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The Bayesian formulation of EIT: Analysis and algorithms
1. | Computing & Mathematical Sciences, California Institute of Technology, Pasadena, CA 91125, United States, United States |
References:
[1] |
A. Adler and W. R. B. Lionheart, Uses and abuses of EIDORS: An extensible software base for EIT, Physiological Measurement, 27 (2006), S25-S42.
doi: 10.1088/0967-3334/27/5/S03. |
[2] |
G. Alessandrini, Stable determination of an inclusion by boundary measurements, Applicable Analysis: An International Journal, 27 (1988), 153-172.
doi: 10.1080/00036818808839730. |
[3] |
M. Bédard, Optimal acceptance rates for Metropolis algorithms: Moving beyond 0.234, Stochastic Processes and their Applications, 118 (2008), 2198-2222.
doi: 10.1016/j.spa.2007.12.005. |
[4] |
A. Beskos, A. Jasra, E. A. Muzaffer and A. M. Stuart, Sequential Monte Carlo methods for Bayesian elliptic inverse problems, Statistics and Computing, 25 (2015), 727-737.
doi: 10.1007/s11222-015-9556-7. |
[5] |
A. Beskos, G. O. Roberts, A. M. Stuart and J. Voss, MCMC methods for diffusion bridges, Stochastics and Dynamics, 8 (2008), 319-350.
doi: 10.1142/S0219493708002378. |
[6] |
V. I. Bogachev, Measure Theory Volume I, Springer, 2007.
doi: 10.1007/978-3-540-34514-5. |
[7] |
L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), R99-R136.
doi: 10.1088/0266-5611/18/6/201. |
[8] |
T. Bui-Thanh and O. Ghattas, An analysis of infinite dimensional bayesian inverse shape acoustic scattering and its numerical approximation, SIAM/ASA Journal on Uncertainty Quantification, 2 (2014), 203-222.
doi: 10.1137/120894877. |
[9] |
S. L. Cotter, G. O. Roberts, A. M. Stuart and D. White, MCMC methods for functions modifying old algorithms to make them faster, Statistical Science, 28 (2013), 424-446.
doi: 10.1214/13-STS421. |
[10] |
M. Dashti and A. M. Stuart, The Bayesian Approach to Inverse Problems, In Handbook of Uncertainty Quantification, Springer, 2016, 1-118.
doi: 10.1007/978-3-319-11259-6_7-1. |
[11] |
S. Duane, A. D. Kennedy, B. J. Pendleton and D. Roweth, Hybrid monte carlo, Physics Letters B, 195 (1987), 216-222.
doi: 10.1016/0370-2693(87)91197-X. |
[12] |
J. N. Franklin, Well posed stochastic extensions of ill posed linear problems}, Journal of Mathematical Analysis and Applications, 31 (1970), 682-716, URL http://adsabs.harvard.edu/abs/1970JIMIA..31..682F.
doi: 10.1016/0022-247X(70)90017-X. |
[13] |
M. Gehre, B. Jin and X. Lu, An analysis of finite element approximation in electrical impedance tomography, Inverse Problems, 30 (2014), 045013, 24pp, URL http://stacks.iop.org/0266-5611/30/i=4/a=045013?key=crossref.284cda5f9645598cb83fccc9d226940a.
doi: 10.1088/0266-5611/30/4/045013. |
[14] |
M. Hairer, A. M. Stuart and S. J. Vollmer, Spectral gaps for a Metropolis-Hastings algorithm in infinite dimensions, The Annals of Applied Probability, 24 (2014), 2455-2490.
doi: 10.1214/13-AAP982. |
[15] |
R. P. Henderson and J. G. Webster, An impedance camera for spatially specific measurements of the thorax, IEEE Transactions on Bio-Medical Engineering, 25 (1978), 250-254.
doi: 10.1109/TBME.1978.326329. |
[16] |
M. A. Iglesias, K. J. H. Law and A. M. Stuart, Ensemble Kalman methods for inverse problems, Inverse Problems, 29 (2013), 045001, 20pp, URL http://stacks.iop.org/0266-5611/29/i=4/a=045001?key=crossref.a70b58896eed5588086da7ad3d04954b.
doi: 10.1088/0266-5611/29/4/045001. |
[17] |
M. A. Iglesias, Y. Lu and A. M. Stuart, A bayesian level set method for geometric inverse problems, Interfaces Free Bound., 18 (2016), 181-217, URL http://arxiv.org/abs/1504.00313.
doi: 10.4171/IFB/362. |
[18] |
J. P. Kaipio, V. Kolehmainen, E. Somersalo and M. Vauhkonen, Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography, Inverse Problems, 16 (2000), 1487-1522.
doi: 10.1088/0266-5611/16/5/321. |
[19] |
J. P. Kaipio, V. Kolehmainen, M. Vauhkonen and E. Somersalo, Inverse problems with structural prior information, Inverse Problems, 15 (1999), 713-729.
doi: 10.1088/0266-5611/15/3/306. |
[20] |
J. P. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Springer, 2005. |
[21] |
R. E. Kass, Markov Chain Monte Carlo in practice: A roundtable discussion, The American Statistician, 52 (1998), 93-100.
doi: 10.2307/2685466. |
[22] |
C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data on manifolds and applications, Analysis and PDE, 6 (2013), 2003-2048.
doi: 10.2140/apde.2013.6.2003. |
[23] |
K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, D-bar method for electrical impedance tomography with discontinuous conductivities, SIAM Journal on Applied Mathematics, 67 (2007), 893-913.
doi: 10.1137/060656930. |
[24] |
K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, Reconstructions of piecewise constant conductivities by the D-bar method for electrical impedance tomography, Journal of Physics: Conference Series, 124 (2008), 012029.
doi: 10.1088/1742-6596/124/1/012029. |
[25] |
K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem, Inverse Problems and Imaging, 3 (2009), 599-624.
doi: 10.3934/ipi.2009.3.599. |
[26] |
V. Kolehmainen, M. Lassas, P. Ola and S. Siltanen, Recovering boundary shape and conductivity in electrical impedance tomography, Inverse Problems and Imaging, 7 (2013), 217-242.
doi: 10.3934/ipi.2013.7.217. |
[27] |
S. Lan, T. Bui-Thanh, M. Christie and M. Girolami, Emulation of Higher-Order Tensors in Manifold Monte Carlo Methods for Bayesian Inverse Problems, Journal of Computational Physics, 308 (2016), 81-101.
doi: 10.1016/j.jcp.2015.12.032. |
[28] |
R. E. Langer, An inverse problem in differential equations, Bulletin of the American Mathematical Society, 39 (1933), 814-820.
doi: 10.1090/S0002-9904-1933-05752-X. |
[29] |
S. Lasanen, Non-gaussian statistical inverse problems. part I: Posterior distributions, Inverse Problems & Imaging, 6 (2012), 215-266.
doi: 10.3934/ipi.2012.6.215. |
[30] |
S. Lasanen, Non-gaussian statistical inverse problems. part II: Posterior convergence for approximated unknowns, Inverse Problems & Imaging, 6 (2012), 267-287.
doi: 10.3934/ipi.2012.6.267. |
[31] |
S. Lasanen, J. M. J. Huttunen and L. Roininen, Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography, Inverse Problems and Imaging, 8 (2014), 561-586, URL http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=9912.
doi: 10.3934/ipi.2014.8.561. |
[32] |
M. Lassas, E. Saksman and S. Siltanen, Discretization-invariant Bayesian inversion and Besov space priors, Inverse Problems and Imaging, 3 (2009), 87-122.
doi: 10.3934/ipi.2009.3.87. |
[33] |
M. S. Lehtinen, L. Paivarinta and E. Somersalo, Linear inverse problems for generalised random variables, Inverse Problems, 5 (1989), 599-612.
doi: 10.1088/0266-5611/5/4/011. |
[34] |
A. Mandelbaum, Linear estimators and measurable linear transformations on a Hilbert space, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 65 (1984), 385-397.
doi: 10.1007/BF00533743. |
[35] |
A. I. Nachman, Reconstructions from boundary measurements, Annals of Mathematics, 128 (1988), 531-576, URL http://cat.inist.fr/?aModele=afficheN&cpsidt=7131706.
doi: 10.2307/1971435. |
[36] |
A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Annals of Mathematics, 143 (1996), 71-96.
doi: 10.2307/2118653. |
[37] |
G. C. Papanicolaou and L. Borcea, Network approximation for transport properties of high contrast materials, SIAM Journal on Applied Mathematics, 58 (1998), 501-539.
doi: 10.1137/S0036139996301891. |
[38] | |
[39] |
D. Schymura, An upper bound on the volume of the symmetric difference of a body and a congruent copy, Advances in Geometry, 14 (2014), 287-298.
doi: 10.1515/advgeom-2013-0029. |
[40] |
E. Somersalo, M. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography, SIAM Journal on Applied Mathematics, 52 (1992), 1023-1040.
doi: 10.1137/0152060. |
[41] |
A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, 19 (2010), 451-559, URL http://webcat.warwick.ac.uk:80/record=b2341209.
doi: 10.1017/S0962492910000061. |
[42] |
J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value, Annals of Mathematics, 125 (1987), 153-169.
doi: 10.2307/1971291. |
[43] |
L. Tierney, Markov chains for exploring posterior distributions, Annals of Statistics, 22 (1994), 1701-1728.
doi: 10.1214/aos/1176325750. |
show all references
References:
[1] |
A. Adler and W. R. B. Lionheart, Uses and abuses of EIDORS: An extensible software base for EIT, Physiological Measurement, 27 (2006), S25-S42.
doi: 10.1088/0967-3334/27/5/S03. |
[2] |
G. Alessandrini, Stable determination of an inclusion by boundary measurements, Applicable Analysis: An International Journal, 27 (1988), 153-172.
doi: 10.1080/00036818808839730. |
[3] |
M. Bédard, Optimal acceptance rates for Metropolis algorithms: Moving beyond 0.234, Stochastic Processes and their Applications, 118 (2008), 2198-2222.
doi: 10.1016/j.spa.2007.12.005. |
[4] |
A. Beskos, A. Jasra, E. A. Muzaffer and A. M. Stuart, Sequential Monte Carlo methods for Bayesian elliptic inverse problems, Statistics and Computing, 25 (2015), 727-737.
doi: 10.1007/s11222-015-9556-7. |
[5] |
A. Beskos, G. O. Roberts, A. M. Stuart and J. Voss, MCMC methods for diffusion bridges, Stochastics and Dynamics, 8 (2008), 319-350.
doi: 10.1142/S0219493708002378. |
[6] |
V. I. Bogachev, Measure Theory Volume I, Springer, 2007.
doi: 10.1007/978-3-540-34514-5. |
[7] |
L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), R99-R136.
doi: 10.1088/0266-5611/18/6/201. |
[8] |
T. Bui-Thanh and O. Ghattas, An analysis of infinite dimensional bayesian inverse shape acoustic scattering and its numerical approximation, SIAM/ASA Journal on Uncertainty Quantification, 2 (2014), 203-222.
doi: 10.1137/120894877. |
[9] |
S. L. Cotter, G. O. Roberts, A. M. Stuart and D. White, MCMC methods for functions modifying old algorithms to make them faster, Statistical Science, 28 (2013), 424-446.
doi: 10.1214/13-STS421. |
[10] |
M. Dashti and A. M. Stuart, The Bayesian Approach to Inverse Problems, In Handbook of Uncertainty Quantification, Springer, 2016, 1-118.
doi: 10.1007/978-3-319-11259-6_7-1. |
[11] |
S. Duane, A. D. Kennedy, B. J. Pendleton and D. Roweth, Hybrid monte carlo, Physics Letters B, 195 (1987), 216-222.
doi: 10.1016/0370-2693(87)91197-X. |
[12] |
J. N. Franklin, Well posed stochastic extensions of ill posed linear problems}, Journal of Mathematical Analysis and Applications, 31 (1970), 682-716, URL http://adsabs.harvard.edu/abs/1970JIMIA..31..682F.
doi: 10.1016/0022-247X(70)90017-X. |
[13] |
M. Gehre, B. Jin and X. Lu, An analysis of finite element approximation in electrical impedance tomography, Inverse Problems, 30 (2014), 045013, 24pp, URL http://stacks.iop.org/0266-5611/30/i=4/a=045013?key=crossref.284cda5f9645598cb83fccc9d226940a.
doi: 10.1088/0266-5611/30/4/045013. |
[14] |
M. Hairer, A. M. Stuart and S. J. Vollmer, Spectral gaps for a Metropolis-Hastings algorithm in infinite dimensions, The Annals of Applied Probability, 24 (2014), 2455-2490.
doi: 10.1214/13-AAP982. |
[15] |
R. P. Henderson and J. G. Webster, An impedance camera for spatially specific measurements of the thorax, IEEE Transactions on Bio-Medical Engineering, 25 (1978), 250-254.
doi: 10.1109/TBME.1978.326329. |
[16] |
M. A. Iglesias, K. J. H. Law and A. M. Stuart, Ensemble Kalman methods for inverse problems, Inverse Problems, 29 (2013), 045001, 20pp, URL http://stacks.iop.org/0266-5611/29/i=4/a=045001?key=crossref.a70b58896eed5588086da7ad3d04954b.
doi: 10.1088/0266-5611/29/4/045001. |
[17] |
M. A. Iglesias, Y. Lu and A. M. Stuart, A bayesian level set method for geometric inverse problems, Interfaces Free Bound., 18 (2016), 181-217, URL http://arxiv.org/abs/1504.00313.
doi: 10.4171/IFB/362. |
[18] |
J. P. Kaipio, V. Kolehmainen, E. Somersalo and M. Vauhkonen, Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography, Inverse Problems, 16 (2000), 1487-1522.
doi: 10.1088/0266-5611/16/5/321. |
[19] |
J. P. Kaipio, V. Kolehmainen, M. Vauhkonen and E. Somersalo, Inverse problems with structural prior information, Inverse Problems, 15 (1999), 713-729.
doi: 10.1088/0266-5611/15/3/306. |
[20] |
J. P. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Springer, 2005. |
[21] |
R. E. Kass, Markov Chain Monte Carlo in practice: A roundtable discussion, The American Statistician, 52 (1998), 93-100.
doi: 10.2307/2685466. |
[22] |
C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data on manifolds and applications, Analysis and PDE, 6 (2013), 2003-2048.
doi: 10.2140/apde.2013.6.2003. |
[23] |
K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, D-bar method for electrical impedance tomography with discontinuous conductivities, SIAM Journal on Applied Mathematics, 67 (2007), 893-913.
doi: 10.1137/060656930. |
[24] |
K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, Reconstructions of piecewise constant conductivities by the D-bar method for electrical impedance tomography, Journal of Physics: Conference Series, 124 (2008), 012029.
doi: 10.1088/1742-6596/124/1/012029. |
[25] |
K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem, Inverse Problems and Imaging, 3 (2009), 599-624.
doi: 10.3934/ipi.2009.3.599. |
[26] |
V. Kolehmainen, M. Lassas, P. Ola and S. Siltanen, Recovering boundary shape and conductivity in electrical impedance tomography, Inverse Problems and Imaging, 7 (2013), 217-242.
doi: 10.3934/ipi.2013.7.217. |
[27] |
S. Lan, T. Bui-Thanh, M. Christie and M. Girolami, Emulation of Higher-Order Tensors in Manifold Monte Carlo Methods for Bayesian Inverse Problems, Journal of Computational Physics, 308 (2016), 81-101.
doi: 10.1016/j.jcp.2015.12.032. |
[28] |
R. E. Langer, An inverse problem in differential equations, Bulletin of the American Mathematical Society, 39 (1933), 814-820.
doi: 10.1090/S0002-9904-1933-05752-X. |
[29] |
S. Lasanen, Non-gaussian statistical inverse problems. part I: Posterior distributions, Inverse Problems & Imaging, 6 (2012), 215-266.
doi: 10.3934/ipi.2012.6.215. |
[30] |
S. Lasanen, Non-gaussian statistical inverse problems. part II: Posterior convergence for approximated unknowns, Inverse Problems & Imaging, 6 (2012), 267-287.
doi: 10.3934/ipi.2012.6.267. |
[31] |
S. Lasanen, J. M. J. Huttunen and L. Roininen, Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography, Inverse Problems and Imaging, 8 (2014), 561-586, URL http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=9912.
doi: 10.3934/ipi.2014.8.561. |
[32] |
M. Lassas, E. Saksman and S. Siltanen, Discretization-invariant Bayesian inversion and Besov space priors, Inverse Problems and Imaging, 3 (2009), 87-122.
doi: 10.3934/ipi.2009.3.87. |
[33] |
M. S. Lehtinen, L. Paivarinta and E. Somersalo, Linear inverse problems for generalised random variables, Inverse Problems, 5 (1989), 599-612.
doi: 10.1088/0266-5611/5/4/011. |
[34] |
A. Mandelbaum, Linear estimators and measurable linear transformations on a Hilbert space, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 65 (1984), 385-397.
doi: 10.1007/BF00533743. |
[35] |
A. I. Nachman, Reconstructions from boundary measurements, Annals of Mathematics, 128 (1988), 531-576, URL http://cat.inist.fr/?aModele=afficheN&cpsidt=7131706.
doi: 10.2307/1971435. |
[36] |
A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Annals of Mathematics, 143 (1996), 71-96.
doi: 10.2307/2118653. |
[37] |
G. C. Papanicolaou and L. Borcea, Network approximation for transport properties of high contrast materials, SIAM Journal on Applied Mathematics, 58 (1998), 501-539.
doi: 10.1137/S0036139996301891. |
[38] | |
[39] |
D. Schymura, An upper bound on the volume of the symmetric difference of a body and a congruent copy, Advances in Geometry, 14 (2014), 287-298.
doi: 10.1515/advgeom-2013-0029. |
[40] |
E. Somersalo, M. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography, SIAM Journal on Applied Mathematics, 52 (1992), 1023-1040.
doi: 10.1137/0152060. |
[41] |
A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, 19 (2010), 451-559, URL http://webcat.warwick.ac.uk:80/record=b2341209.
doi: 10.1017/S0962492910000061. |
[42] |
J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value, Annals of Mathematics, 125 (1987), 153-169.
doi: 10.2307/1971291. |
[43] |
L. Tierney, Markov chains for exploring posterior distributions, Annals of Statistics, 22 (1994), 1701-1728.
doi: 10.1214/aos/1176325750. |
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