Computational efficient single component Gibbs sampling for electrical tomography

Purpose In Bayesian inversion, the Gibbs sampler draws samples from the multivariate posteriori distribution by sequentially sampling from the conditional distributions of the individual parameters. This makes Gibbs sampling a preferable sampling technique with respect to other Markov chain Monte Carlo (MCMC) methods, e.g. the Metropolis Hastings algorithm, yet the evaluation of the conditional distributions is computationally expensive. This paper aims to present an efficient technique based on the Woodbury matrix identity to evaluate the conditional distributions of electrical tomography problems with an underlying finite element (FE) simulation. Design/methodology/approach The approach is based on a modified solution strategy of the used FE sensor simulation model. The computation of the conditional distribution requires the evaluation of the FE model for changes of the electrical material parameter of a single FE. This is formulated as a low rank update of the FE stiffness matrix. Using the modified solution approach, the change of the FE model output can be exactly evaluated by means of the Woodbury matrix identity. Findings Simulation results show a computational speed up of a Gibbs sampler with the proposed scheme for an electrical capacitance tomography example by a factor of at least 30 with respect to a standard implementation, i.e. without the Woodbury matrix identity. The algorithm has a short burn in phase and is able to provide independent samples by each sweep over the state vector. Practical implications The high computation times of MCMC techniques are a major drawback for their application, despite their potential advantages, such as the capability for uncertainty quantification (UQ). The acceleration of the Gibbs sampler with the proposed computational scheme and its potential capability to generate independent samples in the Markov chain provide an important tool for a broader application of Bayesian inversion techniques. Originality/value This paper shows a technique for the efficient evaluation of the conditional distribution for electrical tomography problems with an underlying FE simulation model. The derivation and implementation is coherent with the modified sensor simulation approach. The approach can also be applied within other inverse problems or algorithms, e.g. optimization-based techniques.