Electrical Impedance Tomography using Nonconforming Mesh and Posterior Approximated Regression

Abstract

Electrical impedance tomography (EIT) is an imaging technique where the internal electrical conductivity distribution is reconstructed, using current and voltage measurements on the boundary. The contributions of this thesis are largely in three parts. In the first part of the thesis we present the formulation of the EIT problem using a nonconforming mesh. This is in contrast to the standard approach which uses a conforming mesh. The benefit of employing a nonconforming mesh, however, is that finer discretization can be employed on a localized region which is computationally more efficient. We mainly use the mortar element method as the numerical scheme, which is the state-of-the-art nonconforming finite element method today. In the second part of the thesis we focus on a situation where computational resources are severely limited yet the solutions to inverse problems need to be obtained quickly. Such is the case in process tomography where the solutions need to be obtained in the scale of milliseconds, in an online manner. The inverse problem in this thesis is addressed in the Bayesian statistical setting. The idea presented is to pose the Bayesian inverse problem as a statistical forward problem via the construction of a regression model. The regression model is constructed using samples of the unknowns and the data measurement, drawn from the related likelihood function and prior distribution. We call this the posterior approximated regression model. In the third part of the thesis we are interested in the prior modelling of discretized non-Gaussian random fields. By far the most used type of prior distribution in Bayesian inverse problems is the Gaussian distribution. However Gaussian priors tend to produce smoothing effects on the maximum a posteriori (MAP) estimate which can lead to underestimations of the unknowns. We investigate whether this can be partially resolved by fitting non-Gaussian prior distributions that are longer tailed than the Gaussian. The proposed formulation, method and strategy are verified via various simulated numerically EIT problems on two dimensional domains.