Inverse Problems: Visibility and Invisibility

Uhlmann, Gunther

Journées équations aux dérivées partielles, (2012), p. 1-64 / Harvested from Numdam

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This survey article expands on the lectures given at Biarritz in June, 2012, on “Inverse Problems: Visibility and Invisibility". The first inverse problem we consider is whether one can determine the electrical conductivity of a medium by making voltage and current measurements at the boundary. This is called electrical impedance tomography (EIT) and also Calderón’s problem since the famous analyst proposed it in the mathematical literature [38]. The second is on travel time tomography. The question is whether one can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as a geometry problem, the boundary rigidity problem. Can we determine a Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function between boundary points? These two inverse problems concern visibility, that is whether we can determine the internal properties of a medium by making measurements at the boundary. The last topic of this paper considers the opposite issue: invisibility: Can one make objects invisible to different types of waves, including light?

Publié le : 2012-01-01
DOI : https://doi.org/10.5802/jedp.94

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     author = {Uhlmann, Gunther},
     title = {Inverse Problems: Visibility and Invisibility},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     year = {2012},
     pages = {1-64},
     doi = {10.5802/jedp.94},
     language = {en},
     url = {http://dml.mathdoc.fr/item/JEDP_2012____A11_0}
}

Uhlmann, Gunther. Inverse Problems: Visibility and Invisibility. Journées équations aux dérivées partielles,  (2012), pp. 1-64. doi : 10.5802/jedp.94. http://gdmltest.u-ga.fr/item/JEDP_2012____A11_0/

Bibliographie

[1] Ablowitz, M., Yaacov, D. B. and Fokas, A., On the inverse scattering transform for the Kadomtsev-Petviashvili equation, Studies Appl. Math., 69(1983), 135–143. | MR 715426 | Zbl 0527.35080

[2] Ahlfors, L., Quasiconformal Mappings, Van Nostrand, (1966). | MR 200442

[3] Albin, P, Guillarmou, C., Tzou, L. and Uhlmann, G., Inverse boundary problems for systems in two dimensions, to appear Annales Institut Henri Poincaré. | MR 3085925 | Zbl pre06195029

[4] Alessandrini, G., Stable determination of conductivity by boundary measurements, App. Anal., 27(1988), 153–172. | MR 922775 | Zbl 0616.35082

[5] Alessandrini, G., Singular solutions of elliptic equations and the determination of conductivity by boundary measurements, J. Diff. Equations, 84(1990), 252-272. | MR 1047569 | Zbl 0778.35109

[6] Alessandrini, G., Open issues of stability for the inverse conductivity problem, J. Inverse Ill-Posed Probl., 15(2007), 451–460. | MR 2367859 | Zbl 1221.35443

[7] Alessandrini, G. and Vessella, S., Lipschitz stability for the inverse conductivity problem, Adv. in Appl. Math., 35(2005), 207–241. | MR 2152888 | Zbl 1095.35058

[8] Alexandrova, I., Structure of the Semi-Classical Amplitude for General Scattering Relations, Comm. PDE, 30(2005), 1505-1535. | MR 2182302 | Zbl 1088.81091

[9] 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 2048188 | Zbl 1051.35103

[10] Yu. E. Anikonov, Some Methods for the Study of Multidimensional Inverse Problems , Nauka, Sibirsk Otdel., Novosibirsk (1978). | MR 504333

[11] Astala, K. and Päivärinta, L., Calderón’s inverse conductivity problem in the plane. Annals of Math., 163(2006), 265-299. | MR 2195135 | Zbl 1111.35004

[12] 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 2131051 | Zbl 1129.35483

[13] Bal, G., Hybrid inverse problems and internal functionals, Chapter in Inside Out II, MSRI Publications 60, Cambridge University Press (2012), 271-323 (ed. by G. Uhlmann).

[14] Bal, G., Langmore, I. and Monard, F., Inverse transport with isotropic sources and angularly averaged measurements, Inverse Probl. Imaging, 2(2008),23–42. | MR 2375321 | Zbl 1171.65104

[15] Bal, G., Ren, K., Uhlmann, G, and Zhou, T., Quantitative thermo-acoustics and related problems, Inverse Problems, 27(2011), 055007. | MR 2793826 | Zbl 1217.35207

[16] Bal, G. and Uhlmann, G., Inverse diffusion theory of photoacoustics, Inverse Problems, 26(2010), 085010. | MR 2658827 | Zbl 1197.35311

[17] Bal, G. and Uhlmann, G., Reconstructions for some coupled-physics inverse problems, Applied Mathematics Letters, 25(2012), 1030-1033. | MR 2915119 | Zbl 1252.65185

[18] 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. | MR 3084700 | Zbl 1273.35308

[19] Barber, D. and Brown, B., Applied potential tomography, J. Phys. E, 17(1984), 723–733.

[20] Barceló, T., 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 2373740 | Zbl 1133.35104

[21] Beals, R. and Coifman, R., Transformation spectrales et equation d’evolution non lineares, Seminaire Goulaouic-Meyer-Schwarz, exp. 21, 1981-1982. | Numdam | Zbl 0496.35071

[22] Beals, R. and Coifman, R., Multidimensional inverse scattering and nonlinear PDE, Proc. Symp. Pure Math., 43(1985), American Math. Soc., Providence, 45–70. | Zbl 0575.35011

[23] Belishev, M. I., The Calderón problem for two-dimensional manifolds by the BC-method, SIAM J. Math. Anal., 35(2003), 172–182. | MR 2001471 | Zbl 1048.58019

[24] Belishev, M. and Kurylev, Y, To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Comm. Partial Diff. Eqns., 17(1992), 767–804. | MR 1177292 | Zbl 0812.58094

[25] Borcea, L., Electrical impedance tomography, Inverse Problems, 18(2002), R99–R136. | MR 1955896 | Zbl 1031.35147

[26] Bernstein, I.N. and Gerver, M.L., Conditions on distinguishability of metrics by hodographs. Methods and Algorithms of Interpretation of Seismological Information, Computerized Seismology 13, Nauka, Moscow, 50–73 (in Russian.)

[27] Besson, G., Courtois, G. and Gallot, S., Entropies et rigidités des espaces localement symétriques de courbure strictment négative, Geom. Funct. Anal., 5(1995), 731-799. | MR 1354289 | Zbl 0851.53032

[28] Beylkin, G., Stability and uniqueness of the solution of the inverse kinematic problem in the multidimensional case, J. Soviet Math., 21(1983), 251–254. | Zbl 0507.53018

[29] Blasten, E, Stability and uniqueness for the inverse problem of the Schrödinger equation with potentials in $W^{p, ϵ}$ , arXiv:1106.0632.

[30] Borcea, L., Electrical impedance tomography, Inverse Problems, 18(2002), R99–R136. | MR 1955896 | Zbl 1031.35147

[31] Borcea, L., Druskin, V., Guevara Vasquez, F. and Mamonov, A.V., Resistor network approaches to electrical impedance tomography, Inside Out II, MSRI Publications, Volume 60(2012), 55-118 (G. Uhlmann, editor).

[32] 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 1881563 | Zbl 0991.35104

[33] Brown, R. and Torres, R., Uniqueness in the inverse conductivity problem for conductivities with $3 / 2$ derivatives in $L^{p}, p > 2 n,$ J. Fourier Analysis Appl., 9(2003), 1049-1056. | Zbl 1051.35105

[34] 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 1452176

[35] Bukhgeim, A., Recovering the potential from Cauchy data in two dimensions, J. Inverse Ill-Posed Probl., 16(2008), 19-34. | MR 2387648 | Zbl 1142.30018

[36] Bukhgeim, A. and Uhlmann, G., Recovering a potential from partial Cauchy data, Comm. PDE, 27(2002), 653-668. | MR 1900557 | Zbl 0998.35063

[37] Burago, D. and Ivanov, S., Boundary rigidity and filling volume minimality for metrics close to a Euclidean metric, Annals of Math., 171(2010), 1183-1211 | MR 2630062 | Zbl 1192.53048

[38] 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 590275

[39] Calderón, A. P., Reminiscencias de mi vida matemática, Discurso de investidura de “Doctor Honoris Causa", Universidad Autónoma de Madrid, Publicaciones UAM (1997), 117-125.

[40] 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 203254

[41] Caro, P., Ola, P. and Salo, M., Inverse boundary value problem for Maxwell equations with local data, Comm. PDE, 34(2009), 1425-1464. | MR 2581979 | Zbl 1185.35321

[42] Chanillo S., A problem in electrical prospection and a $n$ -dimensional Borg-Levinson theorem, Proc. AMS, 108(1990), 761–767. | MR 998731 | Zbl 0702.35035

[43] Chen, J. and Yang, Y., Quantitative photo-acoustic tomography with partial data, Inverse Problems, 28(2012), 115014. | MR 2997231 | Zbl 1252.35278

[44] Cheney, M., Isaacson, D., Newell, J. C., Electrical impedance tomography, SIAM Rev., 41(1999), 85–101. | MR 1669729 | Zbl 0927.35130

[45] Creager, K. C., Anisotropy of the inner core from differential travel times of the phases PKP and PKIPK, Nature, 356(1992), 309-314.

[46] Croke, C., Rigidity and the distance between boundary points, J. Differential Geom., 33(1991), 445–464. | MR 1094465 | Zbl 0729.53043

[47] Croke, C., Rigidity for surfaces of non-positive curvature , Comment. Math. Helv., 65(1990), 150-169. | MR 1036134 | Zbl 0704.53035

[48] Croke, C, Dairbekov, D. and Sharafutdinov, V., Local boundary rigidity of a compact Riemannian manifold with curvature bounded above, Trans. Amer. Math. Soc. 352(2000), no. 9, 3937–3956. | MR 1694283 | Zbl 0958.53027

[49] Croke, C. and Kleiner, B., Conjugacy and Rigidity for Manifolds with a Parallel Vector Field, J. Diff. Geom. 39(1994), 659–680. | MR 1274134 | Zbl 0807.53035

[50] Dairbekov, N. and Uhlmann, G., Reconstructing the metric and magnetic field from the scattering relation, Inverse Problems and Imaging, 4(2010), 397-409. | MR 2671103 | Zbl 1202.53042

[51] Dos Santos Ferreira, D., Kenig, C.E., Sjöstrand, J. and Uhlmann, G., On The linearized local Calderón problem, Math. Research Lett., 16(2009), 955-970. | MR 2576684 | Zbl 1198.31003

[52] 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 2287913 | Zbl 1148.35096

[53] 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 2534094 | Zbl 1181.35327

[54] Duistermaat, J.J. and Hörmander, L., Fourier integral operators II, Acta Mathematica, 128(1972), 183-269. | MR 388464 | Zbl 0232.47055

[55] Eisenhart, L., Riemannian geometry, 2nd printing, Princeton University Press, 1949. | MR 35081 | Zbl 0174.53303

[56] Eskin, G., Ralston, J., On the inverse boundary value problem for linear isotropic elasticity, Inverse Problems, 18(2002), 907–921. | MR 1910209 | Zbl 1080.35175

[57] Faddeev D., Growing solutions of the Schrödinger equation , Dokl. Akad. Nauk SSSR, 165(1965), 514–517 (translation in Sov. Phys. Dokl. 10, 1033). | Zbl 0147.09404

[58] Francini, E., Recovering a complex coefficient in a planar domain from the Dirichlet-to-Neumann map, Inverse Problems, 16(2000), 107–119. | MR 1741230 | Zbl 0968.35125

[59] Fridman, B. Kuchment, P., Ma, D. and Papanicolaou, Vassilis G., Solution of the linearized inverse conductivity problem in a half space via integral geometry. Voronezh Winter Mathematical Schools, 85–95, Amer. Math. Soc.Transl. Ser. 2,, 184, 85-95. Amer. Math. Soc., Providence, RI, 1998 | MR 1729927 | Zbl 0904.35096

[60] M. L. Gerver and N. S. Nadirashvili, An isometricity conditions for Riemannian metrics in a disk, Soviet Math. Dokl. 29 (1984), 199–203. | Zbl 0633.53066

[61] Gilbarg D. and Trudinger, N., Elliptic Partial Differential Equations, Interscience Publishers (1964).

[62] Greenleaf, A., Kurylev, Y., Lassas, M. and Uhlmann, G., Cloaking devices, electromagnetic wormholes, and transformation optics, SIAM Rev., 51(2009), 3–33. | MR 2481110 | Zbl 1158.78004

[63] Greenleaf, A., Kurylev, Y., Lassas, M. and Uhlmann, G., Invisibility and inverse problems, Bull. Amer. Math. Soc. (N.S.), 46(2009), 55–97. | MR 2457072 | Zbl 1159.35074

[64] 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 1941812 | Zbl 1061.35165

[65] 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.

[66] Greenleaf, A., Lassas, M. and Uhlmann, G., On nonuniqueness for Calderón’s inverse problem, Math. Res. Lett., 10 (2003), 685-693. | MR 2024725 | Zbl 1054.35127

[67] 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 1838663 | Zbl 1013.35085

[68] M. Gromov, Filling Riemannian manifolds, J. Differential Geometry 18(1983), no. 1, 1–148. | MR 697984 | Zbl 0515.53037

[69] Guillarmou, C. and Sá Barreto, A., Inverse problems for Einstein manifolds, Inverse Problems and Imaging, 3(2009), 1-15. | MR 2558301 | Zbl 1229.58025

[70] 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 1231.35302

[71] Guillarmou, C. and Tzou, L., Calderón inverse problem with partial data on Riemann surfaces, Duke Math. J., 158(2011), 83-120. | MR 2794369 | Zbl 1222.35212

[72] 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 2795512 | Zbl 1260.58011

[73] V. Guillemin, Sojourn times and asymptotic properties of the scattering matrix. Proceedings of the Oji Seminar on Algebraic Analysis and the RIMS Symposium on Algebraic Analysis (Kyoto Univ., Kyoto, 1976). Publ. Res. Inst. Math. Sci. 12(1976/77), supplement, 69–88. | MR 448453 | Zbl 0381.35064

[74] Hähner, P., A periodic Faddeev-type solution operator, J. Differential Equations, 128(1996), 300–308. | MR 1392403 | Zbl 0849.35022

[75] Hanke, M. and Brühl, M., Recent progress in electrical impedance tomography. Special section on imaging, it Inverse Problems, 19(2003),S65–S90. | MR 2036522 | Zbl 1048.92022

[76] Haberman, B. and Tataru, D., Uniqueness in Calderón’s problem with Lipschitz conductivities, to appear Duke Math. J. | MR 3024091 | Zbl 1260.35251

[77] 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 2261266 | Zbl 1106.35133

[78] Henkin, G. and Michel, V., Inverse conductivity problem on Riemann surfaces, J. Geom. Anal., 18(2008), 1033–1052. | MR 2438910 | Zbl 1151.35101

[79] Herglotz, G., Uber die elastizitaet derErde bei beruecksichtigung ihrer variablen dichte, Zeitschr. fur Math. Phys., 52(1905), 275-299.

[80] Holder, D., Electrical Impedance Tomography, Institute of Physics Publishing, Bristol and Philadelphia, 2005.

[81] Holder, D., Isaacson, D., Müller, J. and Siltanen, S., editors, Physiol. Meas., 25(2003) no 1.

[82] Hörmander, L., The analysis of linear partial differential operators, vol. I, Springer-Verlag, Berlin, 1983. | Zbl 0521.35002

[83] 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 2342953 | Zbl 1142.35104

[84] 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 1830379 | Zbl 0988.35168

[85] Ikehata, M., How to draw a picture of an unknown inclusion from boundary measurements. Two mathematical inversion algorithms, J. Inverse Ill-Posed Probl., 7(1999), 255–271. | MR 1694840 | Zbl 0928.35207

[86] Ikehata, M. and Siltanen, S., Numerical method for finding the convex hull of an inclusion in conductivity from boundary measurements, Inverse Problems, 16(2000), 273-296. | MR 1776482 | Zbl 0956.35133

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

[88] Imanuvilov, O., Uhlmann, G. and Yamamoto, M., The Calderón problem with partial data in two dimensions, Journal AMS, 23(2010), 655-691. | MR 2629983 | Zbl 1201.35183

[89] 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 2770947 | Zbl 1256.35203

[90] 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 2999548 | Zbl 1260.35253

[91] Imanuvilov, O., Uhlmann, G. and Yamamoto, M., Inverse boundary problem with Cauchy data on disjoint sets, Inverse Problems, 27(2011), 085007. | MR 2819949 | Zbl 1222.35213

[92] 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 2997011 | Zbl 1264.35281

[93] Isaacson D. and Isaacson E., Comment on Calderón’s paper: “On an inverse boundary value problem”, Math. Comput., 52(1989), 553–559. | MR 962208 | Zbl 0672.65107

[94] Isaacson, D., Müller, J. L., Newell, J. C. and Siltanen, S., Reconstructions of chest phantoms by the d-bar method for electrical impedance tomography, IEEE Transactions on Medical Imaging, 23(2004), 821- 828.

[95] 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.

[96] Isakov, V., On uniqueness in the inverse conductivity problem with local data, Inverse Problems and Imaging, 1(2007), 95-105. | MR 2262748 | Zbl 1125.35113

[97] Isakov, V., On uniqueness in inverse problems for semilinear parabolic equations, Arch. Rat. Mech. Anal., 124(1993) , 1–12. | MR 1233645 | Zbl 0804.35150

[98] Isakov, V., Completeness of products of solutions and some inverse problems for PDE, J. Diff. Equations, 92(1991), 305–317. | MR 1120907 | Zbl 0728.35141

[99] Isakov, V. and Nachman, A., Global uniqueness for a two-dimensional semilinear elliptic inverse problem, Trans. of AMS, 347(1995), 3375–3390. | MR 1311909 | Zbl 0849.35148

[100] Isakov, V. and Sylvester, J., Global uniqueness for a semilinear elliptic inverse problem, Comm. Pure Appl. Math., 47(1994), 1403–1410. | MR 1295934 | Zbl 0817.35126

[101] 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 2102396 | Zbl 1069.35092

[102] Isozaki, H. and Uhlmann, G., Hyperbolic geometric and the local Dirichlet-to-Neumann map, Advances in Math. 188(2004), 294-314. | MR 2087229 | Zbl 1062.35172

[103] 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.

[104] Jossinet, J., The impedivity of freshly excised human breast tissue, Physiol. Meas., 19(1998), 61-75.

[105] Kang, H. and Uhlmann, G., Inverse problems for the Pauli Hamiltonian in two dimensions, Journal of Fourier Analysis and Applications, 10(2004), 201-215. | MR 2054308 | Zbl 1081.35141

[106] Kashiwara, M., On the structure of hyperfunctions, Sagaku no Ayumi, 15(1970), 19–72 (in Japanese).

[107] Kenig, C., Salo, M. and Uhlmann, G., Inverse Problems for the Anisotropic Maxwell’s Equations, Duke Math. J., 157(2011), 369-419. | MR 2783934 | Zbl 1226.35086

[108] Kenig, C., Sjöstrand, J. and Uhlmann, G., The Calderón problem with partial data, Annals of Math., 165(2007), 567-591. | MR 2299741 | Zbl 1127.35079

[109] Knudsen, K., The Calderón problem with partial data for less smooth conductivities, Comm. Partial Differential Equations, 31(2006), 57–71. | MR 2209749 | Zbl 1091.35116

[110] Knudsen, K. and Salo, M., Determining nonsmooth first order terms from partial boundary measurements, Inverse Problems and Imaging, 1(2007), 349-369. | MR 2282273 | Zbl 1122.35152

[111] Kocyigit, I., Acoustic-electric tomography and CGO solutions with internal data, Inverse Problems, 28(2012), 125004. | MR 2997013 | Zbl 1266.78015

[112] 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 2384775 | Zbl 1153.35406

[113] 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 773707 | Zbl 0573.35084

[114] Kohn, R. and Vogelius, M., Determining conductivity by boundary measurements, Comm. Pure Appl. Math., 37(1984), 289–298. | MR 739921 | Zbl 0586.35089

[115] Kohn, R. and Vogelius, M., Determining conductivity by boundary measurements II. Interior results, Comm. Pure Appl. Math., 38(1985), 643–667. | MR 803253 | Zbl 0595.35092

[116] Kolehmainen, V., Lassas, M., Ola, P., Inverse conductivity problem with an imperfectly known boundary, SIAM J. Appl. Math. 66(2005), 365–383. | MR 2203860 | Zbl 1141.35472

[117] Krupchyk, K., Lassas, M. and Uhlmann, G., Inverse problems for differential forms on Riemannian manifolds with boundary", Comm. PDE., 36(2011), 1475-1509. | MR 2825599 | Zbl 1227.35245

[118] 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 2914058 | Zbl 1238.35188

[119] Krupchyk, K., Lassas, M. and Uhlmann, G., Inverse boundary value problems for the polyharmonic operator, Journal Functional Analysis, 262(2012), 1781-1801. | MR 2873860 | Zbl 1239.35184

[120] 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 2873860

[121] Krupchyk, K. and Uhlmann, G., Determining a magnetic Schrödinger operator with a bounded magnetic potential from boundary measurements, preprint.

[122] Lassas, M., Sharafutdinov, V. and Uhlmann, G., Semiglobal boundary rigidity for Riemannian metrics, Math. Annalen 325(2003), 767-793. | MR 1974568 | Zbl pre01944293

[123] Lassas, M. and Uhlmann, G., Determining a Riemannian manifold from boundary measurements, Ann. Sci. École Norm. Sup., 34(2001), 771–787. | Numdam | MR 1862026 | Zbl 0992.35120

[124] 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 2014876 | Zbl 1077.58012

[125] Lee, J. and Uhlmann, G., Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math., 42 (1989), 1097–1112. | MR 1029119 | Zbl 0702.35036

[126] Leonhardt, U., Optical Conformal Mapping, Science 312 (2006), 1777-1780. | MR 2237569 | Zbl 1226.78001

[127] Li, X. and Uhlmann, G., Inverse problems on a slab, Inverse Problems and Imaging, 4(2010), 449-462. | MR 2671106 | Zbl 1200.35331

[128] Mandache, N., Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems, 17(2001), 1435–1444. | MR 1862200 | Zbl 0985.35110

[129] Melrose, R. B., Geometric scattering theory, Cambridge University Press, 1995. | MR 1350074 | Zbl 0849.58071

[130] Michel, R., Sur la rigidité imposée par la longueur des géodésiques, Invent. Math., 65(1981), 71-83. | MR 636880 | Zbl 0471.53030

[131] Michel, R., Restriction de la distance géodésique a un arc et rigidité. Bull. Soc. Math. France, 122(1994), 435–442. | Numdam | MR 1294465 | Zbl 0821.53038

[132] Mukhometov, R. G., The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry (Russian), Dokl. Akad. Nauk SSSR,232(1977), no. 1, 32–35. | MR 431074 | Zbl 0372.53034

[133] Mukhometov, R.G., On one problem of reconstruction of Riemannian metric (Russian), Siberian Math. Journal 22(1981), no. 3, 119–135. | Zbl 0478.53048

[134] Mukhometov, R.G. and Romanov, V.G., On the problem of finding an isotropic Riemannian metric in an $n$ -dimensional space (Russian), Dokl. Akad. Nauk SSSR 243(1978), no. 1, 41–44. | MR 511273 | Zbl 0418.53028

[135] Nachman, A., Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143(1996), 71-96. | MR 1370758 | Zbl 0857.35135

[136] Nachman, A., Reconstructions from boundary measurements, Ann. of Math., 128(1988), 531–576. | MR 970610 | Zbl 0675.35084

[137] Nachman, A. and Ablowitz, N., A multidimensional inverse scattering method, Studies in App. Math., 71(1984), 243–250. | MR 769078 | Zbl 0557.35032

[138] Nachman, A. and Street, B., Reconstruction in the Calderón problem with partial data, Comm. PDE, 35 (2010), 375-390. | MR 2748629 | Zbl 1186.35242

[139] Nagayasu, S., Uhlmann, G. and Wang, J.-N., Depth dependent stability estimate in electrical impedance tomography, Inverse Problems, 25(2009), 075001. | MR 2519853 | Zbl 1172.35514

[140] Nagayasu, S., Uhlmann, G. and Wang, J.-N., Reconstruction of penetrable obstacles in acoustics, SIAM J. Math. Anal., 43(2011), 189-211. | MR 2765688 | Zbl 1234.35315

[141] Nagayasu, S, Uhlmann, G. and Wang, J.-N., Increasing stability for the acoustic equation, to appear Inverse Problems.

[142] 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 1843272 | Zbl 0981.35100

[143] 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 0814.35147

[144] Nakamura G., and Uhlmann, G., Inverse problems at the boundary for an elastic medium, SIAM J. Math. Anal., 26(1995), 263–279. | MR 1320220 | Zbl 0840.35122

[145] 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 1354996 | Zbl 0843.35134

[146] 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. | MR 976992 | Zbl 0689.35098

[147] Novikov, R. G. and Henkin, G. M., The $\bar{\partial}$ -equation in the multidimensional inverse scattering problem, Russ. Math. Surv., 42(1987), 109–180. | MR 896879 | Zbl 0674.35085

[148] Ola, P., Päivärinta, L. and Somersalo, E., An inverse boundary value problem in electrodynamics, Duke Math. J., 70(1993), 617–653. | MR 1224101 | Zbl 0804.35152

[149] Ola, P. and Somersalo, E. , Electromagnetic inverse problems and generalized Sommerfeld potentials, SIAM J. Appl. Math., 56(1996), 1129-1145 | MR 1398411 | Zbl 0858.35138

[150] Otal, J.P., Sur les longueur des géodésiques d’une métrique a courbure négative dans le disque, Comment. Math. Helv. 65(1990), 334–347. | MR 1057248 | Zbl 0736.53042

[151] Paternain, G., Salo, M. and Uhlmann, G., Tensor tomography on surfaces, to appear Inventiones Math. | MR 3069117 | Zbl pre06197114

[152] Paternain, G., Salo, M. and Uhlmann, G., The attenuated ray transform for connections and Higgs fields, Geometric and Functional Analysis (GAFA), 22(2012), 1460-1489. | MR 2989440 | Zbl 1256.53021

[153] Päivärinta, L., Panchenko, A. and Uhlmann, G., Complex geometrical optics for Lipschitz conductivities, Revista Matematica Iberoamericana, 19(2003), 57-72. | MR 1993415 | Zbl 1055.35144

[154] Pendry, J.B., Schurig, D. and Smith, D.R., Controlling electromagnetic fields, Science, 312, 1780 - 1782. | MR 2237570 | Zbl 1226.78003

[155] L. Pestov, V.A. Sharafutdinov, Integral geometry of tensor fields on a manifold of negative curvature, Siberian Math. J. 29 (1988), 427–441. | MR 953028 | Zbl 0675.53048

[156] Pestov, L. and Uhlmann, G., Two dimensional simple Riemannian manifolds with boundary are boundary distance rigid,Annals of Math., 161(2005), 1089-1106. | MR 2153407 | Zbl 1076.53044

[157] Pestov, L., and Uhlmann, G., The boundary distance function and the Dirichlet-to-Neumann map, Math. Research Letters, 11(2004), 285-298. | MR 2067474 | Zbl 1067.53027

[158] Pestov, P., and Uhlmann, G., Characterization of the range and inversion formulas for the geodesic X-ray transform, International Mathematical Research Notices, 80(2004), 4331-4347. | MR 2126628 | Zbl 1075.44003

[159] Petersen P., Riemannian Geometry, Springer-Verlag, 1998. | MR 1480173 | Zbl 1220.53002

[160] Ramm, A. G., Recovery of the potential from fixed energy scattering data, Inverse Problems, 4(1988), 877-886. | MR 965652 | Zbl 0632.35075

[161] 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 2198854 | Zbl 1158.35105

[162] Romanov, V.G., Inverse Problems of Mathematical Physics, VNU Science Press, Utrech, the Netherlands, 1987. | MR 885902

[163] Salo, M., Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field, Comm. PDE, 31(2006), 1639-1666. | MR 2273968 | Zbl 1119.35119

[164] Salo, M., Inverse problems for nonsmooth first order perturbations of the Laplacian, Ann. Acad. Sci. Fenn. Math. Diss., 139(2004), 67 pp. | MR 2105191 | Zbl 1059.35175

[165] 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 2669360 | Zbl 1197.35329

[166] Salo, M. and Wang, J.-N. , Complex spherical waves and inverse problems in unbounded domains, Inverse Problems 22(2006), 2299–2309. | MR 2277543 | Zbl 1106.35140

[167] Santosa, F. and Vogelius, M., A backprojection algorithm for electrical impedance imaging, SIAM J. Appl. Math., 50(1990), 216–243. | MR 1036240 | Zbl 0691.65087

[168] Sharafutdinov, V., Integral Geometry of Tensor Fields, Inverse and Ill-Posed Problems Series, VSP, Utrecht, 1994. | MR 1374572 | Zbl 0883.53004

[169] V.A. Sharafutdinov, Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds, J. Geom. Anal. 17 (2007), 147–187. | MR 2302878 | Zbl 1142.53029

[170] Sharafutdinov, V., Skokan, M. and Uhlmann, G., Regularity of ghosts in tensor tomography, Journal of Geometric Analysis, 15(2005), 517-560. | MR 2190243

[171] Sharafutdinov V. and Uhlmann, G., On deformation boundary rigidity and spectral rigidity for Riemannian surfaces with no focal points, Journal of Differential Geometry, 56 (2001), 93–110. | MR 1863022 | Zbl 1065.53039

[172] Schurig, D., Mock, J., Justice, B., Cummer, S., Pendry, J., Starr, A. and Smith, D., Metamaterial electromagnetic cloak at microwave frequencies, Science, 314(2006), 977-980.

[173] 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.

[174] Sjöstrand, J., Singularités analytiques microlocales, Astérisque, 1985. | Zbl 0524.35007

[175] Sjöstrand, J., Remark on extensions of the Watermelon theorem, Math. Res. Lett., 1(1994), 309–317. | MR 1302646 | Zbl 0842.35003

[176] 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 1181585 | Zbl 0757.65128

[177] Stefanov, P. and Uhlmann, G., Multi-Wave Methods via Ultrasound, Inverse Problems and Applications, Inside Out II, MSRI Publications 60, Cambridge University Press (2012), 271-323 (ed. by G. Uhlmann).

[178] Stefanov, P. and Uhlmann, G., Recent progress on the boundary rigidity problem, Electr. Res. Announc. Amer. Math. Soc., 11(2005), 64-70. | MR 2150946 | Zbl 1113.53027

[179] Stefanov, P. and Uhlmann, G., Rigidity for metrics with the same lengths of geodesics, Math. Res. Lett., 5(1998), 83–96. | MR 1618347 | Zbl 0934.53031

[180] Stefanov P. and Uhlmann, G., Stability estimates for the X-ray transform of tensor fields and boundary rigidity, Duke Math. J., 123(2004), 445–467. | MR 2068966 | Zbl 1058.44003

[181] Stefanov, P. and Uhlmann, G., Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map, International Math. Research Notices, 17(2005), 1047–1061. | MR 2145709 | Zbl 1088.53027

[182] Stefanov, P. and Uhlmann, G, Boundary rigidity and stability for generic simple metrics, Journal Amer. Math. Soc., 18(2005), 975–1003. | MR 2163868 | Zbl 1079.53061

[183] Stefanov, P. and Uhlmann, G, Integral geometry of tensor fields on a class of non-simple Riemannian manifolds, American J. of Math., 130(2008), 239-268. | MR 2382148 | Zbl 1151.53033

[184] Stefanov, P. and Uhlmann, G., Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds, J. Differential Geometry, 82(2009), 383-409. | MR 2520797 | Zbl 1247.53049

[185] Sun, Z., On a quasilinear boundary value problem, Math. Z.,221(1996), 293–305. | MR 1376299 | Zbl 0843.35137

[186] Sun, Z., An inverse boundary value problem for Schrödinger operators with vector potentials, Trans. of AMS, 338 (1993), 953–969. | MR 1179400 | Zbl 0795.35143

[187] Sun, Z., Conjectures in inverse boundary value problems for quasilinear elliptic equations, Cubo, 7(2005), 65–73. | MR 2191048 | Zbl 1100.35123

[188] Sun, Z. and Uhlmann, G., Anisotropic inverse problems in two dimensions, Inverse Problems, 19(2003), 1001-1010. | MR 2024685 | Zbl 1054.35139

[189] Sun, Z. and Uhlmann, G., Generic uniqueness for an inverse boundary value problem, Duke Math. Journal, 62(1991), 131–155. | MR 1104326 | Zbl 0728.35132

[190] Sun, Z. amd Uhlmann, G., Inverse problems in quasilinear anisotropic media, Amer. J. Math., 119(1997), 771-797. | MR 1465069 | Zbl 0886.35176

[191] Sylvester, J., An anisotropic inverse boundary value problem, Comm. Pure Appl. Math., 43(1990), 201–232. | MR 1038142 | Zbl 0709.35102

[192] Sylvester, J. and Uhlmann, G., A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153–169. | MR 873380 | Zbl 0625.35078

[193] 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 820341 | Zbl 0611.35088

[194] Sylvester, J. and Uhlmann, G., Inverse boundary value problems at the boundary – continuous dependence, Comm. Pure Appl. Math., 41 (1988), 197–221. | MR 924684 | Zbl 0632.35074

[195] Sylvester, J. and Uhlmann, G., Inverse problems in anisotropic media, Contemp. Math., 122(1991), 105–117. | MR 1135861 | Zbl 0748.35057

[196] 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. | MR 2484149 | Zbl 1152.35519

[197] Tataru, D., Unique continuation for solutions to PDE’s; between Hörmander’s theorem and Holmgren’s theorem, Comm. P.D.E. 20(1995), 855–884. | MR 1326909 | Zbl 0846.35021

[198] Treves, F., Introduction to Pseudodifferential and Fourier Integral Operators, Vol. 1. Pseudodifferential Operators. The University Series in Mathematics, Plenum Press, New York–London, 1980. | MR 597144 | Zbl 0453.47027

[199] Tsai, T. Y., The Schrödinger equation in the plane, Inverse Problems, 9(1993), 763–787. | Zbl 0797.35140

[200] Tolmasky, C., Exponentially growing solutions for nonsmooth first-order perturbations of the Laplacian, SIAM J. Math. Anal., 29(1998), 116–133. | MR 1617178 | Zbl 0908.35028

[201] Tzou, L., Stability estimates for coefficients of magnetic Schrödinger equation from full and partial measurements, Comm. PDE, 33(2008), 161-184. | MR 2475324 | Zbl 1157.35108

[202] Uhlmann, G., Inverse boundary value problems for partial differential equations, Documenta Mathematica, Extra Volume ICM 98, Vol III(1998) 77–86. | MR 1648142 | Zbl 0906.35111

[203] Uhlmann, G., Inverse boundary value problems and applications, Astérisque, 207(1992), 153–211. | MR 1205179 | Zbl 0787.35123

[204] 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 1743870 | Zbl 0963.35203

[205] Uhlmann, G., Scattering by a metric, Chap. 6.1.5, in Encyclopedia on Scattering, Academic Pr., R. Pike and P. Sabatier, eds. (2002), 1668-1677.

[206] Uhlmann G., The Cauchy data and the scattering relation, Geometric methods in inverse problems and PDE control, 263–287, IMA Vol. Math. Appl., 137, Springer, New York, 2004. | MR 2169908 | Zbl 1061.35175

[207] Uhlmann, G. and Vasy A., Low-energy inverse problems in three-body scattering, Inverse Problems, 18(2002), 719–736. | MR 1910198 | Zbl 1002.35106

[208] 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 2299437 | Zbl 1131.35088

[209] Uhlmann, G. and Wang, J.-N., Reconstruction of discontinuities in systems, SIAM J. Appl. Math., 28(2008), 1026-1044. | MR 2390978 | Zbl 1146.35097

[210] 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 2531555 | Zbl 1173.35123

[211] Wang, J.-N., Stability for the reconstruction of a Riemannian metric by boundary measurements, Inverse Probl., 15(1999), 1177–1192. | MR 1715358 | Zbl 0969.53018

[212] E. Wiechert and K. Zoeppritz, Uber erdbebenwellen, Nachr. Koenigl. Geselschaft Wiss, Goettingen, 4(1907), 415-549.

[213] Zhdanov, M. S. Keller, G. V., The geoelectrical methods in geophysical exploration, Methods in Geochemistry and Geophysics, 31(1994), Elsevier.

[214] Zhou, T., Reconstructing electromagnetic obstacles by the enclosure method, Inverse Problems and Imaging. | MR 2671111 | Zbl 1206.35262

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