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International Journal for Uncertainty Quantification

Impact factor: 1.000

ISSN Print: 2152-5080
ISSN Online: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2015007941
pages 375-392

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

Bledar A. Konomi
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221, USA
Guang Lin
Computational Science & Mathematics Division, Pacific Northwest National Laboratory, Richland, Washington 99352; Department of Mathematics, School of Mechanical Engineering, Purdue University, West Lafayette, Indiana, USA

ABSTRACT

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.