Sobolev Calibration of Imperfect Computer Models

TL;DR

The paper develops Sobolev calibration for imperfect computer models by defining an identifiable calibration parameter through minimizing discrepancies in a Sobolev space $\mathcal{H}_m(\Omega)$. It establishes consistency, $\sqrt{n}$-consistency, asymptotic normality, and semiparametric efficiency, and shows how $L_2$ calibration and Kennedy–O'Hagan calibration are special cases when $m=0$ or for certain kernel choices. The method offers a tunable trade-off between function value accuracy and shape preservation via $m$, with practical computation via RKHS norms and kernel ridge surrogates, and it extends to Gaussian-process physical experiments. Numerical simulations and an ion-channel real-data example illustrate the flexibility and competitive performance of Sobolev calibration, validating its theoretical properties and showcasing its applicability to real-world calibration tasks. Overall, the framework unifies and extends key calibration approaches, enabling principled uncertainty quantification and downstream predictive fidelity control in imperfect computer models.

Abstract

Calibration refers to the statistical estimation of unknown model parameters in computer experiments, such that computer experiments can match underlying physical systems. This work develops a new calibration method for imperfect computer models, Sobolev calibration, which can rule out calibration parameters that generate overfitting calibrated functions. We prove that the Sobolev calibration enjoys desired theoretical properties including fast convergence rate, asymptotic normality and semiparametric efficiency. We also demonstrate an interesting property that the Sobolev calibration can bridge the gap between two influential methods: $L_2$ calibration and Kennedy and O'Hagan's calibration. In addition to exploring the deterministic physical experiments, we theoretically justify that our method can transfer to the case when the physical process is indeed a Gaussian process, which follows the original idea of Kennedy and O'Hagan's. Numerical simulations as well as a real-world example illustrate the competitive performance of the proposed method.

Figures (8)

• Figure 1: Visualization of the calibrated computer models in Example 1 of numerical experiments. The Sobolev calibration, the KO calibration, and the $L_2$ calibration are presented in dotted lines, while the physical experiment is presented in a solid line.
• Figure 2: Visualization of $\delta_1=\sin(10\pi x)$ and $\delta_2=\sin(2\pi x)$ over $\Omega=[0,1]$.
• Figure H.1: Visualization of Example \ref{['dtm-case1']} in the simulation studies. Panel (a) presents discrepancy measured by different normalized norms. The symbols (square, circle, triangle, etc.) together with vertical small dotted lines represent the minimizer of different norms, and the solid lines show $\|f_p(\cdot)-f_s(\cdot,\theta)\|_{\mathcal{H}_m(\Omega)}$ and $\theta^*_{S}$. Panel (b) shows the calibrated computer models (in dotted lines) using the Sobolev calibration, the KO calibration, and the $L_2$ calibration, respectively, together with the physical experiment (in solid line).
• Figure H.2: Visualization of Example \ref{['dtm-case2']} in the simulation studies. Panel (a) presents discrepancy measured by different normalized norms. The symbols (square, circle, triangle, etc.) together with vertical small dotted lines represent the minimizer of different norms, and the solid lines show $\|f_p(\cdot)-f_s(\cdot,\theta)\|_{\mathcal{H}_m(\Omega)}$ and $\theta^*_{S}$. Panel (b) shows the calibrated computer models (in dotted lines) using the Sobolev calibration, the KO calibration, and the $L_2$ calibration, respectively, together with the physical experiment (in solid line).
• Figure H.3: Visualization of the calibrated computer models (in dotted lines) using the Sobolev calibration, the KO calibration, and the $L_2$ calibration, respectively, together with the physical experiment (in solid line).

Theorems & Definitions (21)

• Remark 1
• Proposition 2.1
• Definition 1: Powers of RKHS
• Remark 2
• Proposition 3.1: Consistency of $\widehat{\bm{\theta}}_S$
• Theorem 3.2
• Theorem 3.3
• Corollary 3.4: Theorem 1 of tuowu2014
• Corollary 3.5
• Proposition 4.1: Consistency of $\widehat{\bm{\theta}}_S$