Scientific Computation and Applied & Industrial Mathematics Seminar: Johnathan M. Bardsley

Many solution techniques for inverse problems involve solving an optimization problem using a numerical method. For example, the Tikhonov regularized solution is commonly defined as the minimizer of a penalized least squares function. Uncertainty quantification (UQ), on the other hand, often requires sampling from the Bayesian posterior density function arising from the assumed physical model, measurement error model, and prior probability density function. In this talk, we bring these two computational approaches (numerical methods and sampling) together and present posterior sampling – and specifically Markov Chain Monte Carlo (MCMC) – methods for UQ that utilize existing numerical algorithms for solving inverse problems. In all cases, care is taken to make sure that the MCMC methods presented provide theoretically correct samples from the posterior density function. Moreover, we present MCMC methods for both linear and nonlinear inverse problems.