Journal
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
Volume 298, Issue -, Pages 205-228Publisher
ELSEVIER SCIENCE SA
DOI: 10.1016/j.cma.2015.10.002
Keywords
Karhunen-Loeve expansion; Dimensionality reduction; Markov Chain Monte Carlo; Polynomial Chaos; Bayesian inference
Funding
- King Abdullah University of Science and Technology (KAUST)
- US Department of Energy (DOE), Office of Science, Office of Advanced Scientific Computing Research [DE-SC0008789]
- U.S. Department of Energy (DOE) [DE-SC0008789] Funding Source: U.S. Department of Energy (DOE)
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This paper addresses model dimensionality reduction for Bayesian inference based on prior Gaussian fields with uncertainty in the covariance function hyper-parameters. The dimensionality reduction is traditionally achieved using the Karhunen-Loeve expansion of a prior Gaussian process assuming covariance function with fixed hyper-parameters, despite the fact that these are uncertain in nature. The posterior distribution of the Karhunen-Loeve coordinates is then inferred using available observations. The resulting inferred field is therefore dependent on the assumed hyper-parameters. Here, we seek to efficiently estimate both the field and covariance hyper-parameters using Bayesian inference. To this end, a generalized Karhunen-Loeve expansion is derived using a coordinate transformation to account for the dependence with respect to the covariance hyper-parameters. Polynomial Chaos expansions are employed for the acceleration of the Bayesian inference using similar coordinate transformations, enabling us to avoid expanding explicitly the solution dependence on the uncertain hyper-parameters. We demonstrate the feasibility of the proposed method on a transient diffusion equation by inferring spatially-varying log-diffusivity fields from noisy data. The inferred profiles were found closer to the true profiles when including the hyper-parameters' uncertainty in the inference formulation. (C) 2015 Elsevier B.V. All rights reserved.
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