Reconstruction of singular and degenerate inclusions in Calderón's problem

Henrik Garde; Nuutti Hyvönen

doi:10.3934/ipi.2022021

Reconstruction of singular and degenerate inclusions in Calderón's problem

Henrik Garde^*, Nuutti Hyvönen

^*Corresponding author af dette arbejde

Institut for Matematik

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

Abstract

We consider the reconstruction of the support of an unknown perturbation to a known conductivity coefficient in Calderón's problem. In a previous result by the authors on monotonicity-based reconstruction, the perturbed coefficient is allowed to simultaneously take the values $0$ and $\infty$ in some parts of the domain and values bounded away from $0$ and $\infty$ elsewhere. We generalise this result by allowing the unknown coefficient to be the restriction of an $A_2$-Muckenhoupt weight in parts of the domain, thereby including singular and degenerate behaviour in the governing equation. In particular, the coefficient may tend to $0$ and $\infty$ in a controlled manner, which goes beyond the standard setting of Calderón's problem. Our main result constructively characterises the outer shape of the support of such a general perturbation, based on a local Neumann-to-Dirichlet map defined on an open subset of the domain boundary.

Originalsprog	Engelsk
Tidsskrift	Inverse Problems and Imaging
Vol/bind	16
Nummer	5
Sider (fra-til)	1219-1227
Antal sider	9
ISSN	1930-8337
DOI	https://doi.org/10.3934/ipi.2022021
Status	Udgivet - aug. 2022

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10.3934/ipi.2022021

https://arxiv.org/abs/2106.07764

Citationsformater

@article{70b88dabc83b4210b45351c1bebd8950,

title = "Reconstruction of singular and degenerate inclusions in Calder{\'o}n's problem",

abstract = "We consider the reconstruction of the support of an unknown perturbation to a known conductivity coefficient in Calder{\'o}n's problem. In a previous result by the authors on monotonicity-based reconstruction, the perturbed coefficient is allowed to simultaneously take the values $0$ and $\infty$ in some parts of the domain and values bounded away from $0$ and $\infty$ elsewhere. We generalise this result by allowing the unknown coefficient to be the restriction of an $A_2$-Muckenhoupt weight in parts of the domain, thereby including singular and degenerate behaviour in the governing equation. In particular, the coefficient may tend to $0$ and $\infty$ in a controlled manner, which goes beyond the standard setting of Calder{\'o}n's problem. Our main result constructively characterises the outer shape of the support of such a general perturbation, based on a local Neumann-to-Dirichlet map defined on an open subset of the domain boundary. ",

keywords = "Calder{\'o}n's problem, electrical impedance tomography, monotonicity method, degenerate elliptic problem, singular elliptic problem, Calderon's problem, inclusion detection, SHAPE-RECONSTRUCTION",

author = "Henrik Garde and Nuutti Hyv{\"o}nen",

year = "2022",

month = aug,

doi = "10.3934/ipi.2022021",

language = "English",

volume = "16",

pages = "1219--1227",

journal = "Inverse Problems and Imaging",

issn = "1930-8337",

publisher = "American Institute of Mathematical Sciences (AIMS)",

number = "5",

}

TY - JOUR

T1 - Reconstruction of singular and degenerate inclusions in Calderón's problem

AU - Garde, Henrik

AU - Hyvönen, Nuutti

PY - 2022/8

Y1 - 2022/8

N2 - We consider the reconstruction of the support of an unknown perturbation to a known conductivity coefficient in Calderón's problem. In a previous result by the authors on monotonicity-based reconstruction, the perturbed coefficient is allowed to simultaneously take the values $0$ and $\infty$ in some parts of the domain and values bounded away from $0$ and $\infty$ elsewhere. We generalise this result by allowing the unknown coefficient to be the restriction of an $A_2$-Muckenhoupt weight in parts of the domain, thereby including singular and degenerate behaviour in the governing equation. In particular, the coefficient may tend to $0$ and $\infty$ in a controlled manner, which goes beyond the standard setting of Calderón's problem. Our main result constructively characterises the outer shape of the support of such a general perturbation, based on a local Neumann-to-Dirichlet map defined on an open subset of the domain boundary.

AB - We consider the reconstruction of the support of an unknown perturbation to a known conductivity coefficient in Calderón's problem. In a previous result by the authors on monotonicity-based reconstruction, the perturbed coefficient is allowed to simultaneously take the values $0$ and $\infty$ in some parts of the domain and values bounded away from $0$ and $\infty$ elsewhere. We generalise this result by allowing the unknown coefficient to be the restriction of an $A_2$-Muckenhoupt weight in parts of the domain, thereby including singular and degenerate behaviour in the governing equation. In particular, the coefficient may tend to $0$ and $\infty$ in a controlled manner, which goes beyond the standard setting of Calderón's problem. Our main result constructively characterises the outer shape of the support of such a general perturbation, based on a local Neumann-to-Dirichlet map defined on an open subset of the domain boundary.

KW - Calderón's problem

KW - electrical impedance tomography

KW - monotonicity method

KW - degenerate elliptic problem

KW - singular elliptic problem

KW - Calderon's problem

KW - inclusion detection

KW - SHAPE-RECONSTRUCTION

U2 - 10.3934/ipi.2022021

DO - 10.3934/ipi.2022021

M3 - Journal article

SN - 1930-8337

VL - 16

SP - 1219

EP - 1227

JO - Inverse Problems and Imaging

JF - Inverse Problems and Imaging

IS - 5

ER -

Reconstruction of singular and degenerate inclusions in Calderón's problem

Abstract

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