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

Henrik Garde

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

^*Corresponding author af dette arbejde

Publikation: Bidrag til tidsskrift/Konferencebidrag i tidsskrift /Bidrag til avis › Tidsskriftartikel › Forskning › peer review

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.

Originalsprog	Engelsk
Tidsskrift	Inverse Problems in Science and Engineering
Vol/bind	26
Nummer	1
Sider (fra-til)	33-50
Antal sider	18
ISSN	1741-5977
Status	Udgivet - 2018
Udgivet eksternt	Ja

Citationsformater

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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 Fr{\'e}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, direct reconstruction, ill-posed problem, inverse boundary value problem, monotonicity method",

author = "Henrik Garde",

year = "2018",

language = "English",

volume = "26",

pages = "33--50",

journal = "Inverse Problems in Science and Engineering",

issn = "1741-5977",

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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 - direct reconstruction

KW - ill-posed problem

KW - inverse boundary value problem

KW - monotonicity method

M3 - Journal article

SN - 1741-5977

VL - 26

SP - 33

EP - 50

JO - Inverse Problems in Science and Engineering

JF - Inverse Problems in Science and Engineering

IS - 1

ER -