A Reproducing Kernel Hilbert Space Approach to Functional Calibration of Computer Models

Abstract

This article develops a frequentist solution to the functional calibration problem, where the value of a calibration parameter in a computer model is allowed to vary with the value of control variables in the physical system. The need of functional calibration is motivated by engineering applications where using a constant calibration parameter results in a significant mismatch between outputs from the computer model and the physical experiment. Reproducing kernel Hilbert spaces (RKHS) are used to model the optimal calibration function, defined as the functional relationship between the calibration parameter and control variables that gives the best prediction. This optimal calibration function is estimated through penalized least squares with an RKHS-norm penalty and using physical data. An uncertainty quantification procedure is also developed for such estimates. Theoretical guarantees of the proposed method are provided in terms of prediction consistency and consitency of estimating the optimal calibration function. The proposed method is tested using both real and synthetic data and exhibits more robust performance in prediction and uncertainty quantification than the existing parametric functional calibration method and a state-of-art Bayesian method.

Keywords:

• Calibration
• Computer experiment
• Reproducing kernel Hilbert spaces
• Smoothing splines
• Uncertainty quantification

Supplementary Materials

The supplementary materials contain the proofs of Theorems 1–4.

Acknowledgment

We are grateful to the associate editor and reviewers for many valuable comments which helped significantly improve previous versions of the article.

Additional information

Funding

Shiyuan He’s work was supported by NSFC Project 11801561. Ding’s work was partially supported by NSF grants CMMI-1545038, IIS-1849085, CCF-1934904. Huang’s work was partially supported by NSF grants DMS-1208952, IIS-1900990, CCF-1956219. The first two authors, Tuo and He, made equal contributions to the article.