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Fast sampling from parameterised Gaussian random fields

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Gaussian random fields are popular models for spatially varying uncertainties, arising for instance in geotechnical engineering, hydrology or image processing. A Gaussian random field is fully characterised by its mean function and covariance operator. In more complex models these can also be partially unknown. Then we need to handle a family of Gaussian random fields indexed with hyperparameters. Sampling for a fixed configuration of hyperparameters is already very expensive, as it requires a Cholesky or spectral decomposition of the discretised covariance operator, which is in general a large dense matrix. Sampling from multiple configurations increases the total computational cost severely.     In this report we construct a reduced basis surrogate for parameterised Karhunen-Loève expansions – upon which our sampling procedure relies. The reduced basis is built using snapshots of Karhunen-Loève eigenvectors. In particular, we consider Matern type covariance operators with unknown correlation length and standard deviation. We suggest a linearisation of the covariance function and describe the associated online-offline decomposition. In numerical experiments, we investigate the approximation error of the reduced eigenpairs. As an application, we consider forward uncertainty propagation and Bayesian inversion in an elliptic PDE , where the log of the diffusion coefficient is a parameterised Gaussian random field. We discretise PDE operators and covariance operators with finite elements. In the Bayesian inverse problem we employ a Markov Chain Monte Carlo method on the reduced space to generate samples from the posterior measure. All numerical experiments are done in 2D space with non-separable covariance operators on finite element grids with tens of thousands of degrees of freedom.



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