  Various natural phenomena exhibit spatial extremal dependence at short
spatial distances. However, existing models proposed in the spatial extremes
literature often assume that extremal dependence persists across the entire
domain. This is a strong limitation when modeling extremes over large
geographical domains, and yet it has been mostly overlooked in the literature.
We here develop a more realistic Bayesian framework based on a novel Gaussian
scale mixture model, with the Gaussian process component defined by a
stochastic partial differential equation yielding a sparse precision matrix,
and the random scale component modeled as a low-rank Pareto-tailed or
Weibull-tailed spatial process determined by compactly-supported basis
functions. We show that our proposed model is approximately tail-stationary and
that it can capture a wide range of extremal dependence structures. Its
inherently sparse structure allows fast Bayesian computations in high spatial
dimensions based on a customized Markov chain Monte Carlo algorithm
prioritizing calibration in the tail. We fit our model to analyze heavy monsoon
rainfall data in Bangladesh. Our study shows that our model outperforms natural
competitors and that it fits precipitation extremes well. We finally use the
fitted model to draw inference on long-term return levels for marginal
precipitation and spatial aggregates.
