Probabilistic iterative methods for linear systems

Abstract

This paper presents a probabilistic perspective on iterative methods for approximating the solution x∗∈Rd of a nonsingular linear system Ax∗=b. In the approach a standard iterative method on Rd is lifted to act on the space of probability distributions P(Rd). Classically, an iterative method produces a sequence xm of approximations that converge to x∗. The output of the iterative methods proposed in this paper is, instead, a sequence of probability distributions μm∈P(Rd). The distributional output both provides a "best guess" for x∗, for example as the mean of μm, and also probabilistic uncertainty quantification for the value of x∗ when it has not been exactly determined. Theoretical analysis is provided in the prototypical case of a stationary linear iterative method. In this setting we characterise both the rate of contraction of μm to an atomic measure on x∗ and the nature of the uncertainty quantification being provided. We conclude with an empirical illustration that highlights the insight into solution uncertainty that can be provided by probabilistic iterative methods.

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

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