American Institute of Mathematical Sciences

Advanced
November  2016, 10(4): 1007-1036. doi: 10.3934/ipi.2016030

The Bayesian formulation of EIT: Analysis and algorithms

Matthew M. Dunlop 1, and  Andrew M. Stuart 1,

1. 

Computing & Mathematical Sciences, California Institute of Technology, Pasadena, CA 91125, United States, United States

Received  August 2015 Revised  July 2016 Published  October 2016

We provide a rigorous Bayesian formulation of the EIT problem in an infinite dimensional setting, leading to well-posedness in the Hellinger metric with respect to the data. We focus particularly on the reconstruction of binary fields where the interface between different media is the primary unknown. We consider three different prior models -- log-Gaussian, star-shaped and level set. Numerical simulations based on the implementation of MCMC are performed, illustrating the advantages and disadvantages of each type of prior in the reconstruction, in the case where the true conductivity is a binary field, and exhibiting the properties of the resulting posterior distribution.
Keywords: Inverse problems, Bayesian regularisation, level set methodology., ill-posed problems, electrical impedance tomography, geometric priors.
Mathematics Subject Classification: Primary: 62G05, 65N21; Secondary: 92C5.
Citation: Matthew M. Dunlop, Andrew M. Stuart. The Bayesian formulation of EIT: Analysis and algorithms. Inverse Problems and Imaging, 2016, 10 (4) : 1007-1036. doi: 10.3934/ipi.2016030
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]

M. Salo, Calderón problem,, Lecture Notes., (). 

[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]

M. Salo, Calderón problem,, Lecture Notes., (). 

[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.

[1]

Masoumeh Dashti, Stephen Harris, Andrew Stuart. Besov priors for Bayesian inverse problems. Inverse Problems and Imaging, 2012, 6 (2) : 183-200. doi: 10.3934/ipi.2012.6.183
[2]

Stefan Kindermann. Convergence of the gradient method for ill-posed problems. Inverse Problems and Imaging, 2017, 11 (4) : 703-720. doi: 10.3934/ipi.2017033
[3]

T. J. Sullivan. Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors. Inverse Problems and Imaging, 2017, 11 (5) : 857-874. doi: 10.3934/ipi.2017040
[4]

Ye Zhang, Bernd Hofmann. Two new non-negativity preserving iterative regularization methods for ill-posed inverse problems. Inverse Problems and Imaging, 2021, 15 (2) : 229-256. doi: 10.3934/ipi.2020062
[5]

Sergiy Zhuk. Inverse problems for linear ill-posed differential-algebraic equations with uncertain parameters. Conference Publications, 2011, 2011 (Special) : 1467-1476. doi: 10.3934/proc.2011.2011.1467
[6]

Lassi Roininen, Janne M. J. Huttunen, Sari Lasanen. Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography. Inverse Problems and Imaging, 2014, 8 (2) : 561-586. doi: 10.3934/ipi.2014.8.561
[7]

Matthew A. Fury. Estimates for solutions of nonautonomous semilinear ill-posed problems. Conference Publications, 2015, 2015 (special) : 479-488. doi: 10.3934/proc.2015.0479
[8]

Paola Favati, Grazia Lotti, Ornella Menchi, Francesco Romani. An inner-outer regularizing method for ill-posed problems. Inverse Problems and Imaging, 2014, 8 (2) : 409-420. doi: 10.3934/ipi.2014.8.409
[9]

Matthew A. Fury. Regularization for ill-posed inhomogeneous evolution problems in a Hilbert space. Conference Publications, 2013, 2013 (special) : 259-272. doi: 10.3934/proc.2013.2013.259
[10]

Hiroshi Isozaki. Inverse boundary value problems in the horosphere - A link between hyperbolic geometry and electrical impedance tomography. Inverse Problems and Imaging, 2007, 1 (1) : 107-134. doi: 10.3934/ipi.2007.1.107
[11]

Tan Bui-Thanh, Omar Ghattas. A scalable algorithm for MAP estimators in Bayesian inverse problems with Besov priors. Inverse Problems and Imaging, 2015, 9 (1) : 27-53. doi: 10.3934/ipi.2015.9.27
[12]

Felix Lucka, Katharina Proksch, Christoph Brune, Nicolai Bissantz, Martin Burger, Holger Dette, Frank Wübbeling. Risk estimators for choosing regularization parameters in ill-posed problems - properties and limitations. Inverse Problems and Imaging, 2018, 12 (5) : 1121-1155. doi: 10.3934/ipi.2018047
[13]

Olha P. Kupenko, Rosanna Manzo. On optimal controls in coefficients for ill-posed non-Linear elliptic Dirichlet boundary value problems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1363-1393. doi: 10.3934/dcdsb.2018155
[14]

Eliane Bécache, Laurent Bourgeois, Lucas Franceschini, Jérémi Dardé. Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: The 1D case. Inverse Problems and Imaging, 2015, 9 (4) : 971-1002. doi: 10.3934/ipi.2015.9.971
[15]

Jiangfeng Huang, Zhiliang Deng, Liwei Xu. A Bayesian level set method for an inverse medium scattering problem in acoustics. Inverse Problems and Imaging, 2021, 15 (5) : 1077-1097. doi: 10.3934/ipi.2021029
[16]

Ke Zhang, Maokun Li, Fan Yang, Shenheng Xu, Aria Abubakar. Electrical impedance tomography with multiplicative regularization. Inverse Problems and Imaging, 2019, 13 (6) : 1139-1159. doi: 10.3934/ipi.2019051
[17]

Bastian Gebauer. Localized potentials in electrical impedance tomography. Inverse Problems and Imaging, 2008, 2 (2) : 251-269. doi: 10.3934/ipi.2008.2.251
[18]

Johnathan M. Bardsley. Gaussian Markov random field priors for inverse problems. Inverse Problems and Imaging, 2013, 7 (2) : 397-416. doi: 10.3934/ipi.2013.7.397
[19]

Fabrice Delbary, Rainer Kress. Electrical impedance tomography using a point electrode inverse scheme for complete electrode data. Inverse Problems and Imaging, 2011, 5 (2) : 355-369. doi: 10.3934/ipi.2011.5.355
[20]

Kari Astala, Jennifer L. Mueller, Lassi Päivärinta, Allan Perämäki, Samuli Siltanen. Direct electrical impedance tomography for nonsmooth conductivities. Inverse Problems and Imaging, 2011, 5 (3) : 531-549. doi: 10.3934/ipi.2011.5.531

2020 Impact Factor: 1.639

Tools

Metrics

  • PDF downloads (161)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy