Stochastic kriging is a popular metamodeling technique for constructing response surfaces of complex stochastic simulation models in a variety of disciplines including queueing simulation, financial risk management, insurance product pricing, etc. And it is well known that incorporating gradient information can significantly enhance the prediction accuracy of stochastic kriging, such as the gradient extrapolated stochastic kriging, which converts the gradient information to a “pseudo” observation of the response surface by linear extrapolation to expand the number of kriging points.

However, such an enhancement cannot be scaled trivially to high-dimensional design space, since one needs to invert a covariance matrix of size n(d +1) x n(d +1) that captures the spatial correlations betw...[ Read more ]

Stochastic kriging is a popular metamodeling technique for constructing response surfaces of complex stochastic simulation models in a variety of disciplines including queueing simulation, financial risk management, insurance product pricing, etc. And it is well known that incorporating gradient information can significantly enhance the prediction accuracy of stochastic kriging, such as the gradient extrapolated stochastic kriging, which converts the gradient information to a “pseudo” observation of the response surface by linear extrapolation to expand the number of kriging points.

However, such an enhancement cannot be scaled trivially to high-dimensional design space, since one needs to invert a covariance matrix of size n(d +1) x n(d +1) that captures the spatial correlations between the responses and the gradient estimates at the design points, where d and n are the dimensionality and the number of design points, respectively. Not only is the inversion computationally inefficient, but also numerically unstable since the covariance matrix is often ill-conditioned.

We address the scalability issue via a novel approach without resorting to matrix approximations. By virtue of the so-called Markovian covariance functions, the associated covariance matrix can be invertible analytically, thereby improving both the efficiency and stability dramatically. We extend this methodology to the setting where gradient estimators are available, thereby making the gradient-enhanced stochastic kriging metamodel scalable. Numerical experiments, including analytical cases and practical cases, demonstrate that the idea of integrating gradient information with SK can handle large-scale problems where prior methods fail completely.

In addition, we consider the case that with limited computing budget and non-constant intrinsic variance among large-scale response surface, we can also apply some optimal simulation allocation strategies to improve the prediction accuracy of both stochastic kriging and gradient extrapolated stochastic kriging, numerical experiments are conducted to prove our argument.