Approximations and Surrogates for Computational Inverse Boundary Value Problems

Lauri Mustonen

Approximations and Surrogates for Computational Inverse Boundary Value Problems

Julkaisun otsikon käännös: Approksimaatioita ja surrogaatteja laskennallisille käänteisreuna-arvo-ongelmille

Lauri Mustonen

Tutkimustuotos: Doctoral Thesis › Collection of Articles

Abstrakti

Käänteiset reuna-arvo-ongelmat liittyvät läheisesti kuvantamismenetelmiin, joissa kappaleen sisäisiä ominaisuuksia rekonstruoidaan pintamittausten perusteella. Väitöskirjassa esitellään parannettuja rekonstruktiomenetelmiä ja uusia laskennallisia lähestymistapoja elliptisille ja parabolisille käänteisreuna-arvo-ongelmille. Sovelluskohteina tarkastellaan impedanssitomografiaa eli EIT:tä sekä lämpötomografiaa. Tarkasteltavat käänteisongelmat ovat epälineaarisia ja rekonstruktiot lasketaan pienimmän neliösumman menetelmällä. Neliösumman minimointialgoritmit perustuvat kohdefunktion ja sen osittaisderivaattojen toistuvaan evaluointiin. Väitöskirjassa kohdefunktiot, jotka ovat osittaisdifferentiaaliyhtälöiden ratkaisuja, korvataan polynomisijaismalleilla eli -surrogaateilla, joiden evaluointi ja derivointi on suoraviivaista. EIT:n yhteydessä menetelmää käytetään sähköisten ominaisuuksien lisäksi kappaleen muodon estimointiin. Menetelmän todetaan toimivan lämpötomografiassakin, myös silloin, kun tieto kappaleen muodosta on epävarmaa. Väitöskirjassa esitellään myös uusi logaritminen linearisointimenetelmä EIT:lle. Muuntamalla jännitemittaukset tietyllä logaritmisella tavalla voidaan vähentää sähköisten ominaisuuksien ja mittausten välistä epälineaarisuutta, jolloin rekonstruktio voidaan laskea pienemmällä määrällä minimointiaskeleita. Lisäksi EIT:n niin sanotulle täydelliselle elektrodimallille ehdotetaan uutta muotoilua, joka on yhteensopiva kokeellisten mittausten kanssa, mutta joka lisää ratkaistavan potentiaalin säännöllisyyttä ja siten nopeuttaa numeeristen menetelmien suppenemista.

Julkaisun otsikon käännös	Approksimaatioita ja surrogaatteja laskennallisille käänteisreuna-arvo-ongelmille
Alkuperäiskieli	Englanti
Pätevyys	Tohtorintutkinto
Myöntävä instituutio	Aalto-yliopisto
Valvoja/neuvonantaja	Hyvönen, Nuutti, Vastuuprofessori Hyvönen, Nuutti, Ohjaaja
Kustantaja	Aalto University
Painoksen ISBN	978-952-60-7611-9
Sähköinen ISBN	978-952-60-7610-2
Tila	Julkaistu - 2017
OKM-julkaisutyyppi	G5 Artikkeliväitöskirja

Tutkimusalat

inversio-ongelmat
epävarmuuden kvantifiointi
osittaisdifferentiaaliyhtälöt
epälineaarinen pienimmän neliösumman menetelmä
polynomisurrogaatti
impedanssitomografia
täydellinen elektrodimalli
lämpötomografia

Pääsy asiakirjaan

http://urn.fi/URN:ISBN:978-952-60-7610-2Lisenssi: Määrittelemätön

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@phdthesis{9dbcafc1549747b4b570db3cc89589fa,

title = "Approximations and Surrogates for Computational Inverse Boundary Value Problems",

abstract = "Inverse boundary value problems are closely related to imaging techniques where measurements on the surface are used to estimate, or reconstruct, inner properties of the imaged object. In this thesis, improved reconstruction methods and new computational approaches are presented for elliptic and parabolic inverse boundary value problems. Two imaging applications that are addressed are electrical impedance tomography (EIT) and thermal tomography. The inverse problems that are considered are nonlinear and the reconstructions are sought by least squares minimization. Algorithms for such minimization often rely on iterative evaluation of the target function and its partial derivatives. In this thesis, we approximate these target functions, which themselves are solutions to partial differential equations, with polynomial surrogates that are simple to evaluate and differentiate. In the context of EIT, this method is used to estimate the shape of the object in addition to its electrical properties. The method is also shown to be feasible for thermal tomography, including the case of uncertain object shape. We also present a novel logarithmic linearization method for EIT. Transforming the voltage measurements in a certain logarithmic way reduces the nonlinearity in the relationship between the electrical properties and the measurements, allowing a reconstruction with fewer or only one minimization step. Furthermore, we propose a modification to the complete electrode model for EIT. The new model is shown to be compatible with experimental measurement data, while the increased regularity of the predicted electromagnetic potential improves convergence properties of numerical methods.",

keywords = "inverse problems, uncertainty quantification, partial differential equations, nonlinear least squares, polynomial surrogate, electrical impedance tomography, complete electrode model, thermal tomography, inversio-ongelmat, ep{\"a}varmuuden kvantifiointi, osittaisdifferentiaaliyht{\"a}l{\"o}t, ep{\"a}lineaarinen pienimm{\"a}n neli{\"o}summan menetelm{\"a}, polynomisurrogaatti, impedanssitomografia, t{\"a}ydellinen elektrodimalli, l{\"a}mp{\"o}tomografia, inverse problems, uncertainty quantification, partial differential equations, nonlinear least squares, polynomial surrogate, electrical impedance tomography, complete electrode model, thermal tomography",

author = "Lauri Mustonen",

year = "2017",

language = "English",

isbn = "978-952-60-7611-9",

series = "Aalto University publication series DOCTORAL DISSERTATIONS",

publisher = "Aalto University",

number = "174",

address = "Finland",

school = "Aalto University",

}

TY - BOOK

T1 - Approximations and Surrogates for Computational Inverse Boundary Value Problems

AU - Mustonen, Lauri

PY - 2017

Y1 - 2017

N2 - Inverse boundary value problems are closely related to imaging techniques where measurements on the surface are used to estimate, or reconstruct, inner properties of the imaged object. In this thesis, improved reconstruction methods and new computational approaches are presented for elliptic and parabolic inverse boundary value problems. Two imaging applications that are addressed are electrical impedance tomography (EIT) and thermal tomography. The inverse problems that are considered are nonlinear and the reconstructions are sought by least squares minimization. Algorithms for such minimization often rely on iterative evaluation of the target function and its partial derivatives. In this thesis, we approximate these target functions, which themselves are solutions to partial differential equations, with polynomial surrogates that are simple to evaluate and differentiate. In the context of EIT, this method is used to estimate the shape of the object in addition to its electrical properties. The method is also shown to be feasible for thermal tomography, including the case of uncertain object shape. We also present a novel logarithmic linearization method for EIT. Transforming the voltage measurements in a certain logarithmic way reduces the nonlinearity in the relationship between the electrical properties and the measurements, allowing a reconstruction with fewer or only one minimization step. Furthermore, we propose a modification to the complete electrode model for EIT. The new model is shown to be compatible with experimental measurement data, while the increased regularity of the predicted electromagnetic potential improves convergence properties of numerical methods.

AB - Inverse boundary value problems are closely related to imaging techniques where measurements on the surface are used to estimate, or reconstruct, inner properties of the imaged object. In this thesis, improved reconstruction methods and new computational approaches are presented for elliptic and parabolic inverse boundary value problems. Two imaging applications that are addressed are electrical impedance tomography (EIT) and thermal tomography. The inverse problems that are considered are nonlinear and the reconstructions are sought by least squares minimization. Algorithms for such minimization often rely on iterative evaluation of the target function and its partial derivatives. In this thesis, we approximate these target functions, which themselves are solutions to partial differential equations, with polynomial surrogates that are simple to evaluate and differentiate. In the context of EIT, this method is used to estimate the shape of the object in addition to its electrical properties. The method is also shown to be feasible for thermal tomography, including the case of uncertain object shape. We also present a novel logarithmic linearization method for EIT. Transforming the voltage measurements in a certain logarithmic way reduces the nonlinearity in the relationship between the electrical properties and the measurements, allowing a reconstruction with fewer or only one minimization step. Furthermore, we propose a modification to the complete electrode model for EIT. The new model is shown to be compatible with experimental measurement data, while the increased regularity of the predicted electromagnetic potential improves convergence properties of numerical methods.

KW - inverse problems

KW - uncertainty quantification

KW - partial differential equations

KW - nonlinear least squares

KW - polynomial surrogate

KW - electrical impedance tomography

KW - complete electrode model

KW - thermal tomography

KW - inversio-ongelmat

KW - epävarmuuden kvantifiointi

KW - osittaisdifferentiaaliyhtälöt

KW - epälineaarinen pienimmän neliösumman menetelmä

KW - polynomisurrogaatti

KW - impedanssitomografia

KW - täydellinen elektrodimalli

KW - lämpötomografia

KW - inverse problems

KW - uncertainty quantification

KW - partial differential equations

KW - nonlinear least squares

KW - polynomial surrogate

KW - electrical impedance tomography

KW - complete electrode model

KW - thermal tomography

M3 - Doctoral Thesis

SN - 978-952-60-7611-9

T3 - Aalto University publication series DOCTORAL DISSERTATIONS

PB - Aalto University

ER -

Approximations and Surrogates for Computational Inverse Boundary Value Problems

Abstrakti

Tutkimusalat

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