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Various natural phenomena exhibit spatial extremal dependence at shortspatial distances. However, existing models proposed in the spatial extremesliterature often assume that extremal dependence persists across the entiredomain. This is a strong limitation when modeling extremes over largegeographical domains, and yet it has been mostly overlooked in the literature.We here develop a more realistic Bayesian framework based on a novel Gaussianscale mixture model, with the Gaussian process component defined by astochastic partial differential equation yielding a sparse precision matrix,and the random scale component modeled as a low-rank Pareto-tailed orWeibull-tailed spatial process determined by compactly-supported basisfunctions. We show that our proposed model is approximately tail-stationary andthat it can capture a wide range of extremal dependence structures. Itsinherently sparse structure allows fast Bayesian computations in high spatialdimensions based on a customized Markov chain Monte Carlo algorithmprioritizing calibration in the tail. We fit our model to analyze heavy monsoonrainfall data in Bangladesh. Our study shows that our model outperforms naturalcompetitors and that it fits precipitation extremes well. We finally use thefitted model to draw inference on long-term return levels for marginalprecipitation and spatial aggregates.

Realistic and Fast Modeling of Spatial Extremes over Large Geographical Domains