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.

翻译：贝叶斯推断任务持续面临计算挑战，特别是在高维潜在参数空间普遍存在的时空建模领域。集成嵌套拉普拉斯逼近（INLA）方法学为适用于大规模可加贝叶斯层次模型子类的贝叶斯推断提供了框架。结合随机偏微分方程（SPDE）方法，该框架形成了高效的时空建模方案。本研究在INLA-SPDE方法基础上，提出了一种面向大规模应用的高性能分布式内存变体——INLA-DIST。为执行由Cholesky分解、线性系统求解及选择性矩阵求逆组成的计算核心操作，我们提供了两种数值求解器选项：基于CPU的稀疏库与本文提出的新型GPU加速分块方法。我们利用了所生成精度（逆协方差）矩阵中重复出现的非零块结构，从而能够在稀疏计算环境中使用密集子例程。两种版本的INLA-DIST均具备高度可扩展性，能够对包含数百万潜在参数的模型进行推断。我们通过合成数据集和真实气候数据集应用验证了其精度与性能表现。