Abstract:
This thesis presents a methodology for solving an inverse problem: given a complete description of a physical system, we can synthesise the output data of some measurements. The problem of predicting the result of measurements is called the forward problem. The inverse problem is to estimate the parameters of the system from the measurements. In this thesis we show image reconstruction in electrical impedance tomography (EIT). In EIT, currents are injected into the body through electrodes which are attached on the boundary of the body. The resulting voltages are measured on the electrodes. Using different current patterns and voltage measurements, an approximation for the spatial distribution of conductivity inside the body can be reconstructed. Mathematically the EIT reconstruction problem is a nonlinear ill-posed inverse problem. In our research we use statistical inversion to reconstruct the image. The methods are presented from the Bayesian point of view. The stochastic formulation of an inverse problem requires a resolution in terms of samples from a posterior distribution. We have used a polygonal representation for the unknown insulating inclusion which defines the prior distribution. In the Bayesian paradigm, the problem is to summarise the posterior distribution for the unknown parameter, given the data and the prior information of the object. We used Markov chain Monte Carlo (MCMC) technique to draw samples from the target distribution and performed output analysis over the entire model space. i As this inverse problem is computationally intensive, we have considered the computational work load required to solve the forward problem within the likelihood evaluation. We investigated direct methods in our linear solver and have measured the computational cost for matrix updating when the matrix gets perturbed at each step of the MCMC run. Ultimately we found that we can use an iterative technique to get an approximate solution and can save CPU time. This approximation scheme is reasonable in terms of probabilistic sense and computational efficiency. Also we have established an algorithm which estimates the mean separation time between an approximate chain on the state space and the true chain that is constructed based on exact calculation of the likelihood. In this thesis, we give a general idea of this novel coupling separation algorithm for efficient MCMC and show a simple example of how this algorithm works in practice. We conclude by stating some open questions for future research.