Randomize-Then-Optimize for Sampling and Uncertainty Quantification in Electrical Impedance Tomography

In a typical inverse problem, a spatially distributed parameter in a physical model is estimated from measurements of model output. Since measurements are stochastic in nature, so is any parameter estimate. Moreover, in the Bayesian setting, the choice of regularization corresponds to the definition of the prior probability density function, which in turn is an uncertainty model for the unknown parameters. For both of these reasons, significant uncertainties exist in the solution of an inverse problem. Thus to fully understand the solution, quantifying these uncertainties is important. When the physical model is linear and the error model and prior are Gaussian, the posterior density function is Gaussian with a known mean and covariance matrix. However, the electrical impedance tomography inverse problem is nonlinear, and hence no closed form expression exists for the posterior density. The typical approach for such problems is to sample from the posterior and then use the samples to compute statistics (s...