Discrepancy modeling for model calibration with multivariate output

This paper explores the application of the Kennedy and O'Hagan (KOH) Bayesian framework to the calibration of physics models with multivariate outputs by formulating the problem in a dimension-reduced subspace. The approach in the KOH framework is to calibrate the physics model parameters simultaneously to the parameters of an additive discrepancy (model error) function. It is a known issue that such discrepancy functions may result in non-identifiability between the model parameters and discrepancy function parameters. Three main approaches to avoid this problem have been considered in the literature: (i) careful definition of the parameter priors based on extensive knowledge of the problem physics, (ii) separating the calibration process into more than a single step (referred to as a modular or sequential solution), or (iii) choosing functions that are less flexible than a Gaussian process (GP). By transformation of the problem into a dimension-reduced principal components (PC) space [using PC analysis (PCA)], we explore a fourth approach to this problem. Advantages are dimension reduction of the calibration problem due to fewer outputs, simplified discrepancy functions and priors, and feasibility for the simultaneous solution approach. The additive discrepancy method is limited in the case of future predictions with the model; thus, we instead suggest how the results may be used for model diagnostic purposes. The methods are demonstrated on a simple numerical example and gas turbine engine heat transfer model.