Probabilistic linear solvers: a unifying view

Abstract

Several recent works have developed a new, probabilistic interpretation for numerical algorithms solving linear systems in which the solution is inferred in a Bayesian framework, either directly or by inferring the unknown action of the matrix inverse. These approaches have typically focused on replicating the behaviour of the conjugate gradient method as a prototypical iterative method. In this work,surprisingly general conditions for equivalence of these disparate methods arepresented. We also describe connections between probabilistic linear solvers andprojection methods for linear systems, providing a probabilistic interpretation of afar more general class of iterative methods. In particular, this provides such aninterpretation of the generalised minimum residual method. A probabilistic view ofpreconditioning is also introduced. These developments unify the literature onprobabilistic linear solvers and provide foundational connections to the literatureon iterative solvers for linear systems.

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ORCID for Jonathan Cockayne: orcid.org/0000-0002-3287-199X

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