Title: Sensitivity of uncertainty quantification and Bayesian inverse problems
Author: Bjoern Sprungk (TU Bergakademie Freiberg)

For a probabilistic uncertainty quantification analysis of, e.g., a partial differential equation with uncertain coefficients we need to postulate a probability distribution for the unknown parameters. This distribution is often based on subjective knowledge or statistical estimation given experimental data. In this talk we investigate the sensitivity of the resulting distribution of the random solution with respect to perturbations in the input distribution for the unknown parameters. We prove a local Lipschitz continuity with respect to total variation as well as Wasserstein distance and extend our sensitivity analysis also to risk functionals of quantities of interest of the solution. Here, we provide a novel result for the sensitivity of coherent risk functionals with respect to the underlying probability distribution.
Besides these sensitivity results for the propagation of uncertainty and risk, we also investigate the inverse problem, i. e., Bayesian inference for the unknown coefficients given noisy observations of the solution. Although, well-posedness of Bayesian inverse problems is well-known (with respect to the observational data), we establish the local Lipschitz continuity of the posterior with respect to pertubations of the (again subjective) prior. We consider continuity in Wasserstein distance as well as with respect to several other common metrics for probability measures. However, our explicit bounds indicate a growing sensitivity of Bayesian inference for increasingly informative observational data.