Probabilistic Iterative Methods for Linear Systems

Abstract

This paper presents a probabilistic perspective on iterative methods for approximating the solution x_* R^d of a nonsingular linear system A x_* = b. In the approach a standard iterative method on R^d is lifted to act on the space of probability distributions P(R^d). Classically, an iterative method produces a sequence x_m of approximations that converge to x_*. The output of the iterative methods proposed in this paper is, instead, a sequence of probability distributions _m P(R^d). The distributional output both provides a "best guess" for x_*, for example as the mean of _m, 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 _m 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.

Tasks

Reproductions