Solving the inverse three-dimensional continuous model of electrical resistance tomography using the method of fundamental solutions and the Markov chain Monte Carlo approach.

This paper discusses the three-dimensional continuous model of electrical resistance tomography for detecting rigid inclusions embedded in a bounded background medium. The mathematical model is governed by Laplace’s equation subject to a homogeneous Dirichlet boundary condition on the unknown rigid inclusion and Cauchy data on the outer boundary. The forward solver uses the meshless method of fundamental solutions, which is a discrete variant of the single layer potential in the direct form of the boundary element method for harmonic functions, but with source points shifted outside the solution domain such that singularities in the fundamental solution are avoided. The inverse solution is based on the Bayesian approach and Markov chain Monte Carlo (MCMC) estimation technique. The MCMC is used not only for estimating the desirable model parameters, but also for uncertainty and reliability assessment. Numerical examples are investigated to demonstrate the effectiveness and the accuracy of the proposed approach.