Rapid Uncertainty Quantification for Nonlinear Inverse Problems

In this thesis, we study a fast approximate inference method based on a technique called "Expectation Propagation" for exploring the posterior probability distribution arising from the Bayesian formulation of nonlinear inverse problems. It is capable of efficiently delivering reliable posterior mean and covariance estimates, thereby providing a solution to the inverse problem together with quantified uncertainties. Some theoretical properties of the iterative algorithm are discussed, and an efficient implementation for an important class of problems of projection type is described. The method is illustrated with two typical nonlinear inverse problems, electrical impedance tomography with complete electrode model and inverse scattering, under sparsity constraints. Numerical results for both with experimental data are presented, and compared with those by a Markov chain Monte Carlo method. The results indicate that the method is accurate and computationally highly efficient.