  Projected kernel calibration is known to be theoretically superior, its loss
function is abbreviated as PK loss function. In this work, we prove the uniform
convergence of PK loss function and show that (1) when the sample size is
large, any local minimum point and local maximum point of the $L_2$ loss
between the true process and the computer models is a local minimum point of
the PK loss function; (2) all the local minimum values of the PK loss function
converge to the same value. These theoretical results imply that it is
extremely hard for the projected kernel calibration to identify the global
minimum point of the $L_2$ loss which is defined as the optimal value of the
calibration parameters. To solve this problem, a frequentist method, called the
penalized projected kernel calibration method is proposed. As a frequentist
method, the proposed method is proved to be semi-parametric efficient. On the
other hand, the proposed method has a natural bayesian version, which allows
users to calculate the credible region of the calibration parameters without
using a large sample approximation. Through extensive simulation studies and a
real-world case study, we show that the proposed calibration can accurately
estimate the calibration parameters, and compare favorably to alternative
calibration methods regardless of the sample size.
