Scalable Stochastic Kriging with Markovian Covariances

Stochastic kriging is a popular technique for simulation metamodeling due to its flexibility and analytical tractability. Its computational bottleneck is the inversion of a covariance matrix, which takes $O(n^3)$ time in general and becomes prohibitive for large $n$, where $n$ is the number of design points. Moreover, the covariance matrix is often ill-conditioned for large $n$, and thus the inversion is prone to numerical instability, resulting in erroneous parameter estimation and prediction. These two numerical issues preclude the use of stochastic kriging at a large scale. This paper presents a novel approach to address them. We construct a class of covariance functions, called Markovian covariance functions (MCFs), which have two properties: (i) the associated covariance matrices can be inverted analytically, and (ii) the inverse matrices are sparse. With the use of MCFs, the inversion-related computational time is reduced to $O(n^2)$ in general, and can be further reduced by orders of magnitude with additional assumptions on the simulation errors and design points. The analytical invertibility also enhance the numerical stability dramatically. The key in our approach is that we identify a general functional form of covariance functions that can induce sparsity in the corresponding inverse matrices. We also establish a connection between MCFs and linear ordinary differential equations. Such a connection provides a flexible, principled approach to constructing a wide class of MCFs. Extensive numerical experiments demonstrate that stochastic kriging with MCFs can handle large-scale problems in an both computationally efficient and numerically stable manner.