A Bayesian Conjugate Gradient Method

TL;DR

This paper introduces a Bayesian conjugate gradient method that provides probabilistic error estimates for solving large linear systems, enhancing the classical approach with statistical insights.

Contribution

It presents a novel Bayesian framework that generalizes the conjugate gradient method, offering improved error estimation and analysis within Krylov subspace methods.

Findings

The Bayesian approach yields meaningful error estimates in challenging systems.

The method demonstrates effective performance in medical imaging applications.

A contraction result for the posterior is established.

Abstract

A fundamental task in numerical computation is the solution of large linear systems. The conjugate gradient method is an iterative method which offers rapid convergence to the solution, particularly when an effective preconditioner is employed. However, for more challenging systems a substantial error can be present even after many iterations have been performed. The estimates obtained in this case are of little value unless further information can be provided about the numerical error. In this paper we propose a novel statistical model for this numerical error set in a Bayesian framework. Our approach is a strict generalisation of the conjugate gradient method, which is recovered as the posterior mean for a particular choice of prior. The estimates obtained are analysed with Krylov subspace methods and a contraction result for the posterior is presented. The method is then analysed in a simulation study as well as being applied to a challenging problem in medical imaging.