A Theoretical Framework of the Scaled Gaussian Stochastic Process in Prediction and Calibration

Abstract

Model calibration or data inversion is one of the fundamental tasks in uncertainty quantification. In this work, we study the theoretical properties of the scaled Gaussian stochastic process (S-GaSP) for modeling the discrepancy between reality and the imperfect mathematical model. We establish an explicit connection between the Gaussian stochastic process (GaSP) and S-GaSP through the orthogonal series representation. The predictive mean estimator in the S-GaSP calibration model converges to reality at the same rate as the GaSP with suitable choices of the regularization and scaling parameters. We also show that the calibrated mathematical model in the S-GaSP calibration converges to the one that minimizes the loss between reality and the mathematical model, whereas the GaSP model with other, widely used covariance functions does not have this property. Numerical examples confirm the excellent finite sample performance of our approaches.