), 655-691. | MR | Zbl

[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. | MR | Zbl

[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. | MR

[52] Imanuvilov, O., Uhlmann, G. and Yamamoto, M., Inverse boundary problem with Cauchy data on disjoint sets, Inverse Problems, 27 (2011), 085007. | MR | Zbl

[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. | MR

[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.

[55] Imanuvilov, O. and Yamamoto, M., Inverse boundary value for Schrödinger equation in two dimensions, arXiv:1211.1419v1.

[56] Imanuvilov, O. and Yamamoto, M., Uniqueness for inverse boundary problems by Dirichlet-to-Neumann map on arbitrary subboundaries, preprint, arXiv:1303.2159.

[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. | MR | Zbl

[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. | MR | Zbl

[61] Isozaki, H. and Uhlmann, G., Hyperbolic geometric and the local Dirichlet-to-Neumann map, Advances in Math. 188 (2004), 294-314. | MR | Zbl

[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.

[65] Kenig, C. and Salo, M., Recent progress in the Calderón problem with partial data, preprint, arXiv:1302.4218.

[66] Kenig, C., Salo, M. and Uhlmann, G., Inverse problems for the anisotropic Maxwell equations", Duke Math. J., 157 (2011), 369-419. | MR | Zbl

[67] Kenig, C., Sjöstrand, J. and Uhlmann, G., The Calderón problem with partial data, Annals of Math., 165 (2007), 567-591. | MR | Zbl

[68] Knudsen, K., The Calderón problem with partial data for less smooth conductivities, Comm. Partial Differential Equations, 31 (2006), 57–71. | MR | Zbl

[69] Knudsen, K. and Salo, M., Determining nonsmooth first order terms from partial boundary measurements, Inverse Problems and Imaging, 1 (2007), 349-369. | MR | Zbl

[70] Kocyigit, I., Acoustic-electric tomography and CGO solutions with internal data, Inverse Problems, 28 (2012), 125004. | MR

[71] Kohn, R., Shen, H., Vogelius, M. and Weinstein, M., Cloaking via change of variables in Electrical Impedance Tomography, Inverse Problems 24 (2008), 015016 (21pp). | MR | Zbl

[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). | MR | Zbl

[73] Kohn, R. and Vogelius, M., Determining conductivity by boundary measurements, Comm. Pure Appl. Math., 37 (1984), 289–298. | MR | Zbl

[74] Kohn, R. and Vogelius, M., Determining conductivity by boundary measurements II. Interior results, Comm. Pure Appl. Math., 38 (1985), 643–667. | MR | Zbl

[75] Krupchyk, K., Lassas, M. and Uhlmann, G., Inverse problems for differential forms on Riemannian manifolds with boundary, Comm. PDE., 36 (2011), 1475-1509. | MR | Zbl

[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. | MR | Zbl

[77] Krupchyk, K., Lassas, M. and Uhlmann, G., Inverse boundary value problems for the polyharmonic operator, Journal Functional Analysis, 262 (2012), 1781-1801. | MR | Zbl

[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. | MR

[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. | Numdam | MR | Zbl

[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. | MR | Zbl

[82] Lee, J. and Uhlmann, G., Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math., 42 (1989), 1097–1112. | MR | Zbl

[83] Li, X. and Uhlmann, G., Inverse problems on a slab, Inverse Problems and Imaging, 4 (2010), 449-462. | MR | Zbl

[84] Mandache, N., Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems, 17 (2001), 1435–1444. | MR | Zbl

[85] Nachman, A., Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), 71-96. | MR | Zbl

[86] Nachman, A., Reconstructions from boundary measurements, Ann. of Math., 128 (1988), 531–576. | MR | Zbl

[87] Nachman, A. and Street, B., Reconstruction in the Calderón problem with partial data, Comm. PDE, 35 (2010), 375-390. | MR | Zbl

[88] Nagayasu, S., Uhlmann, G. and Wang, J.-N., Depth dependent stability estimate in electrical impedance tomography, Inverse Problems, 25 (2009), 075001. | MR | Zbl

[89] Nagayasu, S., Uhlmann, G. and Wang, J.-N., Reconstruction of penetrable obstacles in acoustics, SIAM J. Math. Anal., 43 (2011), 189-211. | MR | Zbl

[90] Nagayasu, S, Uhlmann, G. and Wang, J.-N., Increasing stability for the acoustic equation, Inverse Problems, 29 (2013), 020012.

[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. | MR | Zbl

[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. | Zbl

[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. | MR | Zbl

[94] Novikov R. G., Multidimensional inverse spectral problems for the equation -Δψ+(v(x)-Eu(x))ψ=0, Funktsionalny Analizi Ego Prilozheniya, 22 (1988), 11-12, Translation in Functional Analysis and its Applications, 22 (1988) 263–272. | MR | Zbl

[95] Ola, P., Päivärinta, L. and Somersalo, E., An inverse boundary value problem in electrodynamics, Duke Math. J., 70 (1993), 617–653. | MR | Zbl

[96] Ola, P. and Somersalo, E. , Electromagnetic inverse problems and generalized Sommerfeld potentials, SIAM J. Appl. Math., 56 (1996), 1129-1145 | MR | Zbl

[97] Päivärinta, L., Panchenko, A. and Uhlmann, G., Complex geometrical optics for Lipschitz conductivities, Revista Matematica Iberoamericana, 19 (2003), 57-72. | MR | Zbl

[98] Pestov, L. and Uhlmann, G., Two dimensional simple Riemannian manifolds with boundary are boundary distance rigid,Annals of Math., 161 (2005), 1089-1106. | MR | Zbl

[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. | MR | Zbl

[100] Salo, M., Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field, Comm. PDE, 31 (2006), 1639-1666. | MR | Zbl

[101] Salo, M., Inverse problems for nonsmooth first order perturbations of the Laplacian, Ann. Acad. Sci. Fenn. Math. Diss., 139 (2004), 67 pp. | MR | Zbl

[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. | MR | Zbl

[103] Salo, M. and Wang, J.-N. , Complex spherical waves and inverse problems in unbounded domains, Inverse Problems 22 (2006), 2299–2309. | MR | Zbl

[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. | MR | Zbl

[106] Sun, Z. and Uhlmann, G., Anisotropic inverse problems in two dimensions, Inverse Problems, 19 (2003), 1001-1010. | MR | Zbl

[107] Sun, Z. and Uhlmann, G., Generic uniqueness for an inverse boundary value problem, Duke Math. Journal, 62 (1991), 131–155. | MR | Zbl

[108] Sylvester, J., An anisotropic inverse boundary value problem, Comm. Pure Appl. Math., 43 (1990), 201–232. | MR | Zbl

[109] Sylvester, J. and Uhlmann, G., A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153–169. | MR | Zbl

[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. | MR | Zbl

[111] Sylvester, J. and Uhlmann, G., Inverse boundary value problems at the boundary – continuous dependence, Comm. Pure Appl. Math., 41 (1988), 197–221. | MR | Zbl

[112] Sylvester, J. and Uhlmann, G., Inverse problems in anisotropic media, Contemp. Math., 122 (1991), 105–117. | MR | Zbl

[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. | MR | Zbl

[114] Tzou, L., Stability estimates for coefficients of magnetic Schrödinger equation from full and partial measurements, Comm. PDE, 33 (2008), 161-184. | MR | Zbl

[115] Uhlmann, G., Calderón’s problem and electrical impedance tomography, Inverse Problems, 25th Anniversary Volume, 25 (2009), 123011 (39pp.) | Zbl

[116] Uhlmann, G., Editor of Inside Out II: Inverse Problems and Applications, MSRI Publications 60, Cambridge University Press (2012).

[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. | MR | Zbl

[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. | MR | Zbl

[119] Uhlmann, G. and Wang, J.-N., Reconstruction of discontinuities in systems, SIAM J. Appl. Math., 28 (2008), 1026-1044. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[123] Zou, Y. and Guo, Z, A review of electrical impedance techniques for breast cancer detection, Med. Eng. Phys., 25 (2003), 79-90.

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