Efficient Markov chain Monte Carlo sampling for electrical impedance tomography

Erfang Ma

Erfang Ma Department of Mathematical Sciences, Xi’an Jiaotong Liverpool University

Abstract

This paper studies electrical impedance tomography (EIT) using Bayesian inference [1]. The resulting posterior distribution is sampled by Markov chain Monte Carlo (MCMC) [2]. This paper studies a toy model of EIT as the one presented in [3], and focuses on efficient MCMC sampling for this model. First, this paper analyses the computation of forward map of EIT which is the bottleneck of each MCMC update. The forward map is computed by the finite element method [4]. Here its exact computation was conducted up to five times more efficient, by updating the Cholesky factor of the stiffness matrix [5]. Since the forward map computation takes up nearly all the CPU time in each MCMC update, the overall efficiency of MCMC algorithms can be improved almost to the same amount. The forward map can also be computed approximately by local linearisation, and this approximate computation is much more efficient than the exact one. Without loss of efficiency, this approximate computation is more accurate here, after a log transformation is introduced into the local linearisation process. Later on, this improvement of accuracy will play an important role when the approximate computation of forward map will be employed for devising efficient MCMC algorithms. Second, the paper presents two novel MCMC algorithms for sampling the posterior distribution in the toy model of EIT. The two algorithms are made within the 'multiple prior update' [6] and 'the delayed-acceptance Metropolis-Hastings' [7] schemes respectively. Both of them have MCMC proposals that are made of localized updates, so that the forward map computation in each MCMC update can be made efficient by updating the Cholesky factor of the stiffness matrix. Both algorithms' performances are compared to that of the standard single-site Metropolis [8], which is considered hard to surpass [3]. The algorithm of 'multiple prior update' is found to be six times more efficient, while 'the delayed-acceptance Metropolis-Hastings' with single-site update is at least twice more efficient.

Keywords

electrical impedance tomography, Bayesian inference, Markov chain Monte Carlo.,

References

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