BayesCG As An Uncertainty Aware Version of CG

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. Our CG-based implementation of BayesCG under a structure-exploiting prior distribution represents an 'uncertainty-aware' version of CG. Its output consists of CG iterates and posterior covariances that can be propagated to subsequent computations. The covariances have low-rank and are maintained in factored form. This allows easy generation of accurate samples to probe uncertainty in downstream computations. Numerical experiments confirm the effectiveness of the low-rank posterior covariances.Comment: 33 Pages including supplementary material (main paper is 23 pages, supplement is 10 pages). Computer codes are available at https://github.com/treid5/ProbNumCG_Sup