Statistical properties of BayesCG under the Krylov prior

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 po...

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Bibliographic Details
Published in:Numerische Mathematik 2023-12, Vol.155 (3-4), p.239-288
Main Authors: Reid, Tim W., Ipsen, Ilse C. F., Cockayne, Jon, Oates, Chris J.
Format: Article
Language:English
Subjects:
Calibration
Conjugate gradient method
Linear equations
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Matrices (mathematics)
Numerical Analysis
Numerical and Computational Physics
Simulation
Statistical analysis
Theoretical
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Summary: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.
ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-023-01375-7