Journal
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
Volume 327, Issue -, Pages 147-172Publisher
ELSEVIER SCIENCE SA
DOI: 10.1016/j.cma.2017.08.016
Keywords
Infinite-dimensional Bayesian inverse problems; Curse of dimensionality; Hessian-based adaptive sparse quadrature; Sparse grid; Gaussian prior; Dimension-independent convergence analysis
Funding
- DARPA's EQUiPS program [W911NF-15-2-0121]
- National Science Foundation [CBET-1508713, ACI-1550593]
- DOE [DE-SC0010518, DE-SC0009286]
- Direct For Computer & Info Scie & Enginr
- Office of Advanced Cyberinfrastructure (OAC) [1550593] Funding Source: National Science Foundation
- U.S. Department of Energy (DOE) [DE-SC0009286] Funding Source: U.S. Department of Energy (DOE)
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In this work we propose and analyze a Hessian-based adaptive sparse quadrature to compute infinite-dimensional integrals with respect to the posterior distribution in the context of Bayesian inverse problems with Gaussian prior. Due to the concentration of the posterior distribution in the domain of the prior distribution, a prior-based parametrization and sparse quadrature may fail to capture the posterior distribution and lead to erroneous evaluation results. By using a parametrization based on the Hessian of the negative log-posterior, the adaptive sparse quadrature can effectively allocate the quadrature points according to the posterior distribution. A dimension-independent convergence rate of the proposed method is established under certain assumptions on the Gaussian prior and the integrands. Dimension-independent and faster convergence than O(N-1/2) is demonstrated for a linear as well as a nonlinear inverse problem whose posterior distribution can be effectively approximated by a Gaussian distribution at the MAP point. Published by Elsevier B.V.
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