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.

翻译：校正是指通过统计方法估计计算机实验中的未知模型参数，使计算机实验能够匹配底层物理系统。本文针对非完美计算机模型提出一种新的校正方法——Sobolev校正，该方法能够排除生成过拟合校正函数的校正参数。我们证明了Sobolev校正具有理想的渐近性质，包括快速收敛率、渐近正态性和半参数有效性。我们还揭示了一个有趣特性：Sobolev校正能够弥合两种主流方法——$L_2$校正与Kennedy和O'Hagan校正之间的差距。除了探索确定性物理实验外，我们从理论上证明该方法可推广至物理过程为高斯过程的情形，这与Kennedy和O'Hagan的原始思想一脉相承。数值模拟与真实世界案例均验证了所提方法的优越性能。