Probabilistic iterative methods for linear systems
Probabilistic iterative methods for linear systems
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.
Cockayne, Jon
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Ipsen, Ilse C.F.
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Oates, Chris J.
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Reid, Tim W.
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Cockayne, Jon
da87c8b2-fafb-4856-938d-50be8f0e4a5b
Ipsen, Ilse C.F.
83eae4c2-19d4-4f74-9d16-4146a63d2c4c
Oates, Chris J.
faae6d14-7a66-4ca3-a6ba-daaf1938e164
Reid, Tim W.
8ab4ae9b-b21e-4fa4-ba9c-8a4bf9bc7cc9
Cockayne, Jon, Ipsen, Ilse C.F., Oates, Chris J. and Reid, Tim W.
(2021)
Probabilistic iterative methods for linear systems.
Journal of Machine Learning Research, 22.
(In Press)
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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Probabilistic Iterative Methods for Linear Systems
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Accepted/In Press date: 2 August 2021
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Local EPrints ID: 451487
URI: http://eprints.soton.ac.uk/id/eprint/451487
ISSN: 1532-4435
PURE UUID: 88d74419-0200-463c-8e21-b17e4d87d2eb
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Date deposited: 30 Sep 2021 16:34
Last modified: 06 Nov 2024 03:04
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Contributors
Author:
Ilse C.F. Ipsen
Author:
Chris J. Oates
Author:
Tim W. Reid
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