Some approximation methods for Bayesian inversion of electrical impedance tomography

Abstract

Electrical impedance tomography (EIT) is a non-invasive imaging technique where the conductivity of an object is inferred through measurements on electrodes attached to its surface. EIT is well-known as a highly ill-posed nonlinear inverse problem, where the forward problem is modelled by an elliptic partial differential equation (PDE). Bayesian inferences using Markov chain Monte Carlo (MCMC) are computationally expensive because, for each iteration of MCMC, we need to solve a PDE. We propose and analyse the convergence rate of some approximation methods to reduce the computational cost of Bayesian computation in EIT. 1) Using multivariate Lagrange interpolation, we approximate the PDE forward solver by a polynomial surrogate. The set of interpolating nodes is chosen adaptively based on the importance of parameters. 2) We use a multi-level MCMC algorithm to approximate the posterior expectation. 3) We approximate the posterior distribution using adaptive mesh refinement to solve forward PDEs.