Probabilistic numerical linear solvers

Abstract: We formulate iterative methods for the solution of nonsingular linear systems as statistical inference processes by modeling the epistemic uncertainty in the iterates due to a limited computational budget. The goal is to obtain well-calibrated uncertainty that is more insightful than traditional worst-case bounds, and to produce a probabilistic description of the error that can be propagated coherently through a computational pipeline.

Our Bayesian Conjugate Gradient Method (BayesCG) for real symmetric positive-definite linear systems posits a prior distribution for the solution, and conditions on the finite amount of information obtained during the iterations to produce a posterior distribution that reflects the reduced uncertainty. The following topics will be addressed: (i) choice of prior for fast convergence and well-calibrated uncertainty; (ii) error estimation through test statistics that mitigate the effect of BayesCG's nonlinear dependence on the solution; and (iii) numerical stability to maintain positive semi-definiteness of the posteriors, and prevent convergence slow down from loss of orthogonality in residuals and search directions.

This is joint work with Jon Cockayne (http://www.joncockayne.com/), Chris J. Oates (http://oates.work/), and Timothy W. Reid (https://math.sciences.ncsu.edu/people/twreid/).

numerical analysis

Audience: researchers in the topic

E-NLA - Online seminar series on numerical linear algebra

Series comments: E-NLA is an online seminar series dedicated to topics in Numerical Linear Algebra. Talks take place on Wednesdays at 4pm (Central European Time) via Zoom and are initially scheduled on a weekly basis.

To join the seminar, please complete the sign up form at the bottom of the webpage. Information about how to connect to the conference call will be circulated via email to all registered attendees.