Search Menu
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information
Search Scholarly Publications

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
Citation Metrics:
Cited by: 10 works
Citation information provided by
Web of Science

References (50)

Tenth SPE Comparative Solution Project: A Comparison of Upscaling Techniques journal August 2001
The multi-level Monte Carlo finite element method for a stochastic Brinkman Problem journal March 2013
Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients journal January 2011
The Construction of the Coarse de Rham Complexes with Improved Approximation Properties journal January 2014
P3DFFT: A Framework for Parallel Computations of Fourier Transforms in Three Dimensions journal January 2012
A Multilevel, Hierarchical Sampling Technique for Spatially Correlated Random Fields journal January 2017
Nonlinear multigrid solvers exploiting AMGe coarse spaces with approximation properties journal October 2018
OpenCL Based Parallel Algorithm for RBF-PUM Interpolation journal April 2017
Fast and Exact Simulation of Stationary Gaussian Processes through Circulant Embedding of the Covariance Matrix journal July 1997
Multilevel Monte Carlo Methods book January 2001
Bayesian Spatial Modelling with R - INLA journal January 2015
Element agglomeration coarse Raviart-Thomas spaces with improved approximation properties: COARSE RAVIART-THOMAS SPACES WITH IMPROVED APPROXIMATION PROPERTIES journal January 2012
A Computational Framework for Infinite-Dimensional Bayesian Inverse Problems Part I: The Linearized Case, with Application to Global Seismic Inversion journal January 2013
Perfect spatial hashing journal July 2006
Spatial variability and uncertainty in groundwater flow parameters: A geostatistical approach journal April 1979
Mixed Finite Element Methods and Applications book January 2013
Multilevel Techniques Lead to Accurate Numerical Upscaling and Scalable Robust Solvers for Reservoir Simulation conference February 2015
On Stationary Processes in the Plane journal January 1954
Multilevel Monte Carlo methods journal April 2015
Multilevel Monte Carlo Path Simulation journal June 2008
Finite Element Error Analysis of Elliptic PDEs with Random Coefficients and Its Application to Multilevel Monte Carlo Methods journal January 2013
Randomized algorithms for generalized Hermitian eigenvalue problems with application to computing Karhunen-Loève expansion: RANDOMIZED ALGORITHMS FOR GHEP journal November 2015
In order to make spatial statistics computationally feasible, we need to forget about the covariance function: SPDES, GMRFS, AND KERNEL METHODS journal October 2011
PFFT: An Extension of FFTW to Massively Parallel Architectures journal January 2013
Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients journal March 2013
Exact de Rham Sequences of Spaces Defined on Macro-Elements in Two and Three Spatial Dimensions journal January 2008
Inverse problems: A Bayesian perspective journal May 2010
An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach: Link between Gaussian Fields and Gaussian Markov Random Fields journal August 2011
Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients journal April 2011
Upscaling of Mixed Finite Element Discretization Problems by the Spectral AMGe Method journal January 2016
A Parallel Approach to the Variational Transfer of Discrete Fields between Arbitrarily Distributed Unstructured Finite Element Meshes journal January 2016
Scheduling Massively Parallel Multigrid for Multilevel Monte Carlo Methods journal January 2017
Interior penalty preconditioners for mixed finite element approximations of elliptic problems journal April 1996
Numerical Multilevel Upscaling for Incompressible Flow in Reservoir Simulation: An Element-Based Algebraic Multigrid (AMGe) Approach journal January 2017
Random numbers for large-scale distributed Monte Carlo simulations journal June 2007
A Note on Preconditioning for Indefinite Linear Systems journal January 2000
A Block Circulant Embedding Method for Simulation of Stationary Gaussian Random Fields on Block-Regular Grids journal January 2015
An Algorithm for Non-Matching Grid Projections with Linear Complexity book January 2009
Spatial Variation book January 2019
Perfect spatial hashing conference January 2006
The multi-level Monte Carlo finite element method for a stochastic Brinkman Problem text January 2013
On Stationary Processes in the Plane journal December 1954
Spatial Variation. journal January 1988
Spatial variation null January 2015
The multi-level Monte Carlo finite element method for a stochastic Brinkman problem text January 2011
Multi-Level Monte Carlo Finite Element Method for elliptic PDEs with stochastic coefficients text January 2011
Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients preprint January 2012
Scheduling massively parallel multigrid for multilevel Monte Carlo methods preprint January 2016
P3DFFT: a framework for parallel computations of Fourier transforms in three dimensions text January 2019
Random numbers for large scale distributed Monte Carlo simulations text January 2006

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

Similar Records

Related Subjects

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