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Mathematics of Computation

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Bayesian inverse problems with non-commuting operators

Author: Peter Mathé
Journal: Math. Comp. 88 (2019), 2897-2912
MSC (2010): Primary 62G05, 65J20; Secondary 62F15, 47A57
Published electronically: April 25, 2019
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Abstract: The Bayesian approach to ill-posed operator equations in Hilbert space recently gained attraction. In this context, and when the prior distribution is Gaussian, then two operators play a significant role, the one which governs the operator equation, and the one which describes the prior covariance. Typically it is assumed that these operators commute. Here we extend this analysis to non-commuting operators, replacing the commutativity assumption by a link condition. We discuss its relation to the commuting case, and we indicate that this allows us to use interpolation type results to obtain tight bounds for the contraction of the posterior Gaussian distribution towards the data generating element.

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Peter Mathé
Affiliation: Weierstrass Institute, Mohrenstrasse 39, 10117 Berlin, Germany

Received by editor(s): March 9, 2018
Received by editor(s) in revised form: November 29, 2018, and February 12, 2019
Published electronically: April 25, 2019
Article copyright: © Copyright 2019 American Mathematical Society