# Exact shape-reconstruction by one-step linearization in electrical impedance tomography

Bastian Harrach, Jin Keun Seo

Research output: Contribution to journalArticle

35 Citations (Scopus)

### Abstract

For electrical impedance tomography (EIT), the linearized reconstruction method using the Fréchet derivative of the Neumann-to-Dirichlet map with respect to the conductivity has been widely used in the last three decades. However, few rigorous mathematical results are known regarding the errors caused by the linear approximation. In this work we prove that linearizing the inverse problem of EIT does not lead to shape errors for piecewise-analytic conductivities. If a solution of the linearized equations exists, then it has the same outer support as the true conductivity change, no matter how large the latter is. Under an additional definiteness condition we also show how to approximately solve the linearized equation so that the outer support converges toward the right one. Our convergence result is global and also applies for approximations by noisy finitedimensional data. Furthermore, we obtain bounds on how well the linear reconstructions and the true conductivity difference agree on the boundary of the outer support.

Original language English 1505-1518 14 SIAM Journal on Mathematical Analysis 42 4 https://doi.org/10.1137/090773970 Published - 2010 Sep 2

### Fingerprint

Shape Reconstruction
Electrical Impedance Tomography
Acoustic impedance
Linearization
Conductivity
Tomography
Inverse problems
Derivatives
Dirichlet-to-Neumann Map
Linear Approximation
Convergence Results
Inverse Problem
Converge
Derivative
Approximation

### All Science Journal Classification (ASJC) codes

• Analysis
• Computational Mathematics
• Applied Mathematics

### Cite this

title = "Exact shape-reconstruction by one-step linearization in electrical impedance tomography",
abstract = "For electrical impedance tomography (EIT), the linearized reconstruction method using the Fr{\'e}chet derivative of the Neumann-to-Dirichlet map with respect to the conductivity has been widely used in the last three decades. However, few rigorous mathematical results are known regarding the errors caused by the linear approximation. In this work we prove that linearizing the inverse problem of EIT does not lead to shape errors for piecewise-analytic conductivities. If a solution of the linearized equations exists, then it has the same outer support as the true conductivity change, no matter how large the latter is. Under an additional definiteness condition we also show how to approximately solve the linearized equation so that the outer support converges toward the right one. Our convergence result is global and also applies for approximations by noisy finitedimensional data. Furthermore, we obtain bounds on how well the linear reconstructions and the true conductivity difference agree on the boundary of the outer support.",
author = "Bastian Harrach and Seo, {Jin Keun}",
year = "2010",
month = "9",
day = "2",
doi = "10.1137/090773970",
language = "English",
volume = "42",
pages = "1505--1518",
journal = "SIAM Journal on Mathematical Analysis",
issn = "0036-1410",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "4",

}

In: SIAM Journal on Mathematical Analysis, Vol. 42, No. 4, 02.09.2010, p. 1505-1518.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Exact shape-reconstruction by one-step linearization in electrical impedance tomography

AU - Harrach, Bastian

AU - Seo, Jin Keun

PY - 2010/9/2

Y1 - 2010/9/2

N2 - For electrical impedance tomography (EIT), the linearized reconstruction method using the Fréchet derivative of the Neumann-to-Dirichlet map with respect to the conductivity has been widely used in the last three decades. However, few rigorous mathematical results are known regarding the errors caused by the linear approximation. In this work we prove that linearizing the inverse problem of EIT does not lead to shape errors for piecewise-analytic conductivities. If a solution of the linearized equations exists, then it has the same outer support as the true conductivity change, no matter how large the latter is. Under an additional definiteness condition we also show how to approximately solve the linearized equation so that the outer support converges toward the right one. Our convergence result is global and also applies for approximations by noisy finitedimensional data. Furthermore, we obtain bounds on how well the linear reconstructions and the true conductivity difference agree on the boundary of the outer support.

AB - For electrical impedance tomography (EIT), the linearized reconstruction method using the Fréchet derivative of the Neumann-to-Dirichlet map with respect to the conductivity has been widely used in the last three decades. However, few rigorous mathematical results are known regarding the errors caused by the linear approximation. In this work we prove that linearizing the inverse problem of EIT does not lead to shape errors for piecewise-analytic conductivities. If a solution of the linearized equations exists, then it has the same outer support as the true conductivity change, no matter how large the latter is. Under an additional definiteness condition we also show how to approximately solve the linearized equation so that the outer support converges toward the right one. Our convergence result is global and also applies for approximations by noisy finitedimensional data. Furthermore, we obtain bounds on how well the linear reconstructions and the true conductivity difference agree on the boundary of the outer support.

UR - http://www.scopus.com/inward/record.url?scp=77956066305&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77956066305&partnerID=8YFLogxK

U2 - 10.1137/090773970

DO - 10.1137/090773970

M3 - Article

AN - SCOPUS:77956066305

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JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

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