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Artificial boundary conditions and domain truncation in electrical impedance tomography. Part II: Stochastic extension of the boundary map
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Artificial boundary conditions and domain truncation in electrical impedance tomography. Part I: Theory and preliminary results
1. | Case Western Reserve University, Department of Mathematics, Applied Mathematics, and Statistics, Cleveland, OH 44106, United States, United States, United States |
2. | University of Auckland, Department of Mathematics, Auckland, New Zealand, New Zealand |
3. | University of Eastern Finland, Department of Applied Physics, Kuopio, Finland |
4. | Rensselaer Polytechnic Institute, Department of Mathematics, Troy, NY 12180, United States |
5. | Case Western Reserve University, Department of Radiology, Cleveland, OH 44106, United States |
References:
[1] |
D. Calvetti, J. P. Kaipio and E. Somersalo, Aristotelian prior boundary conditions, Int. J. Math. Comp. Sci., 1 (2006), 63-81. |
[2] |
D. Calvetti and E. Somersalo, Statistical compensation of boundary clutter in image deblurring, Inverse Problems, 21 (2005), 1697-1714.
doi: 10.1088/0266-5611/21/5/012. |
[3] |
D. Calvetti and E. Somersalo, Introduction to Bayesian Scientific Computing - Ten Lectures on Subjective Computing, Springer Verlag, New York, 2007. |
[4] |
D. Calvetti, P. J, Hadwin, J. M. J. Huttunen, J. P. Kaipio and E. Somersalo, Artificial boundary conditions and domain truncation in electrical impedance tomography. Part II: Computational results, Inv. Probl. Imaging., 12 (2015). |
[5] |
D. Givoli, Numerical Methods for Problems in Infinite Domains, Elsevier, Amsterdam, 1992. |
[6] |
D. Givoli, Recent advances in the DtN FE method, Arch. Comput. Meth. Engin., 6 (1999), 71-116.
doi: 10.1007/BF02736182. |
[7] |
P. Grisvard, Elliptic Boundary Value Problems in Non-Smooth Domains, SIAM, Philadelphia, 2011; Original publication by Pitman Publishing Inc., Marschfied, Mass., 1985. |
[8] |
M. J. Grote and J. B. Keller, On nonreflecting boundary conditions, J. Comp. Phys., 122 (1995), 231-243.
doi: 10.1006/jcph.1995.1210. |
[9] |
M. J. Grote and C. Kirsch, Dirichlet-to-Neumann boundary conditions for multiple scattering problems, J. Comp. Phys., 201 (2004), 630-650.
doi: 10.1016/j.jcp.2004.06.012. |
[10] |
E. Jonsson, Partial Dirichlet to Neumann Maps in the Approximate Reconstruction of Conductivity Distribution, PhD Thesis, Rensselaer Polytechnic Institute, Troy, NY, 1997. |
[11] |
E. Jonsson, Electrical conductivity reconstruction using nonlocal boundary conditions, SIAM J. Appl. Math., 59 (1999), 1582-1598.
doi: 10.1137/S0036139997327770. |
[12] |
T.-J. Kao, G. J. Saulnier, H. Xia, C. Tamma, J. C. Newell and D. Isaacson, A compensated radiolucent electrode array for combined EIT and mammography, Physiol. Meas., 28 (2007), S291-S299.
doi: 10.1088/0967-3334/28/7/S22. |
[13] |
B. S. Kim, G. Boverman, J. C. Newell, G. J. Saulnier and D. Isaacson, The complete electrode model for EIT in a mammography geometry, Physiol. Meas., 28 (2007), S57-S69.
doi: 10.1088/0967-3334/28/7/S05. |
[14] |
J. B. Keller and D. Givoli, Exact non-reflecting boundary conditions, J. Comp. Phys., 82 (1989), 172-192.
doi: 10.1016/0021-9991(89)90041-7. |
[15] |
M. H. Loke and R. D. Barker, Practical techniques for 3D resistivity surveys and data inversion, Geophys. Prospecting, 44 (1996), 499-523. |
[16] |
D. Calvetti, D. McGivney and E. Somersalo, Left and right preconditioning for electrical impedance tomography with structural information, Inverse Problems, 28 (2012), 055015, 26pp.
doi: 10.1088/0266-5611/28/5/055015. |
[17] |
D. McGivney, D. Calvetti and E. Somersalo, Quantitative imaging with electrical impedance spectroscopy, Phys. Med. Biol., 57 (2012), p7289.
doi: 10.1088/0031-9155/57/22/7289. |
[18] |
D. McGivney, Statistical Preconditioners and Quantitative Imaging in Electrical Impedance Tomography, PhD Thesis, Case Western Reserve University, Cleveland, 2013. |
[19] |
A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford University Press, Oxford, 1999. |
[20] |
E. Somersalo, D. Isaacson and M. Cheney, Existence and uniqueness for electrode models for electric current computed tomography, SIAM J. Appl. Math. 52 (1992), 1023-1040.
doi: 10.1137/0152060. |
[21] |
H. Triebel, Interpolation Theory, Function Spaces Differential Operators, 2nd ed. Barth, Heidelberg-Leipzig, 1995. |
[22] |
Y. Zou and Z. Guo, A review of electrical impedance techniques for breast cancer detection, Med. Eng. Phys., 25 (2003), 79-90.
doi: 10.1016/S1350-4533(02)00194-7. |
show all references
References:
[1] |
D. Calvetti, J. P. Kaipio and E. Somersalo, Aristotelian prior boundary conditions, Int. J. Math. Comp. Sci., 1 (2006), 63-81. |
[2] |
D. Calvetti and E. Somersalo, Statistical compensation of boundary clutter in image deblurring, Inverse Problems, 21 (2005), 1697-1714.
doi: 10.1088/0266-5611/21/5/012. |
[3] |
D. Calvetti and E. Somersalo, Introduction to Bayesian Scientific Computing - Ten Lectures on Subjective Computing, Springer Verlag, New York, 2007. |
[4] |
D. Calvetti, P. J, Hadwin, J. M. J. Huttunen, J. P. Kaipio and E. Somersalo, Artificial boundary conditions and domain truncation in electrical impedance tomography. Part II: Computational results, Inv. Probl. Imaging., 12 (2015). |
[5] |
D. Givoli, Numerical Methods for Problems in Infinite Domains, Elsevier, Amsterdam, 1992. |
[6] |
D. Givoli, Recent advances in the DtN FE method, Arch. Comput. Meth. Engin., 6 (1999), 71-116.
doi: 10.1007/BF02736182. |
[7] |
P. Grisvard, Elliptic Boundary Value Problems in Non-Smooth Domains, SIAM, Philadelphia, 2011; Original publication by Pitman Publishing Inc., Marschfied, Mass., 1985. |
[8] |
M. J. Grote and J. B. Keller, On nonreflecting boundary conditions, J. Comp. Phys., 122 (1995), 231-243.
doi: 10.1006/jcph.1995.1210. |
[9] |
M. J. Grote and C. Kirsch, Dirichlet-to-Neumann boundary conditions for multiple scattering problems, J. Comp. Phys., 201 (2004), 630-650.
doi: 10.1016/j.jcp.2004.06.012. |
[10] |
E. Jonsson, Partial Dirichlet to Neumann Maps in the Approximate Reconstruction of Conductivity Distribution, PhD Thesis, Rensselaer Polytechnic Institute, Troy, NY, 1997. |
[11] |
E. Jonsson, Electrical conductivity reconstruction using nonlocal boundary conditions, SIAM J. Appl. Math., 59 (1999), 1582-1598.
doi: 10.1137/S0036139997327770. |
[12] |
T.-J. Kao, G. J. Saulnier, H. Xia, C. Tamma, J. C. Newell and D. Isaacson, A compensated radiolucent electrode array for combined EIT and mammography, Physiol. Meas., 28 (2007), S291-S299.
doi: 10.1088/0967-3334/28/7/S22. |
[13] |
B. S. Kim, G. Boverman, J. C. Newell, G. J. Saulnier and D. Isaacson, The complete electrode model for EIT in a mammography geometry, Physiol. Meas., 28 (2007), S57-S69.
doi: 10.1088/0967-3334/28/7/S05. |
[14] |
J. B. Keller and D. Givoli, Exact non-reflecting boundary conditions, J. Comp. Phys., 82 (1989), 172-192.
doi: 10.1016/0021-9991(89)90041-7. |
[15] |
M. H. Loke and R. D. Barker, Practical techniques for 3D resistivity surveys and data inversion, Geophys. Prospecting, 44 (1996), 499-523. |
[16] |
D. Calvetti, D. McGivney and E. Somersalo, Left and right preconditioning for electrical impedance tomography with structural information, Inverse Problems, 28 (2012), 055015, 26pp.
doi: 10.1088/0266-5611/28/5/055015. |
[17] |
D. McGivney, D. Calvetti and E. Somersalo, Quantitative imaging with electrical impedance spectroscopy, Phys. Med. Biol., 57 (2012), p7289.
doi: 10.1088/0031-9155/57/22/7289. |
[18] |
D. McGivney, Statistical Preconditioners and Quantitative Imaging in Electrical Impedance Tomography, PhD Thesis, Case Western Reserve University, Cleveland, 2013. |
[19] |
A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford University Press, Oxford, 1999. |
[20] |
E. Somersalo, D. Isaacson and M. Cheney, Existence and uniqueness for electrode models for electric current computed tomography, SIAM J. Appl. Math. 52 (1992), 1023-1040.
doi: 10.1137/0152060. |
[21] |
H. Triebel, Interpolation Theory, Function Spaces Differential Operators, 2nd ed. Barth, Heidelberg-Leipzig, 1995. |
[22] |
Y. Zou and Z. Guo, A review of electrical impedance techniques for breast cancer detection, Med. Eng. Phys., 25 (2003), 79-90.
doi: 10.1016/S1350-4533(02)00194-7. |
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