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Statistical properties of BayesCG under the Krylov prior

Lookup NU author(s): Professor Chris Oates

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Abstract

© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature. We analyse the calibration of BayesCG under the Krylov prior. BayesCG is a probabilistic numeric extension of the Conjugate Gradient (CG) method for solving systems of linear equations with real symmetric positive definite coefficient matrix. In addition to the CG solution, BayesCG also returns a posterior distribution over the solution. In this context, a posterior distribution is said to be ‘calibrated’ if the CG error is well-described, in a precise distributional sense, by the posterior spread. Since it is known that BayesCG is not calibrated, we introduce two related weaker notions of calibration, whose departures from exact calibration can be quantified. Numerical experiments confirm that, under low-rank approximate Krylov posteriors, BayesCG is only slightly optimistic and exhibits the characteristics of a calibrated solver, and is computationally competitive with CG.

Publication metadata

Author(s): Reid TW, Ipsen ICF, Cockayne J, Oates CJ

Publication type: Article

Publication status: Published

Journal: Numerische Mathematik

Year: 2023

Volume: 155

Pages: 239-288

Print publication date: 01/12/2023

Online publication date: 12/10/2023

Acceptance date: 14/09/2023

ISSN (print): 0029-599X

ISSN (electronic): 0945-3245

Publisher: Springer Nature

URL: https://doi.org/10.1007/s00211-023-01375-7

DOI: 10.1007/s00211-023-01375-7

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