BayesCG As An Uncertainty Aware Version of CG

Abstract

The Bayesian Conjugate Gradient method (BayesCG) is a probabilistic generalization of the Conjugate Gradient method (CG) for solving linear systems with real symmetric positive definite coefficient matrices. We present a CG-based implementation of BayesCG with a structure-exploiting prior distribution. The BayesCG output consists of CG iterates and posterior covariances that can be propagated to subsequent computations. The covariances are low-rank and maintained in factored form. This allows easy generation of accurate samples to probe uncertainty in subsequent computations. Numerical experiments confirm the effectiveness of the posteriors and their low-rank approximations.Comment: 31 Pages including supplementary material (main paper is 22 pages, supplement is 9 pages). Computer codes are available at https://github.com/treid5/ProbNumCG_Sup