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Algorithms for Kullback--Leibler Approximation of Probability Measures in Infinite Dimensions
SIAM Journal on Scientific Computing, v 37(6), pp A2733-A2757
08 Aug 2014
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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).\ud \ud \ud \ud \ud
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36 citations in Scopus
Details
- Title
- Algorithms for Kullback--Leibler Approximation of Probability Measures in Infinite Dimensions
- Creators
- Frank J. Pinski - University of CincinnatiGideon Simpson - Drexel UniversityAndrew M. Stuart - University of WarwickHendrik Weber - University of Warwick
- Publication Details
- SIAM Journal on Scientific Computing, v 37(6), pp A2733-A2757
- Publisher
- SIAM
- Grant note
- 0967140 / National Science Foundation (nsf_________::NSF) 0002085 / National Science Foundation (nsf_________::NSF)
- Resource Type
- Other
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000367019400012
- Scopus ID
- 2-s2.0-84953283212
- Other Identifier
- 991019168739304721
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- Collaboration types
- Domestic collaboration
- International collaboration
- Web of Science research areas
- Mathematics, Applied