Application of Markov Chain Monte Carlo Methods in Electrical Impedance Tomography

Abstract

This thesis discusses the application of Markov chain Monte Carlo (MCMC) methods in electrical impedance tomography (EIT). This topic arises in the Bayesian approach to the reconstruction problem of EIT. This study is a derivative of the work of Higdon et al. (2011).

First, the computation of a forward map in EIT is discussed, since this computation is very important for the application of MCMC methods in EIT. The computation involves solving a boundary value problem using the finite element method (FEM). The convergence of the FEM approximation is studied numerically, and compelling evidence is provided for the validity of this approximation. Numerical experiments have shown that updating the Cholesky factor of stiffness matrix in FEM, can make the forward map computation much more efficient. The approximate computation of forward map is also discussed. This approximation uses a local linearisation of forward map, and is useful in making efficient MCMC proposals. The accuracy of this approximation is improved dramatically after a log-transformation on the variable of conductivity is introduced into this approximation. The improvement on the accuracy is later on shown to be crucial for this approximation to be employed in making an efficient proposal.

Second, this thesis extends the work of Higdon et al. (2011). The same EIT model as in Higdon et al. (2011) is considered here. As in Higdon et al. (2011), a Bayesian approach is employed to reconstruct the conductivity distribution. The resulting posterior distribution is sampled by single site Metropolis, Random walk Metropolis (RWM), differential evolution MCMC (DE-MCMC), and Directional Random Walk Metropolis (DRWM). Our results on the performance of these MCMC algorithms are generally consistent with those of Higdon et al. (2011). Single site Metropolis, RWM and DRWM have also been analysed on sampling a high-dimensional Gaussian distribution. For their performance, elegant formulae for the acceptance rate for these MCMC samplers are obtained. These formulae indicate these algorithms perform similarly in this case as well, thus provide valuable insights into these MCMC algorithms.

Finally and most importantly, this study comes up with some novel MCMC algorithms that dramatically outperform the standard single site Metropolis as in Higdon et al. (2011), for sampling the posterior distribution in EIT. According to Higdon et al. (2011), the performance of standard single site Metropolis was hard to surpass for this application. However, some of the novel MCMC algorithms in this thesis, when equipped with efficient computation of forward map, can be as many as 100 times more efficient than the standard single site Metropolis. These novel algorithms could be employed to explore posterior distribution of similar kind in other applications, e.g., electrical capacitance tomography. The success of these algorithms in this thesis indicates that they would also achieve equal success there.