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Algorithms for Kullback--Leibler approximation of probability measures in infinite dimensions
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Pinski, F. J., Simpson, G., Stuart, A. M. and Weber, Hendrik (2015) Algorithms for Kullback--Leibler approximation of probability measures in infinite dimensions. SIAM Journal on Scientific Computing, 37 (6). A2733-A2757. doi:10.1137/14098171X ISSN 1064-8275.
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Official URL: http://dx.doi.org/10.1137/14098171X
Abstract
In this paper we study algorithms to find a Gaussian approximation to a target measure defined on a Hilbert space of functions; the target measure itself is defined via its density with respect to a reference Gaussian measure. We employ the Kullback--Leibler divergence as a distance and find the best Gaussian approximation by minimizing this distance. It then follows that the approximate Gaussian must be equivalent to the Gaussian reference measure, defining a natural function space setting for the underlying calculus of variations problem. We introduce a computational algorithm which is well-adapted to the required minimization, seeking to find the mean as a function, and parameterizing the covariance in two different ways: through low rank perturbations of the reference covariance and through Schrödinger potential perturbations of the inverse reference covariance. Two applications are shown: to a nonlinear inverse problem in elliptic PDEs and to a conditioned diffusion process. These Gaussian approximations also serve to provide a preconditioned proposal distribution for improved preconditioned Crank--Nicolson Monte Carlo--Markov chain sampling of the target distribution. This approach is not only well-adapted to the high dimensional setting, but also behaves well with respect to small observational noise (resp., small temperatures) in the inverse problem (resp., conditioned diffusion).
Item Type: | Journal Article | ||||||
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Subjects: | Q Science > QA Mathematics | ||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
Library of Congress Subject Headings (LCSH): | Gaussian measures, Hilbert space, Algorithms | ||||||
Journal or Publication Title: | SIAM Journal on Scientific Computing | ||||||
Publisher: | Society for Industrial and Applied Mathematics | ||||||
ISSN: | 1064-8275 | ||||||
Official Date: | 17 November 2015 | ||||||
Dates: |
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Volume: | 37 | ||||||
Number: | 6 | ||||||
Page Range: | A2733-A2757 | ||||||
DOI: | 10.1137/14098171X | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Date of first compliant deposit: | 3 March 2017 | ||||||
Date of first compliant Open Access: | 3 March 2017 | ||||||
Funder: | United States. Department of Energy, National Science Foundation (U.S.) (NSF), Engineering and Physical Sciences Research Council (EPSRC), European Research Council (ERC), Great Britain. Office for Nuclear Regulation (ONR) | ||||||
Grant number: | DE-SC0002085 (United States. Department of Energy), OISE-0967140 (NSF) |
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