Performing Bayesian inference on large spatio-temporal models requires extracting inverse elements of large sparse precision matrices for marginal variances, as well as estimating model hyperparameters. Although direct matrix factorizations can be used for the inversion, such methods fail to scale well for distributed problems when run on large computing clusters. On the contrary, Krylov subspace methods for the selected inversion have been gaining traction. We propose a parallel hybrid approach based on domain decomposition, which extends the Rao-Blackwellized Monte Carlo estimator for distributed precision matrices. Our approach exploits the strength of Krylov subspace methods as global solvers and efficiency of direct factorizations as base case solvers to compute the marginal variances and the derivatives required for hyperparameter estimation using a divide-and-conquer strategy. By introducing subdomain overlaps, one can achieve greater accuracy at an increased computational effort with little to no additional communication. We demonstrate the speed improvements and efficient hyperparameter inference on both simulated models and a massive US daily temperature data.

翻译：对大规模时空模型进行贝叶斯推断时，需从大型稀疏精度矩阵中提取逆矩阵元素以计算边际方差，并估计模型超参数。尽管直接矩阵分解可用于求逆，但此类方法在大型计算集群上运行时难以扩展。相比之下，基于Krylov子空间的选择性求逆方法正逐渐受到关注。本文提出一种基于区域分解的并行混合方法，扩展了分布式精度矩阵的Rao-Blackwellized蒙特卡洛估计器。该方法融合了Krylov子空间方法作为全局求解器的优势，以及直接分解作为基础求解器的高效性，通过分治策略计算边际方差及超参数估计所需的导数。通过引入子区域重叠，可在几乎不增加通信开销的前提下，以更高计算代价实现更高精度。我们在模拟模型及海量美国日温度数据上验证了该方法的提速效果与高效超参数推断能力。