On applying stochastic Galerkin finite element method to electrical impedance tomography

Aalto University, P.O. Box 11000, FI-00076 Aalto www.aalto.fi Author Matti Leinonen Name of the doctoral dissertation On applying stochastic Galerkin finite element method to electrical impedance tomography Publisher School of Science Unit Department of Mathematics and Systems Analysis Series Aalto University publication series DOCTORAL DISSERTATIONS 135/2015 Field of research Mathematics Manuscript submitted 12 May 2015 Date of the defence 6 November 2015 Permission to publish granted (date) 14 August 2015 Language English Monograph Article dissertation (summary + original articles) 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.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.