Recent progress in the conductivity reconstruction in Calderón’s problem
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In this work, we study a nonlinear inverse problem for an elliptic partial differential equation known as the Calderón problem or the inverse conductivity problem. We give a short survey on the reconstruction question of conductivity from measurements on the boundary by covering the main currently known results regarding the isotropic problem with full data in two and higher dimensions. We present Nachman’s reconstruction procedure and summarize the theoretical progress of the technique to more recent results in the field. An open problem of significant interest is proposed to check whether it is possible to extend the method for Lipschitz conductivities.
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G. Alessandrini, Singular solutions of elliptic equations and the determination of conductivity by boundary measurements, J. Differential Equations. 84(2) (1990), 252–272.
G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal. 27(1-3) (1988), 153–172.
H. Ammari and G. Uhlmann, Reconstruction of the potential from partial Cauchy data for the Schrödinger equation, Indiana Univ. Math. J. 53(1) (2004), 169–183.
M. Aoudj, On the main aspects of the inverse conductivity problem, Journal of Nonlinear Modeling and Analysis. 4(1) (2022), 1–17.
Y. M. Assylbekov, Reconstruction in the partial data Calderón problem on admissible manifolds, Inverse Probl. Imaging. 11(3) (2017), 455–476.
K. Astala, T. Iwaniec, and G. Martin, Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton Mathematical Series, Princeton, 2009.
K. Astala, M. Lassas and L. Päivärinta, Calderón’s inverse problem for anisotropic conductivity in the plane, Comm. Partial Differential Equations. 30(1-2) (2005), 207–224.
K. Astala and L. Päivärinta, Calderón’s inverse conductivity problem in the plane, Ann. of Math. 163(1) (2006), 265–299.
L. Borcea, Electrical impedance tomography, Inverse Problems. 18(6) (2002), R99–R136.
J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geom. Funct. Anal. 3(2) (1993), 107–156.
B. H. Brown, Electrical impedance tomography (eit): a review, J. Med. Eng. Technol. 27(3) (2003), 97–108.
R. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Comm. Partial Differential Equations. 22(5-6) (1997), 1009–1027.
R. Brown, Recovering the conductivity at the boundary from the Dirichlet to Neumann map: a pointwise result, J. Inverse Ill-Posed Probl. 9(6) (2001), 567–574.
R. M. Brown and R. H. Torres, Uniqueness in the inverse conductivity problem for conductivities with 3/2 derivatives in Lp, p > 2n, J. Fourier Anal. Appl. 9(6) (2003), 563–574.
A. P. Calderón, On an inverse boundary value problem, in:Seminar on Numerical Analysis and its Applications to Continuum Physics, 1980, pp. 65–73.
P. Caro and K. Rogers, Global uniqueness for the Calderón problem with Lipschitz conductivities, Forum Math. Pi. 4(e2) 28 (2016).
M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography, SIAM Review. 41(1) (1999), 85–101.
L. D. Faddeev, Growing solutions to Schrödinger’s equation, Doklady Akademii nauk SSSR. 165(3) (1965), 514–517.
J. Feldman, M. Salo and G. Uhlmann, The Calderón Problem – An Introduction to Inverse Problems, Spring-Verlag, New York, 2010.
A. García and G. Zhang, Reconstruction from boundary measurements for less regular conductivities, Inverse Problems. 32 (11) (2016), Article ID 115015, 20 pages.
H. Garde, Reconstruction of piecewise constant layered conductivities in electrical impedance tomography, Comm. Partial Differential Equations. 45(9) (2020), 1118–1133.
B. Haberman, Uniqueness in Calderón’s problem for conductivities with unbounded gradient, Comm. Math. Phys. 340(2) (2015), 639–659.
B. Haberman and D. Tataru, Uniqueness in Calderón problem with Lipschitz conductivities, Duke Math. 162(3) (2013), 497–516.
S. Ham, Y. Kwon and S. Lee, Uniqueness in the Calderón problem and bilinear restriction estimates, J. Funct. Anal. 281(8) (2021), Article ID 109119, 58 pages.
M. Hanke and M. Bröhl, Recent progress in electrical impedance tomography, Inverse Problems. 19(6) (2003), S65–S90.
C. Kenig and M. Salo, Recent progress in the Calderón problem with partial data, Inverse Problems and Applications. 165(2) (2017), 567–591.
C. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data. Ann. of Math. 165(2) (2007), 567–591.
K. Knudsen and A. Tamasan, Reconstruction of less regular conductivities in the plane, Comm. Partial Differential Equations. 29(3-4) (2005), 361–381.
R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Comm. Pure Appl. Math. 37(3 )(1984), 289–298.
M. Lassas and G. Uhlmann, On determining a Riemannian manifold from the Dirichlet-toNeumann map, Ann. Sci. Ec. Norm. Supér. 34(5) (2001), 771–787.
G. Lytle, P. Perry and S. Siltanen, Nachman’s reconstruction method for the Calderón problem with discontinuous conductivities, Inverse problems. 36(3) (2019), 035018. 15 pages. DOI:10.1088/1361-6420/ab5a12
J. L. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications, Computational Science & Engineering, SIAM, Philadelphia, PA, 2012. DOI:10.1137/1.9781611972344
A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. 143(2) (1996), 71–96.
A. I. Nachman, Reconstructions from boundary measurements, Ann. of Math. 128(3) (1988), 531–576.
A. Nachman and B. Street, Reconstruction in the Calderón problem with partial data, Comm. Partial Differential Equations. 35(2) (2010), 375–390.
H. M. Nguyen and D. Spirn, Recovering a potential from cauchy data via complex geometrical optics solutions, (2014) preprint arXiv:1403.2255
R. G. Novikov, A multidimensional inverse spectral problem for the equation −Dy + (v(x) − Eu(x))y = 0. Funct. Anal. Appl. 22(4) (1988), 263–272.
R. G. Novikov and G. M. Khenkin, The partial-equation in the multi-dimensional inverse scattering problem, Russian Math. Surveys. 42(3) (1987), 109–180.
F. Ponc-Vanegas, The bilinear strategy for Calderón’s problem, Rev. Mat. Iberoam. 37(6) (2021), 2119–2160.
S. Siltanen, J. L. Mueller and D. Isaacson, An implementation of the reconstruction algorithm of A. Nachman for the 2D inverse conductivity problem, Inverse Problems. 16(3) (2000), 681–699.
E. M. Stein and R. Shakarchi, Fourier Analysis an Introduction, Princeton University Press, Princeton, Oxford, 2003.
J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. 125(1) (1987), 153–169.
J. Sylvester and G. Uhlmann, Inverse boundary value problems at the boundary-continuous dependence, Comm. Pure Appl. Math. 41(2) (1988), 197–219.
A. Tarikere, Reconstruction of rough conductivities from boundary meuserments, (2020), preprint arXiv:2001.05155v1
D. Tataru, The Xqs spaces and unique continuation for solutions to the semilinear wave equation, Comm. Partial Differential Equations. 21(5-6) (1996), 841–887.
G. Uhlmann, Electrical impedance tomography and Calderón’s problem, Inverse Problems. 25(12) (2009), 123011, 42 pages.