  Generating large-scale samples of stationary random fields is of great
importance in the fields such as geomaterial modeling and uncertainty
quantification. Traditional methodologies based on covariance matrix
decomposition have the diffculty of being computationally expensive, which is
even more serious when the dimension of the random field is large. This paper
proposes an effcient stochastic realization approach for sampling Gaussian
stationary random fields from a systems and control point of view.
Specifically, we take the exponential and Gaussian covariance functions as
examples and make a decoupling assumption when there are multiple dimensions.
Then a rational spectral density is constructed in each dimension using
techniques from covariance extension, and the corresponding autoregressive
moving-average (ARMA) model is obtained via spectral factorization. As a
result, samples of the random field with a specific covariance function can be
generated very effciently in the space domain by implementing the ARMA
recursion using a white noise input. Such a procedure is computationally cheap
due to the fact that the constructed ARMA model has a low order. Furthermore,
the same method is integrated to multiscale simulations where interpolations of
the generated samples are achieved when one zooms into finer scales. Both
theoretical analysis and simulation results show that our approach performs
favorably compared with covariance matrix decomposition methods.
