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Scalable hierarchical PDE sampler for generating spatially correlated random fields using nonmatching meshes
Author(s) -
Osborn Sarah,
Zulian Patrick,
Benson Thomas,
Villa Umberto,
Krause Rolf,
Vassilevski Panayot S.
Publication year - 2018
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2146
Subject(s) - polygon mesh , solver , embedding , scalability , computer science , discretization , algorithm , random field , uncertainty quantification , monte carlo method , gaussian , sampling (signal processing) , mathematical optimization , mathematics , computational science , artificial intelligence , mathematical analysis , statistics , physics , computer graphics (images) , filter (signal processing) , quantum mechanics , machine learning , computer vision , database
Summary This work describes a domain embedding technique between two nonmatching meshes used for generating realizations of spatially correlated random fields with applications to large‐scale sampling‐based uncertainty quantification. The goal is to apply the multilevel Monte Carlo (MLMC) method for the quantification of output uncertainties of PDEs with random input coefficients on general and unstructured computational domains. We propose a highly scalable, hierarchical sampling method to generate realizations of a Gaussian random field on a given unstructured mesh by solving a reaction–diffusion PDE with a stochastic right‐hand side. The stochastic PDE is discretized using the mixed finite element method on an embedded domain with a structured mesh, and then, the solution is projected onto the unstructured mesh. This work describes implementation details on how to efficiently transfer data from the structured and unstructured meshes at coarse levels, assuming that this can be done efficiently on the finest level. We investigate the efficiency and parallel scalability of the technique for the scalable generation of Gaussian random fields in three dimensions. An application of the MLMC method is presented for quantifying uncertainties of subsurface flow problems. We demonstrate the scalability of the sampling method with nonmatching mesh embedding, coupled with a parallel forward model problem solver, for large‐scale 3D MLMC simulations with up to 1.9·10 9 unknowns.