  In this paper a new approach for constructing \emph{multivariate} Gaussian
random fields (GRFs) using systems of stochastic partial differential equations
(SPDEs) has been introduced and applied to simulated data and real data. By
solving a system of SPDEs, we can construct multivariate GRFs. On the
theoretical side, the notorious requirement of non-negative definiteness for
the covariance matrix of the GRF is satisfied since the constructed covariance
matrices with this approach are automatically symmetric positive definite.
Using the approximate stochastic weak solutions to the systems of SPDEs,
multivariate GRFs are represented by multivariate Gaussian \emph{Markov} random
fields (GMRFs) with sparse precision matrices. Therefore, on the computational
side, the sparse structures make it possible to use numerical algorithms for
sparse matrices to do fast sampling from the random fields and statistical
inference. Therefore, the \emph{big-n} problem can also be partially resolved
for these models. These models out-preform existing multivariate GRF models on
a commonly used real dataset.
