A Bayesian framework for local calibration of expensive computational models through non-isometric matching

We study statistical calibration, i.e., adjusting features of a computational model that are not observable or controllable in its associated physical system. We focus on local, or functional, calibration, which arises in many manufacturing processes where the unobservable features, called calibration variables, are a function of the input variables. A major challenge in many applications is that computational models are expensive and can only be evaluated a limited number of times. Furthermore, without making strong assumptions, the calibration variables are not identifiable. We propose Bayesian non-isometric matching calibration (BNMC) that allows calibration of expensive computational models with only a limited number of samples taken from a computational model and its associated physical system. BNMC replaces the computational model with a dynamic Gaussian Process (GP) whose parameters are trained in the calibration procedure. To resolve the identifiability issue, we present the calibration problem from a geometric perspective of non-isometric curve to surface matching, which enables us to take advantage of combinatorial optimization techniques to extract necessary information for constructing prior distributions. Our numerical experiments demonstrate that in terms of prediction accuracy BNMC outperforms, or is comparable to, other existing calibration frameworks.

Sparse Pseudo-input Local Kriging for Large Non-stationary Spatial Datasets with Exogenous Variables

We study large-scale spatial systems that contain exogenous variables, e.g. environmental factors that are significant predictors in spatial processes. Building predictive models for such processes involve two major challenges. First, the spatial processes are highly non-stationary and characterized by a heterogeneous covariance structure primarily due to the presence of exogenous variables. Second, it is inefficient to apply full Kriging because of the large numbers of observations present. The new theorems proposed in this paper form the basis for a new partitioning policy and a method, which we call Sparse Pseudo-input Local Kriging (SPLK). The proposed method handles heterogeneity in covariance and computational complexity by utilizing hyperplanes to partition a domain into smaller subdomains and then applying a sparse approximation of the full Kriging to each subdomain. We also develop a procedure to find the optimal hyperplanes. We impose continuity constraints on the boundaries of the neighboring subdomains to alleviate the problem of discontinuity of the global predictor. Numerical experiments demonstrate that SPLK outperforms, or is comparable to, the algorithms commonly applied to spatial datasets.