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1. | University of Genoa, Department of Mathematics, Via Dodecaneso 35, 16146 Genoa, Italy |
2. | Danmarks Tekniske Universitet, Department of Applied Mathematics and Computer Science, Matematiktorvet, Building 303 B, DK - 2800 Kgs. Lyngby |
References:
[1] |
A. Abubakar, T. M. Habashy, M. Li and J. Liu, Inversion algorithms for large-scale geophysical electromagnetic, Inverse Problems, 25 (2009), 123012.
doi: 10.1088/0266-5611/25/12/123012. |
[2] |
G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172.
doi: 10.1080/00036818808839730. |
[3] |
G. Alessandrini, Singular solutions of elliptic equations and the determination of conductivity by boundary measurements, J. Differential Equations, 84 (1990), 252-272.
doi: 10.1016/0022-0396(90)90078-4. |
[4] |
K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Ann. of Math. (2), 163 (2006), 265-299.
doi: 10.4007/annals.2006.163.265. |
[5] |
E. Anderson, Z. Bai, C. Bischof, L. S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney and D. Sorensen, LAPACK Users' Guide, 3rd edition, Society for Industrial and Applied Mathematics, Philadelphia, 1999.
doi: 10.1137/1.9780898719604. |
[6] |
J. Bikowski, K. Knudsen and J. L. Mueller, Direct numerical reconstruction of conductivities in three dimensions using scattering transforms, Inverse Problems, 27 (2011), 015002, 19pp.
doi: 10.1088/0266-5611/27/1/015002. |
[7] |
G. Boverman, T.-J. Kao, D. Isaacson and G. J. Saulnier, An implementation of Calderón's Method for 3-D Limited-View EIT, IEEE Trans. Med. Imaging, 28 (2009), 1073-1082.
doi: 10.1109/tmi.2009.2012892. |
[8] |
R. M. Brown and G. A. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Comm. Partial Differential Equations, 22 (1997), 1009-1027.
doi: 10.1080/03605309708821292. |
[9] |
Alberto-P. Calderón, On an inverse boundary value problem, In Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), 65-73. Soc. Brasil. Mat., Rio de Janeiro, 1980. |
[10] |
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences, 3rd edition, Springer, New York, 2013.
doi: 10.1007/978-1-4614-4942-3. |
[11] |
H. Cornean, K. Knudsen and S. Siltanen, Towards a $d$-bar reconstruction method for three-dimensional EIT, J. Inverse Ill-Posed Probl., 14 (2006), 111-134.
doi: 10.1515/156939406777571102. |
[12] |
T. A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method, ACM Transactions on Mathematical Software, 30 (2004), 165-195.
doi: 10.1145/992200.992205. |
[13] |
T. A. Davis and I. S. Duff, An unsymmetric-pattern multifrontal method for sparse LU factorization, SIAM Journal on Matrix Analysis and Applications, 18 (1997), 140-158.
doi: 10.1137/S0895479894246905. |
[14] |
T. A. Davis and I. S. Duff, A combined unifrontal/multifrontal method for unsymmetric sparse matrices, ACM Transactions on Mathematical Software, 25 (1999), 1-20.
doi: 10.1145/305658.287640. |
[15] |
T. A. Davis and I. S. Duff, Algorithm 832: Umfpack v4.3-an unsymmetric-pattern multifrontal method, ACM Transactions on Mathematical Software, 30 (2004), 196-199.
doi: 10.1145/992200.992206. |
[16] |
F. Delbary, P. C. Hansen and K. Knudsen, Electrical impedance tomography: 3D reconstructions using scattering transforms, Appl. Anal., 91 (2012), 737-755.
doi: 10.1080/00036811.2011.598863. |
[17] |
F. Delbary and K. Knudsen, Regularized electrical impedance tomography using truncated scattering transforms,, In preparation., ().
|
[18] |
F. Delbary and R. Kress, Electrical impedance tomography with point electrodes, J. Integral Equations Appl., 22 (2010), 193-216.
doi: 10.1216/JIE-2010-22-2-193. |
[19] |
F. Delbary and R. Kress, Electrical impedance tomography using a point electrode inverse scheme for complete electrode data, Inverse Probl. Imaging, 5 (2011), 355-369.
doi: 10.3934/ipi.2011.5.355. |
[20] |
M. Frigo and S. G. Johnson, The design and implementation of FFTW3, Proceedings of the IEEE, 93 (2005), 1-19.
doi: 10.1109/JPROC.2004.840301. |
[21] |
M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, P. Alken, M. Booth and F. Rossi, GNU Scientific Library Reference Manual, 3rd edition, Network Theory Ltd, Bristol, 2009. |
[22] |
M. Ganesh and I. G. Graham, A high-order algorithm for obstacle scattering in three dimensions, J. Comput. Phys., 198 (2004), 211-242.
doi: 10.1016/j.jcp.2004.01.007. |
[23] |
C. Geuzaine and R. J.-F., Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities, International Journal for Numerical Methods in Engineering, 79 (2009), 1309-1331.
doi: 10.1002/nme.2579. |
[24] |
K. Hesse and I. H. Sloan, Cubature over the sphere $S^2$ in Sobolev spaces of arbitrary order, J. Approx. Theory, 141 (2006), 118-133.
doi: 10.1016/j.jat.2006.01.004. |
[25] |
K. Hesse and I. H. Sloan, Hyperinterpolation on the sphere, in Frontiers in interpolation and approximation, vol. 282 of Pure Appl. Math. (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, (2007), 213-248. |
[26] |
D. S. Holder, Electrical Impedance Tomography: Methods, History and Applications, Series in Medical Physics and Biomedical Engineering, CRC Press, Boca Raton, 2004.
doi: 10.1201/9781420034462. |
[27] |
O. Ivanyshyn and R. Kress, Inverse scattering for surface impedance from phase-less far field data, J. Comput. Phys., 230 (2011), 3443-3452.
doi: 10.1016/j.jcp.2011.01.038. |
[28] |
K. Knudsen, A new direct method for reconstructing isotropic conductivities in the plane, Physiological Measurement, 24 (2003), 391-401.
doi: 10.1088/0967-3334/24/2/351. |
[29] |
K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem, Inverse Probl. Imaging, 3 (2009), 599-624.
doi: 10.3934/ipi.2009.3.599. |
[30] |
K. Knudsen and J. L. Mueller, The Born approximation and Calderón's method for reconstruction of conductivities in 3-D, Discrete Contin. Dyn. Syst., II (2011), 844-853. |
[31] |
K. Knudsen and A. Tamasan, Reconstruction of less regular conductivities in the plane, Comm. Partial Differential Equations, 29 (2004), 361-381.
doi: 10.1081/pde-120030401. |
[32] |
R. Kress, Linear Integral Equations, vol. 82 of Applied Mathematical Sciences, 3rd edition, Springer, New York, 2014.
doi: 10.1007/978-1-4614-9593-2. |
[33] |
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. |
[34] |
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. |
[35] |
J. L. Mueller and S. Siltanen, Direct reconstructions of conductivities from boundary measurements, SIAM J. Sci. Comput., 24 (2003), 1232-1266.
doi: 10.1137/S1064827501394568. |
[36] |
J. L. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications, Computational Science & Engineering, SIAM, Philadelphia, 2012.
doi: 10.1137/1.9781611972344. |
[37] |
A. Nachman, J. Sylvester and G. Uhlmann, An n-dimensional Borg-Levinson theorem, Comm. Math. Phys., 115 (1988), 595-605, http://projecteuclid.org/euclid.cmp/1104161086.
doi: 10.1007/BF01224129. |
[38] |
A. I. Nachman, Reconstructions from boundary measurements, Ann. of Math. (2), 128 (1988), 531-576.
doi: 10.2307/1971435. |
[39] |
A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2), 143 (1996), 71-96.
doi: 10.2307/2118653. |
[40] |
R. G. Newton, Inverse Schrödinger Scattering in Three Dimensions, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1989.
doi: 10.1007/978-3-642-83671-8. |
[41] |
R. G. Novikov, A multidimensional inverse spectral problem for the equation $-\Delta\psi +(v(x)-Eu(x))\psi=0$, Funktsional. Anal. i Prilozhen., 22 (1988), 11-22.
doi: 10.1007/bf01077418. |
[42] |
S. Siltanen, J. Mueller and D. Isaacson, Erratum: "An implementation of the reconstruction algorithm of A. Nachman for the 2D inverse conductivity problem [Inverse Problems 16 (2000), 681-699; MR:1766226 (2001g:35269)], Inverse Problems, 17 (2001), 1561-1563.
doi: 10.1088/0266-5611/17/5/501. |
[43] |
S. Siltanen, J. Mueller and D. Isaacson, An implementation of the reconstruction algorithm of A. Nachman for the 2D inverse conductivity problem, Inverse Problems, 16 (2000), 681-699.
doi: 10.1088/0266-5611/16/3/310. |
[44] |
O. Steinbach and W. L. Wendland, On C. Neumann's method for second-order elliptic systems in domains with non-smooth boundaries, J. Math. Anal. Appl., 262 (2001), 733-748.
doi: 10.1006/jmaa.2001.7615. |
[45] |
J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2), 125 (1987), 153-169.
doi: 10.2307/1971291. |
[46] |
G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 123011, 39pp.
doi: 10.1088/0266-5611/25/12/123011. |
[47] |
L. Wienert, Die Numerische Approximation von Randintegraloperatoren Für die Helmholtzgleichung im $\mathbbR^3$, 1990,, Thesis (Ph.D.)-Georg-August-Universität Göttingen., ().
|
[48] |
T. A. York, \doititle{Status of electrical tomography in industrial applications}, J. Electron. Imaging., 10 (2001), 608-619.
doi: 10.1117/1.1377308. |
show all references
References:
[1] |
A. Abubakar, T. M. Habashy, M. Li and J. Liu, Inversion algorithms for large-scale geophysical electromagnetic, Inverse Problems, 25 (2009), 123012.
doi: 10.1088/0266-5611/25/12/123012. |
[2] |
G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172.
doi: 10.1080/00036818808839730. |
[3] |
G. Alessandrini, Singular solutions of elliptic equations and the determination of conductivity by boundary measurements, J. Differential Equations, 84 (1990), 252-272.
doi: 10.1016/0022-0396(90)90078-4. |
[4] |
K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Ann. of Math. (2), 163 (2006), 265-299.
doi: 10.4007/annals.2006.163.265. |
[5] |
E. Anderson, Z. Bai, C. Bischof, L. S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney and D. Sorensen, LAPACK Users' Guide, 3rd edition, Society for Industrial and Applied Mathematics, Philadelphia, 1999.
doi: 10.1137/1.9780898719604. |
[6] |
J. Bikowski, K. Knudsen and J. L. Mueller, Direct numerical reconstruction of conductivities in three dimensions using scattering transforms, Inverse Problems, 27 (2011), 015002, 19pp.
doi: 10.1088/0266-5611/27/1/015002. |
[7] |
G. Boverman, T.-J. Kao, D. Isaacson and G. J. Saulnier, An implementation of Calderón's Method for 3-D Limited-View EIT, IEEE Trans. Med. Imaging, 28 (2009), 1073-1082.
doi: 10.1109/tmi.2009.2012892. |
[8] |
R. M. Brown and G. A. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Comm. Partial Differential Equations, 22 (1997), 1009-1027.
doi: 10.1080/03605309708821292. |
[9] |
Alberto-P. Calderón, On an inverse boundary value problem, In Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), 65-73. Soc. Brasil. Mat., Rio de Janeiro, 1980. |
[10] |
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences, 3rd edition, Springer, New York, 2013.
doi: 10.1007/978-1-4614-4942-3. |
[11] |
H. Cornean, K. Knudsen and S. Siltanen, Towards a $d$-bar reconstruction method for three-dimensional EIT, J. Inverse Ill-Posed Probl., 14 (2006), 111-134.
doi: 10.1515/156939406777571102. |
[12] |
T. A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method, ACM Transactions on Mathematical Software, 30 (2004), 165-195.
doi: 10.1145/992200.992205. |
[13] |
T. A. Davis and I. S. Duff, An unsymmetric-pattern multifrontal method for sparse LU factorization, SIAM Journal on Matrix Analysis and Applications, 18 (1997), 140-158.
doi: 10.1137/S0895479894246905. |
[14] |
T. A. Davis and I. S. Duff, A combined unifrontal/multifrontal method for unsymmetric sparse matrices, ACM Transactions on Mathematical Software, 25 (1999), 1-20.
doi: 10.1145/305658.287640. |
[15] |
T. A. Davis and I. S. Duff, Algorithm 832: Umfpack v4.3-an unsymmetric-pattern multifrontal method, ACM Transactions on Mathematical Software, 30 (2004), 196-199.
doi: 10.1145/992200.992206. |
[16] |
F. Delbary, P. C. Hansen and K. Knudsen, Electrical impedance tomography: 3D reconstructions using scattering transforms, Appl. Anal., 91 (2012), 737-755.
doi: 10.1080/00036811.2011.598863. |
[17] |
F. Delbary and K. Knudsen, Regularized electrical impedance tomography using truncated scattering transforms,, In preparation., ().
|
[18] |
F. Delbary and R. Kress, Electrical impedance tomography with point electrodes, J. Integral Equations Appl., 22 (2010), 193-216.
doi: 10.1216/JIE-2010-22-2-193. |
[19] |
F. Delbary and R. Kress, Electrical impedance tomography using a point electrode inverse scheme for complete electrode data, Inverse Probl. Imaging, 5 (2011), 355-369.
doi: 10.3934/ipi.2011.5.355. |
[20] |
M. Frigo and S. G. Johnson, The design and implementation of FFTW3, Proceedings of the IEEE, 93 (2005), 1-19.
doi: 10.1109/JPROC.2004.840301. |
[21] |
M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, P. Alken, M. Booth and F. Rossi, GNU Scientific Library Reference Manual, 3rd edition, Network Theory Ltd, Bristol, 2009. |
[22] |
M. Ganesh and I. G. Graham, A high-order algorithm for obstacle scattering in three dimensions, J. Comput. Phys., 198 (2004), 211-242.
doi: 10.1016/j.jcp.2004.01.007. |
[23] |
C. Geuzaine and R. J.-F., Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities, International Journal for Numerical Methods in Engineering, 79 (2009), 1309-1331.
doi: 10.1002/nme.2579. |
[24] |
K. Hesse and I. H. Sloan, Cubature over the sphere $S^2$ in Sobolev spaces of arbitrary order, J. Approx. Theory, 141 (2006), 118-133.
doi: 10.1016/j.jat.2006.01.004. |
[25] |
K. Hesse and I. H. Sloan, Hyperinterpolation on the sphere, in Frontiers in interpolation and approximation, vol. 282 of Pure Appl. Math. (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, (2007), 213-248. |
[26] |
D. S. Holder, Electrical Impedance Tomography: Methods, History and Applications, Series in Medical Physics and Biomedical Engineering, CRC Press, Boca Raton, 2004.
doi: 10.1201/9781420034462. |
[27] |
O. Ivanyshyn and R. Kress, Inverse scattering for surface impedance from phase-less far field data, J. Comput. Phys., 230 (2011), 3443-3452.
doi: 10.1016/j.jcp.2011.01.038. |
[28] |
K. Knudsen, A new direct method for reconstructing isotropic conductivities in the plane, Physiological Measurement, 24 (2003), 391-401.
doi: 10.1088/0967-3334/24/2/351. |
[29] |
K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem, Inverse Probl. Imaging, 3 (2009), 599-624.
doi: 10.3934/ipi.2009.3.599. |
[30] |
K. Knudsen and J. L. Mueller, The Born approximation and Calderón's method for reconstruction of conductivities in 3-D, Discrete Contin. Dyn. Syst., II (2011), 844-853. |
[31] |
K. Knudsen and A. Tamasan, Reconstruction of less regular conductivities in the plane, Comm. Partial Differential Equations, 29 (2004), 361-381.
doi: 10.1081/pde-120030401. |
[32] |
R. Kress, Linear Integral Equations, vol. 82 of Applied Mathematical Sciences, 3rd edition, Springer, New York, 2014.
doi: 10.1007/978-1-4614-9593-2. |
[33] |
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. |
[34] |
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. |
[35] |
J. L. Mueller and S. Siltanen, Direct reconstructions of conductivities from boundary measurements, SIAM J. Sci. Comput., 24 (2003), 1232-1266.
doi: 10.1137/S1064827501394568. |
[36] |
J. L. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications, Computational Science & Engineering, SIAM, Philadelphia, 2012.
doi: 10.1137/1.9781611972344. |
[37] |
A. Nachman, J. Sylvester and G. Uhlmann, An n-dimensional Borg-Levinson theorem, Comm. Math. Phys., 115 (1988), 595-605, http://projecteuclid.org/euclid.cmp/1104161086.
doi: 10.1007/BF01224129. |
[38] |
A. I. Nachman, Reconstructions from boundary measurements, Ann. of Math. (2), 128 (1988), 531-576.
doi: 10.2307/1971435. |
[39] |
A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2), 143 (1996), 71-96.
doi: 10.2307/2118653. |
[40] |
R. G. Newton, Inverse Schrödinger Scattering in Three Dimensions, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1989.
doi: 10.1007/978-3-642-83671-8. |
[41] |
R. G. Novikov, A multidimensional inverse spectral problem for the equation $-\Delta\psi +(v(x)-Eu(x))\psi=0$, Funktsional. Anal. i Prilozhen., 22 (1988), 11-22.
doi: 10.1007/bf01077418. |
[42] |
S. Siltanen, J. Mueller and D. Isaacson, Erratum: "An implementation of the reconstruction algorithm of A. Nachman for the 2D inverse conductivity problem [Inverse Problems 16 (2000), 681-699; MR:1766226 (2001g:35269)], Inverse Problems, 17 (2001), 1561-1563.
doi: 10.1088/0266-5611/17/5/501. |
[43] |
S. Siltanen, J. Mueller and D. Isaacson, An implementation of the reconstruction algorithm of A. Nachman for the 2D inverse conductivity problem, Inverse Problems, 16 (2000), 681-699.
doi: 10.1088/0266-5611/16/3/310. |
[44] |
O. Steinbach and W. L. Wendland, On C. Neumann's method for second-order elliptic systems in domains with non-smooth boundaries, J. Math. Anal. Appl., 262 (2001), 733-748.
doi: 10.1006/jmaa.2001.7615. |
[45] |
J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2), 125 (1987), 153-169.
doi: 10.2307/1971291. |
[46] |
G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 123011, 39pp.
doi: 10.1088/0266-5611/25/12/123011. |
[47] |
L. Wienert, Die Numerische Approximation von Randintegraloperatoren Für die Helmholtzgleichung im $\mathbbR^3$, 1990,, Thesis (Ph.D.)-Georg-August-Universität Göttingen., ().
|
[48] |
T. A. York, \doititle{Status of electrical tomography in industrial applications}, J. Electron. Imaging., 10 (2001), 608-619.
doi: 10.1117/1.1377308. |
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