# Direct and Inverse Problems for PDEs with Random Coefficients - Abstract

**Scheichl, Robert**

*Multilevel sampling techniques for Bayesian inference*

Large-scale PDE-constrained Bayesian inference is an inherently difficult and computationally intensive problem of huge interest in a vast array of applications. Sampling methods are among the most accurate and promising methods for these problems, as their cost is dimension independent thus rendering them very suitable for the infinite-dimensional PDE setting. However, most standard sampling approaches (such as Metropolis-Hastings MCMC) are very slow to converge and hence intractable for realistic applications. Nevertheless, typical approximation schemes to numerically solve the underlying PDEs are naturally hierarchical, paving the way for powerful multilevel sampling methods, for example within a Metropolis-Hastings setting or as ratio estimators for Bayes' formula. Through a clever use of the model hierarchies, they offer the accuracy of “gold-standard” classical MCMC estimators at a fraction of the cost, avoiding dimension truncation, Gaussian approximations or linearisations. We present theory and numerical experiments confirming these massive improvements on a typical model problem from subsurface flow.