We present a Conjugate Gradient (CG) implementation of the probabilistic numerical solver BayesCG , whose error estimates are a fully integrated design feature, easy to compute, and competitive with the best existing estimators. More specifically, we extend BayesCG to singular prior covariances, derive recursions for the posterior covariances, express the posteriors as projections, and establish that BayesCG retains the minimization properties over Krylov spaces regardless of the singular priors. We introduce a possibly singular Krylov prior covariance, under which the BayesCG posterior means coincide with the CG iterates and the posteriors can be computed efficiently. Because of its factored form, the Krylov prior is amenable to low-rank approximation, which produces an efficient BayesCG implementation as a CG method. We also introduce a probabilistic error estimator, the `S-statistic'. Although designed for sampling from BayesCG posteriors, its mean and variance under approximate Krylov priors can be computed with CG. An approximation of the S-statistic by a `95 percent credible interval' avoids the cost of sampling altogether .  Numerical experiments illustrate that the resulting error estimates are competitive with the best existing methods and are easy to compute .
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 .
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We present a Conjugate Gradient
The Bayesian Conjugate Gradient method (BayesCG) is a probabilistic generalization of the Conjugate Gradient method
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implementation of the probabilistic numerical solver BayesCG , whose error estimates are a fully integrated design feature, easy to compute, and competitive with the best existing estimators. More specifically, we extend BayesCG to singular prior covariances, derive recursions for the posterior covariances, express the posteriors as projections, and establish that BayesCG retains the minimization properties over Krylov spaces regardless of the singular priors. We introduce a possibly singular Krylov prior covariance, under which the BayesCG posterior means coincide with the
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
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the posteriors can be computed efficiently. Because of its factored form, the Krylov prior is amenable to
posterior covariances that can be propagated to subsequent computations. The covariances are
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approximation, which produces an efficient BayesCG implementation as a CG method. We also introduce a probabilistic error estimator, the `S-statistic'. Although designed for sampling from BayesCG posteriors, its mean and variance under approximate Krylov priors can be computed with CG. An approximation of the S-statistic by a `95 percent credible interval' avoids the cost of sampling altogether
and maintained in factored form
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This allows easy generation of accurate samples to probe uncertainty in subsequent computations.
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illustrate that the resulting error estimates are competitive with the best existing methods and are easy to compute
confirm the effectiveness of the posteriors and their low-rank approximations