The issue of this work is to discuss numerical methods and computational techniques that can be applied to the discretized mathematical model of the EIT problem and also to use them in connection with some demonstrative numerical simulations. The computational work that has to be performed before resulting in a proper solution is usually large and can often be diminished remarkably by optimizing the efficiency of the applied numerical methods.
The major interest is concentrated on solving the inverse problem in a statistical sense by treating the conductivity as a random variable with some posterior probability distribution and by employing Markov chain Monte Carlo (MCMC) sampling methods for estimating the properties of the posterior distribution. The purpose is to develop such a Monte Carlo algorithm that finding a proper approximative solution would necessitate as small sample enesembles as possible. Drawing a sample from the posterior distribution demands for solving one or more forward problems, i.e. linear systems. Consequently, another important issue is to discover an effective linear algebraic method of solving the forward problem. Statistical solutions are measured against least-squares solutions which appear more frequently in literature.
In the simulations, we restrict ourselves to cases where the conductivity distribution is given in most parts of the body and only a relatively small anomaly is sought. The need for a method of locating small perturbations arises in connection with various real world applications of EIT such as detecting and classifying tumors from breast tissue.
Summarizing the findings, due to the strong ill-conditioned nature
and non-linearity of the inverse problem it is often difficult to
obtain any appropriate numerical solutions.Workability of the
least-squares approach depends on the applied regularization
method. It is difficult to construct regularization method
favoring arbitrary structures, e.g. strongly discontinuous
conductivities. The statistical model is preferable to the
least-squares approach only if there is accurate enough a priori
knowledge available.
Kenrick Bingham 16.10.2003 |