  Bayesian inference tasks continue to pose a computational challenge. This
especially holds for spatial-temporal modeling where high-dimensional latent
parameter spaces are ubiquitous. The methodology of integrated nested Laplace
approximations (INLA) provides a framework for performing Bayesian inference
applicable to a large subclass of additive Bayesian hierarchical models. In
combination with the stochastic partial differential equations (SPDE) approach
it gives rise to an efficient method for spatial-temporal modeling. In this
work we build on the INLA-SPDE approach, by putting forward a performant
distributed memory variant, INLA-DIST, for large-scale applications. To perform
the arising computational kernel operations, consisting of Cholesky
factorizations, solving linear systems, and selected matrix inversions, we
present two numerical solver options, a sparse CPU-based library and a novel
blocked GPU-accelerated approach which we propose. We leverage the recurring
nonzero block structure in the arising precision (inverse covariance) matrices,
which allows us to employ dense subroutines within a sparse setting. Both
versions of INLA-DIST are highly scalable, capable of performing inference on
models with millions of latent parameters. We demonstrate their accuracy and
performance on synthetic as well as real-world climate dataset applications.
