Comparison of linear and non-linear monotonicity-based shape reconstruction using exact matrix characterizations

Henrik Garde*

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

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Abstract

Detecting inhomogeneities in the electrical conductivity is a special case of the inverse problem in electrical impedance tomography, that leads to fast direct reconstruction methods. One such method can, under reasonable assumptions, exactly characterize the inhomogeneities based on monotonicity properties of either the Neumann-to-Dirichlet map (non-linear) or its Fréchet derivative (linear). We give a comparison of the non-linear and linear approach in the presence of measurement noise, and show numerically that the two methods give essentially the same reconstruction in the unit disk domain. For a fair comparison, exact matrix characterizations are used when probing the monotonicity relations to avoid errors from numerical solution to PDEs and numerical integration. Using a special factorization of the Neumann-to-Dirichlet map also makes the non-linear method as fast as the linear method in the unit disk geometry.
Original languageEnglish
JournalInverse Problems in Science and Engineering
Volume26
Issue number1
Pages (from-to)33-50
ISSN1741-5977
DOIs
Publication statusPublished - 2018

Keywords

  • Electrical impedance tomography
  • Monotonicity method
  • Inverse boundary value problem
  • Ill-posed problem
  • Direct reconstruction methods

Cite this

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title = "Comparison of linear and non-linear monotonicity-based shape reconstruction using exact matrix characterizations",
abstract = "Detecting inhomogeneities in the electrical conductivity is a special case of the inverse problem in electrical impedance tomography, that leads to fast direct reconstruction methods. One such method can, under reasonable assumptions, exactly characterize the inhomogeneities based on monotonicity properties of either the Neumann-to-Dirichlet map (non-linear) or its Fr{\~A}{\circledC}chet derivative (linear). We give a comparison of the non-linear and linear approach in the presence of measurement noise, and show numerically that the two methods give essentially the same reconstruction in the unit disk domain. For a fair comparison, exact matrix characterizations are used when probing the monotonicity relations to avoid errors from numerical solution to PDEs and numerical integration. Using a special factorization of the Neumann-to-Dirichlet map also makes the non-linear method as fast as the linear method in the unit disk geometry.",
keywords = "Electrical impedance tomography, Monotonicity method, Inverse boundary value problem, Ill-posed problem, Direct reconstruction methods",
author = "Henrik Garde",
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language = "English",
volume = "26",
pages = "33--50",
journal = "Inverse Problems in Science and Engineering",
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publisher = "CRC Press/Balkema",
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}

Comparison of linear and non-linear monotonicity-based shape reconstruction using exact matrix characterizations. / Garde, Henrik.

In: Inverse Problems in Science and Engineering, Vol. 26, No. 1, 2018, p. 33-50.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - Comparison of linear and non-linear monotonicity-based shape reconstruction using exact matrix characterizations

AU - Garde, Henrik

PY - 2018

Y1 - 2018

N2 - Detecting inhomogeneities in the electrical conductivity is a special case of the inverse problem in electrical impedance tomography, that leads to fast direct reconstruction methods. One such method can, under reasonable assumptions, exactly characterize the inhomogeneities based on monotonicity properties of either the Neumann-to-Dirichlet map (non-linear) or its Fréchet derivative (linear). We give a comparison of the non-linear and linear approach in the presence of measurement noise, and show numerically that the two methods give essentially the same reconstruction in the unit disk domain. For a fair comparison, exact matrix characterizations are used when probing the monotonicity relations to avoid errors from numerical solution to PDEs and numerical integration. Using a special factorization of the Neumann-to-Dirichlet map also makes the non-linear method as fast as the linear method in the unit disk geometry.

AB - Detecting inhomogeneities in the electrical conductivity is a special case of the inverse problem in electrical impedance tomography, that leads to fast direct reconstruction methods. One such method can, under reasonable assumptions, exactly characterize the inhomogeneities based on monotonicity properties of either the Neumann-to-Dirichlet map (non-linear) or its Fréchet derivative (linear). We give a comparison of the non-linear and linear approach in the presence of measurement noise, and show numerically that the two methods give essentially the same reconstruction in the unit disk domain. For a fair comparison, exact matrix characterizations are used when probing the monotonicity relations to avoid errors from numerical solution to PDEs and numerical integration. Using a special factorization of the Neumann-to-Dirichlet map also makes the non-linear method as fast as the linear method in the unit disk geometry.

KW - Electrical impedance tomography

KW - Monotonicity method

KW - Inverse boundary value problem

KW - Ill-posed problem

KW - Direct reconstruction methods

U2 - 10.1080/17415977.2017.1290088

DO - 10.1080/17415977.2017.1290088

M3 - Journal article

VL - 26

SP - 33

EP - 50

JO - Inverse Problems in Science and Engineering

JF - Inverse Problems in Science and Engineering

SN - 1741-5977

IS - 1

ER -