A theoretical framework of the scaled Gaussian stochastic process in prediction and calibration

TL;DR

This paper develops a theoretical framework for the scaled Gaussian stochastic process (S-GaSP) in model calibration, showing its convergence properties and advantages over traditional Gaussian processes in uncertainty quantification.

Contribution

It establishes the explicit connection between GaSP and S-GaSP, demonstrating the superior convergence and calibration properties of S-GaSP in uncertainty quantification tasks.

Findings

Predictive mean estimator in S-GaSP converges at the same rate as GaSP.

Calibrated S-GaSP models converge to the true model minimizing L2 loss.

Numerical examples show S-GaSP's excellent finite sample performance.

Abstract

Model calibration or data inversion is one of fundamental tasks in uncertainty quantification. In this work, we study the theoretical properties of the scaled Gaussian stochastic process (S-GaSP), to model the discrepancy between reality and imperfect mathematical models. We establish the explicit connection between Gaussian stochastic process (GaSP) and S-GaSP through the orthogonal series representation. The predictive mean estimator in the S-GaSP calibration model converges to the reality at the same rate as the GaSP with a suitable choice of the regularization and scaling parameters. We also show the calibrated mathematical model in the S-GaSP calibration converges to the one that minimizes the $L_2$ loss between the reality and 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 compared to a few recent approaches.