E. Bruce Pitman, Abani K. Patra, and Keith Dalbey

Fast Construction an Emulators via Localization

Abstract

To make a Bayesian prediction of the chances of a volcanic hazard impacting a particular region requires an estimate of the mass flow consequent to an eruption, for tens of thousands of input parameters. These inputs include physical parameters, computational factors, and spatial locations. Mass flow estimates can be determined by computer simulations, which are often too slow to be used for all the necessary input evaluations. Statistical emulators provide a very fast procedure for estimating the mass flow, along with a measure of the error in that estimate. But construction of many classical emulators, such as the GAussian Stochastic Process emulator requires inversion of a covariance matrix whose dimension is equal to the number of inputs – again, too slow to be useful. To speed up the emulator construction, some down sample the input space, which ignores expensive and potentially important simulation results. Others propose truncating the covariance to a small-width diagonal band, which is easy to invert. Here we propose an alternative method. We construct a localized emulator around every point at which the mass flow is to be estimated, and tie these localized processes together in a hierarchical fashion. We show how this approach fits into a theory of Gauss-Markov Random Fields, to demonstrate the efficacy of the approach.