On applying stochastic Galerkin finite element method to electrical impedance tomography

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School of Science | Doctoral thesis (article-based) | Defence date: 2015-11-06
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Date

2015

Department

Matematiikan ja systeemianalyysin laitos
Department of Mathematics and Systems Analysis

Major/Subject

Mcode

Degree programme

Language

en

Pages

38 + app. 129

Series

Aalto University publication series DOCTORAL DISSERTATIONS, 135/2015

Abstract

In this thesis, a new solution strategy based on stochastic Galerkin finite element method is introduced for the complete electrode model of electrical impedance tomography. The method allows writing an analytical approximation for the solution to the inverse problem of electrical impedance tomography in the setting of Bayesian inversion with the help of multivariate orthogonal polynomials. If the measurement setting, i.e., geometry, priors, etc., is known (well) in advance, most computations required by the introduced method can be performed and stored before the actual measurement. The formation of the approximative solution to the inverse problem, i.e., the posterior probability density, is practically free of charge once the measurements are available. Subsequently, estimates for the quantities of interest can typically be obtained by either minimizing an explicitly known polynomial or integrating a known analytical expression. In addition, some advances in the development of numerical solvers for parametric partial differential equations in the setting of generalized Polynomial Chaos and stochastic Galerkin finite element method are presented.

Väitöskirjassa esitellään uusi stokastiseen Galerkinin elementtimenetelmään perustuva ratkaisutekniikka impedanssitomografian täydelliselle elektrodimallille. Tekniikka mahdollistaa analyyttisen approksimaation kirjoittamisen impedanssitomografian käänteisongelman bayesiläiselle ratkaisulle usean muuttujan ortogonaalipolynomien avulla. Mikäli impedanssitomografian mittausasetelma tunnetaan ennakkoon, voidaan suurin osa menetelmän vaatimista laskutoimituksista suorittaa ja varastoida ennen varsinaisen mittauksen suorittamista. Mittausten jälkeen approksimaatio inversio-ongelman ratkaisulle eli niin sanotulle posterioritiheysfunktiolle voidaan muodostaa ilman raskasta laskentaa. Tämän jälkeen haluttuja ratkaisuestimaatteja voidaan tyypillisesti laskea joko minimoimalla eksplisiittisesti tunnettua polynomia tai integroimalla tunnettua analyyttistä lauseketta. Lisäksi väitöskirjassa kehitetään yleistettyyn polynomikaaokseen ja stokastiseen Galerkinin elementtimenetelmään pohjautuvia ratkaisumenetelmiä parametririippuville osittaisdifferentiaaliyhtälöille.

Description

Supervising professor

Hyvönen, Nuutti, Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland

Thesis advisor

Hyvönen, Nuutti, Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland
Hakula, Harri, Dr, Aalto University, Department of Mathematics and Systems Analysis, Finland

Keywords

inverse problems, electrical impedance tomography, complete electrode model, stochastic Galerkin finite element method, generalized Polynomial Chaos, stochastic spectral methods, Bayesian inversion, stochastic elliptic partial differential equations, inversio-ongelmat, impedanssitomografia, täydellinen elektrodimalli, stokastinen Galerkinin elementtimenetelmä, yleistetty polynomikaaos, stokastiset spektraalimenetelmät, bayesiläinen inversio, stokastiset elliptiset osittaisdifferentiaaliyhtälöt

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Parts

  • [Publication 1]: Leinonen, M., Hakula, H., and Hyvönen, N. Application of stochastic Galerkin FEM to the complete electrode model of electrical impedance tomography. Journal of Computational Physics, Vol. 269, pp. 181–200, 2014.
    DOI: 10.1016/j.jcp.2014.03.011 View at publisher
  • [Publication 2]: Hakula, H., Hyvönen, N., and Leinonen, M. Reconstruction algorithm based on stochastic Galerkin finite element method for electrical impedance tomography. Inverse Problems, Vol. 30 (6), 065006, 17 pp., 2014.
    DOI: 10.1088/0266-5611/30/6/065006 View at publisher
  • [Publication 3]: Hyvönen, N., and Leinonen, M. Stochastic Galerkin finite element method with local conductivity basis for electrical impedance tomography. Accepted for publication in SIAM/ASA Journal on Uncertainty Quantification, 21 pp., 2015.
  • [Publication 4]: Hakula, H., and Leinonen, M. On efficient construction of stochastic moment matrices. arXiv:1502.07562, 39 pp., 2015.

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https://urn.fi/URN:ISBN:978-952-60-6381-2

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[diss] Perustieteiden korkeakoulu / SCI