  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.
