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Title: Scalable hierarchical PDE sampler for generating spatially correlated random fields using nonmatching meshes: Scalable hierarchical PDE sampler using nonmatching meshes

Journal Article · · Numerical Linear Algebra with Applications
DOI: https://doi.org/10.1002/nla.2146 · OSTI ID:1438783
ORCiD logo [1];  [2];  [1]; ORCiD logo [3];  [2];  [4]
  1. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Center for Applied Scientific c Computing
  2. Univ. della Svizzera Italiana, Lugano (Switzerland). Inst. of Computational Science
  3. Univ. of Texas, Austin, TX (United States). Inst. for Computational Engineering and Sciences
  4. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Center for Applied Scientific c Computing; Portland State Univ., Portland, OR (United States). Fariborz Maseeh Dept. of Mathematics and Statistics
+ Show Author Affiliations

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.

Research Organization:
Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
Sponsoring Organization:
USDOE; Swiss Commission for Technology and Innovation; Swiss National Science Foundation (SNSF); US Army Research Office (ARO)
Grant/Contract Number:
AC52-07NA27344; DMS-1619640; W911NF-15-1-0590
OSTI ID:
1438783
Alternate ID(s):
OSTI ID: 1432429
Report Number(s):
LLNL-JRNL-731006
Journal Information:
Numerical Linear Algebra with Applications, Vol. 25, Issue 3; ISSN 1070-5325
Publisher:
WileyCopyright Statement
Country of Publication:
United States
Language:
English

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Cited By (2)

Multilevel approximation of Gaussian random fields: Fast simulation journal December 2019
Multigrid Methods 2017: Multigrid Methods 2017 journal February 2018

Figures / Tables (9)

Fig. 1(p. 15)figure Fig. 1
Fig. 2(p. 16)figure Fig. 2
Fig. 3(p. 16)figure Fig. 3
Table 1(p. 17)table Table 1
Fig. 4(p. 18)figure Fig. 4
Fig. 5(p. 19)figure Fig. 5
Fig. 6(p. 19)figure Fig. 6
Table 2(p. 20)table Table 2
Fig. 7(p. 21)figure Fig. 7

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97 MATHEMATICS AND COMPUTING
multilevel methods
PDEs with random input data
PDE sampler
non-matching meshes
H(div) problems
mixed nite elements
uncertainty quanti cation
multilevel Monte Carlo