Parallel Selected Inversion for Space-Time Gaussian Markov Random Fields

TL;DR

This paper introduces a parallel hybrid method combining domain decomposition and Krylov subspace techniques to efficiently perform Bayesian inference on large space-time Gaussian Markov Random Fields, enabling scalable computation of marginal variances and hyperparameters.

Contribution

It presents a novel divide-and-conquer approach that integrates Krylov subspace methods with direct factorizations for scalable Bayesian inference in large spatio-temporal models.

Findings

Achieves significant speed improvements over traditional methods.

Enables efficient hyperparameter estimation on massive datasets.

Demonstrates effectiveness on simulated and real US temperature data.

Abstract

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.