Statistical properties of BayesCG under the Krylov prior

Reid, Tim W., Ipsen, Ilse C.F., Cockayne, Jon and Oates, Chris J. (2023) Statistical properties of BayesCG under the Krylov prior. Numerische Mathematik, 155 (3-4), 239-288. (doi:10.1007/s00211-023-01375-7).

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Abstract

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.

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2208.03885 - Accepted Manuscript

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Accepted/In Press date: 14 September 2023

e-pub ahead of print date: 12 October 2023

Published date: December 2023

Additional Information: Funding Information: The work was supported in part by NSF Grant DMS-1745654 (TWR, ICFI), NSF Grant DMS-1760374 and DOE Grant DE-SC0022085 (ICFI), and the Lloyd’s Register Foundation Programme on Data Centric Engineering at the Alan Turing Institute (CJO)