November  2009, 3(4): 599-624. doi: 10.3934/ipi.2009.3.599

Regularized D-bar method for the inverse conductivity problem


Department of Mathematics, Technical University of Denmark


Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 (Gustaf Hallstromin katu 2b) FI-00014


Department of Mathematics, Colorado State University, Fort Collins, CO 80523,, United States


Department of Mathematics and Statistics, University of Helsinki, Finland

Received  January 2009 Revised  September 2009 Published  October 2009

A strategy for regularizing the inversion procedure for the two-dimensional D-bar reconstruction algorithm based on the global uniqueness proof of Nachman [Ann. Math. 143 (1996)] for the ill-posed inverse conductivity problem is presented. The strategy utilizes truncation of the boundary integral equation and the scattering transform. It is shown that this leads to a bound on the error in the scattering transform and a stable reconstruction of the conductivity; an explicit rate of convergence in appropriate Banach spaces is derived as well. Numerical results are also included, demonstrating the convergence of the reconstructed conductivity to the true conductivity as the noise level tends to zero. The results provide a link between two traditions of inverse problems research: theory of regularization and inversion methods based on complex geometrical optics. Also, the procedure is a novel regularized imaging method for electrical impedance tomography.
Citation: Kim Knudsen, Matti Lassas, Jennifer L. Mueller, Samuli Siltanen. Regularized D-bar method for the inverse conductivity problem. Inverse Problems & Imaging, 2009, 3 (4) : 599-624. doi: 10.3934/ipi.2009.3.599

Luca Rondi. On the regularization of the inverse conductivity problem with discontinuous conductivities. Inverse Problems & Imaging, 2008, 2 (3) : 397-409. doi: 10.3934/ipi.2008.2.397


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


Misha Perepelitsa. An ill-posed problem for the Navier-Stokes equations for compressible flows. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 609-623. doi: 10.3934/dcds.2010.26.609


Victor Isakov. On uniqueness in the inverse conductivity problem with local data. Inverse Problems & Imaging, 2007, 1 (1) : 95-105. doi: 10.3934/ipi.2007.1.95


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


Alfredo Lorenzi, Luca Lorenzi. A strongly ill-posed integrodifferential singular parabolic problem in the unit cube of $\mathbb{R}^n$. Evolution Equations & Control Theory, 2014, 3 (3) : 499-524. doi: 10.3934/eect.2014.3.499


Ville Kolehmainen, Matti Lassas, Petri Ola, Samuli Siltanen. Recovering boundary shape and conductivity in electrical impedance tomography. Inverse Problems & Imaging, 2013, 7 (1) : 217-242. doi: 10.3934/ipi.2013.7.217


Albert Clop, Daniel Faraco, Alberto Ruiz. Stability of Calderón's inverse conductivity problem in the plane for discontinuous conductivities. Inverse Problems & Imaging, 2010, 4 (1) : 49-91. doi: 10.3934/ipi.2010.4.49


Jie Chen, Maarten de Hoop. The inverse problem for electroseismic conversion: Stable recovery of the conductivity and the electrokinetic mobility parameter. Inverse Problems & Imaging, 2016, 10 (3) : 641-658. doi: 10.3934/ipi.2016015


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


Peter I. Kogut, Olha P. Kupenko. On optimal control problem for an ill-posed strongly nonlinear elliptic equation with $p$-Laplace operator and $L^1$-type of nonlinearity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1273-1295. doi: 10.3934/dcdsb.2019016


Christodoulos E. Athanasiadis, Vassilios Sevroglou, Konstantinos I. Skourogiannis. The inverse electromagnetic scattering problem by a mixed impedance screen in chiral media. Inverse Problems & Imaging, 2015, 9 (4) : 951-970. doi: 10.3934/ipi.2015.9.951


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


Dong liu, Ville Kolehmainen, Samuli Siltanen, Anne-maria Laukkanen, Aku Seppänen. Estimation of conductivity changes in a region of interest with electrical impedance tomography. Inverse Problems & Imaging, 2015, 9 (1) : 211-229. doi: 10.3934/ipi.2015.9.211


Gen Nakamura, Päivi Ronkanen, Samuli Siltanen, Kazumi Tanuma. Recovering conductivity at the boundary in three-dimensional electrical impedance tomography. Inverse Problems & Imaging, 2011, 5 (2) : 485-510. doi: 10.3934/ipi.2011.5.485


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


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


Victor Isakov, Joseph Myers. On the inverse doping profile problem. Inverse Problems & Imaging, 2012, 6 (3) : 465-486. doi: 10.3934/ipi.2012.6.465


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 & Imaging, 2018, 12 (5) : 1121-1155. doi: 10.3934/ipi.2018047


Guozhi Dong, Bert Jüttler, Otmar Scherzer, Thomas Takacs. Convergence of Tikhonov regularization for solving ill-posed operator equations with solutions defined on surfaces. Inverse Problems & Imaging, 2017, 11 (2) : 221-246. doi: 10.3934/ipi.2017011

2020 Impact Factor: 1.639


  • PDF downloads (130)
  • HTML views (0)
  • Cited by (89)

[Back to Top]