Analyticity of point measurements in inverse conductivity and scattering problems

Otto Seiskari

Analyticity of point measurements in inverse conductivity and scattering problems

Julkaisun otsikon käännös: Pistemittausten analyyttisyys käänteisjohtavuus- ja sirontaongelmissa

Otto Seiskari

Tutkimustuotos: Doctoral Thesis › Collection of Articles

Abstrakti

Väitöskirjassa tutkitaan käänteisjohtavuusongelmia ja Helmholtzin yhtälön sirontaongelmia distributionaalisilla reuna-arvoilla. Impedanssitomografian (EIT) tapauksessa nämä voidaan tulkita mittauksina pistemäisillä elektrodeilla. Työssä esitellään uusi, käänteissirontateorian kaukokentän kaltainen käsite, EIT:n bisweep- data, jota voidaan hyödyntää käänteisjohtavuusongelmien teoriassa. Erityisesti näytetään, että bisweep-data on suhteellisen Neumann-to-Dirichlet-kuvauksen Schwartzin ydin, minkä avulla todistetaan uusia osittaisdatatuloksia Calderónin ongelmalle. Vastaavia tekniikoita sovelletaan myös sirontateoriassa ja osoitetaan kaukokenttäkuvauksen yhdistetty analyyttisyys. Lisäksi väitöskirjassa tutkitaan toista hiljattain esiteltyä käsitettä, kaukokentän takaisinsirontadatan tyyppistä EIT:n sweep-dataa ja laaditaan numeerinen menetelmä pienten epähomogeenisuuksien paikantamiseen sweep-datan avulla. Työssä näytetään myös, kuinka bisweep-datan ja konformikuvausten avulla tietyt numeeriset käänteisjohtavuusongelmat voidaan palauttaa paloittain sileistä tason alueista yksikkökiekkoon.

Julkaisun otsikon käännös	Pistemittausten analyyttisyys käänteisjohtavuus- ja sirontaongelmissa
Alkuperäiskieli	Englanti
Pätevyys	Tohtorintutkinto
Myöntävä instituutio	Aalto-yliopisto
Valvoja/neuvonantaja	Hyvönen, Nuutti, Vastuuprofessori Hyvönen, Nuutti, Ohjaaja
Kustantaja	Aalto University
Painoksen ISBN	978-952-60-5383-7
Sähköinen ISBN	978-952-60-5384-4
Tila	Julkaistu - 2013
OKM-julkaisutyyppi	G5 Artikkeliväitöskirja

Tutkimusalat

inversio-ongelmat
impedanssitomografia
pistemittaukset
Calderónin ongelma
osittainen data
(bi)sweep-data
elliptiset reuna-arvo-ongelmat
sironta
kaukokenttä

Pääsy asiakirjaan

http://urn.fi/URN:ISBN:978-952-60-5384-4

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@phdthesis{7a5c2c18ee02463b8411d3d2f77115e3,

title = "Analyticity of point measurements in inverse conductivity and scattering problems",

abstract = "Inverse conductivity and Helmholtz scattering problems with distributional boundary values are studied. In the context of electrical impedance tomography (EIT), the considered concepts can be interpreted in terms of measurements involving point-like electrodes. The notion of bisweep data of EIT, analogous to the far-field pattern in scattering theory, is introduced and applied in the theory of inverse conductivity problems. In particular, it is shown that bisweep data are the Schwartz kernel of the relative Neumann-to-Dirichlet map, and this result is employed in proving new partial data results for Calderon's problem. Similar techniques are also applied in the scattering context in order to prove the joint analyticity of the far-field pattern. Another recent concept, sweep data of EIT, analogous to the far-field backscatter data, is studied further, and a numerical method for locating small inhomogeneities from sweep data is presented. It is also demonstrated how bisweep data and conformal maps can be used to reduce certain numerical inverse conductivity problems in piecewise smooth plane domains to equivalent problems in the unit disk.",

keywords = "inverse problems, electrical impedance tomography, point measurements, Calder{\'o}n problem, partial data, (bi)sweep data, elliptic boundary value problems, scattering, far-field pattern, inversio-ongelmat, impedanssitomografia, pistemittaukset, Calder{\'o}nin ongelma, osittainen data, (bi)sweep-data, elliptiset reuna-arvo-ongelmat, sironta, kaukokentt{\"a}, inverse problems, electrical impedance tomography, point measurements, Calder{\'o}n problem, partial data, (bi)sweep data, elliptic boundary value problems, scattering, far-field pattern",

author = "Otto Seiskari",

year = "2013",

language = "English",

isbn = "978-952-60-5383-7",

series = "Aalto University publication series DOCTORAL DISSERTATIONS",

publisher = "Aalto University",

number = "165",

address = "Finland",

school = "Aalto University",

}

TY - BOOK

T1 - Analyticity of point measurements in inverse conductivity and scattering problems

AU - Seiskari, Otto

PY - 2013

Y1 - 2013

N2 - Inverse conductivity and Helmholtz scattering problems with distributional boundary values are studied. In the context of electrical impedance tomography (EIT), the considered concepts can be interpreted in terms of measurements involving point-like electrodes. The notion of bisweep data of EIT, analogous to the far-field pattern in scattering theory, is introduced and applied in the theory of inverse conductivity problems. In particular, it is shown that bisweep data are the Schwartz kernel of the relative Neumann-to-Dirichlet map, and this result is employed in proving new partial data results for Calderon's problem. Similar techniques are also applied in the scattering context in order to prove the joint analyticity of the far-field pattern. Another recent concept, sweep data of EIT, analogous to the far-field backscatter data, is studied further, and a numerical method for locating small inhomogeneities from sweep data is presented. It is also demonstrated how bisweep data and conformal maps can be used to reduce certain numerical inverse conductivity problems in piecewise smooth plane domains to equivalent problems in the unit disk.

AB - Inverse conductivity and Helmholtz scattering problems with distributional boundary values are studied. In the context of electrical impedance tomography (EIT), the considered concepts can be interpreted in terms of measurements involving point-like electrodes. The notion of bisweep data of EIT, analogous to the far-field pattern in scattering theory, is introduced and applied in the theory of inverse conductivity problems. In particular, it is shown that bisweep data are the Schwartz kernel of the relative Neumann-to-Dirichlet map, and this result is employed in proving new partial data results for Calderon's problem. Similar techniques are also applied in the scattering context in order to prove the joint analyticity of the far-field pattern. Another recent concept, sweep data of EIT, analogous to the far-field backscatter data, is studied further, and a numerical method for locating small inhomogeneities from sweep data is presented. It is also demonstrated how bisweep data and conformal maps can be used to reduce certain numerical inverse conductivity problems in piecewise smooth plane domains to equivalent problems in the unit disk.

KW - inverse problems

KW - electrical impedance tomography

KW - point measurements

KW - Calderón problem

KW - partial data

KW - (bi)sweep data

KW - elliptic boundary value problems

KW - scattering

KW - far-field pattern

KW - inversio-ongelmat

KW - impedanssitomografia

KW - pistemittaukset

KW - Calderónin ongelma

KW - osittainen data

KW - (bi)sweep-data

KW - elliptiset reuna-arvo-ongelmat

KW - sironta

KW - kaukokenttä

KW - inverse problems

KW - electrical impedance tomography

KW - point measurements

KW - Calderón problem

KW - partial data

KW - (bi)sweep data

KW - elliptic boundary value problems

KW - scattering

KW - far-field pattern

M3 - Doctoral Thesis

SN - 978-952-60-5383-7

T3 - Aalto University publication series DOCTORAL DISSERTATIONS

PB - Aalto University

ER -

Analyticity of point measurements in inverse conductivity and scattering problems

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