LOW-COST MULTI-DIMENSIONAL GAUSSIAN PROCESS WITH APPLICATION TO UNCERTAINTY QUANTIFICATION

Computer codes simulating physical systems often have responses that consist of a set of distinct outputs that evolve in space and time and depend on many uncertain input parameters. The high dimensional nature of these computer codes makes the computations of Gaussian process (GP)-based emulators infeasible, even for a small number of simulation runs. In this paper we develop a covariance function for the GP to explicitly treat the covariance among distinct output variables, input variables, spatial domain, and temporal domain and also allows for Bayesian inference at low computational cost. We base our analysis on a modified version of the linear model of coregionalization (LMC). The proper use of the conditional representation of the multivariate output and the separable model for different domains leads to a Kronecker product representation of the covariance matrix. Moreover, we introduce a nugget to the model which leads to better statistical properties (regarding predictive accuracy) of the multivariate GP without adding to the overall computational complexity. Finally, the prior specification of the LMC parameters allows for an efficient Markov chain Monte Carlo (MCMC) algorithm. Our approach is demonstrated on the Kraichnan-Orszag problem and Flow through randomly heterogeneous porous media.