Probabilistic Iterative Methods for Linear Systems
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This paper presents a probabilistic perspective on iterative methods for approximating the solution 𝐱_* ∈ℝ^d of a nonsingular linear system 𝐀𝐱_* = 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 μ_m ∈𝒫(ℝ^d). The distributional output both provides a "best guess" for 𝐱_*, for example as the mean of μ_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 μ_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.
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