  The article studies non-Gaussian extensions of a recently discovered link
between certain Gaussian random fields, expressed as solutions to stochastic
partial differential equations (SPDEs), and Gaussian Markov random fields. The
focus is on non-Gaussian random fields with Mat\'ern covariance functions, and
in particular we show how the SPDE formulation of a Laplace moving average
model can be used to obtain an efficient simulation method as well as an
accurate parameter estimation technique for the model. This should be seen as a
demonstration of how these techniques can be used, and generalizations to more
general SPDEs are readily available.
