Performing a Bayesian inference on large spatio-temporal models requires extracting inverse elements of large sparse precision matrices for marginal variances. 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 using a divide-and-conquer strategy. By introducing subdomain overlaps, one can achieve a greater accuracy at an increased computational effort with little to no additional communication. We demonstrate the speed improvements on both simulated models and a massive US daily temperature data.

翻译：针对大规模时空模型执行贝叶斯推断时，需要从大型稀疏精度矩阵中提取逆矩阵元素以计算边际方差。尽管直接矩阵分解可用于求逆，但在大型计算集群上处理分布式问题时，此类方法难以有效扩展。相比之下，基于Krylov子空间方法的选点求逆技术日益受到关注。本文提出一种基于区域分解的并行混合方法，将Rao-Blackwellized蒙特卡洛估计器扩展至分布式精度矩阵。该方法利用Krylov子空间方法作为全局求解器的优势，以及直接分解作为局部求解器的高效性，通过分治策略计算边际方差。通过引入子区域重叠，可在几乎无需额外通信的情况下，以增加计算开销为代价获得更高精度。我们在模拟模型和庞大的美国日气温数据集上验证了速度改进效果。