Bayesian Uncertainty Propagation Using Gaussian Processes

Classic non-intrusive uncertainty propagation techniques, typically, require a significant number of model evaluations in order to yield convergent statistics. In practice, however, the computational complexity of the underlying computer codes limits significantly the number of observations that one can actually make. In such situations the estimates produced by classic approaches cannot be trusted since the limited number of observations induces additional epistemic uncertainty. The goal of this chapter is to highlight how the Bayesian formalism can quantify this epistemic uncertainty and provide robust predictive intervals for the statistics of interest with as few simulations as one has available. It is shown how the Bayesian formalism can be materialized by employing the concept of a Gaussian process (GP). In addition, several practical aspects that depend on the nature of the underlying response surface, such as the treatment of spatiotemporal variation, and multi-output responses are discussed. The practicality of the approach is demonstrated by propagating uncertainty through a dynamical system and an elliptic partial differential equation.