2020/12/23 by Jon Cockayne, Cockayne, Jon, Ilse C. F. Ipsen +5
Computer Science · Decision Sciences · Physics and Astronomy · #FOS: Computer and information sciences #FOS: Mathematics #Gaussian Processes and Bayesian Inference #Machine Learning (cs.LG) #Methodology (stat.ME) #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design
paper · doi:10.48550/arxiv.2012.12615
openalex publication_date 2020/12/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper presents a probabilistic perspective on iterative methods for approximating the solution x_* ∈ ℝd of a nonsingular linear system A x_* = 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.