skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

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

Abstract

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. Here, we demonstrate the scalability of the sampling method with nonmatching mesh embedding, coupled with a parallelmore » forward model problem solver, for large-scale 3D MLMC simulations with up to 1.9·109 unknowns.« less

Authors:
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
Publication Date:
Research Org.:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Sponsoring Org.:
USDOE; Swiss Commission for Technology and Innovation; Swiss National Science Foundation (SNSF); US Army Research Office (ARO)
OSTI Identifier:
1438783
Alternate Identifier(s):
OSTI ID: 1432429
Report Number(s):
LLNL-JRNL-731006
Journal ID: ISSN 1070-5325
Grant/Contract Number:  
AC52-07NA27344; DMS-1619640; W911NF-15-1-0590
Resource Type:
Accepted Manuscript
Journal Name:
Numerical Linear Algebra with Applications
Additional Journal Information:
Journal Volume: 25; Journal Issue: 3; Journal ID: ISSN 1070-5325
Publisher:
Wiley
Country of Publication:
United States
Language:
English
Subject:
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

Citation Formats

Osborn, Sarah, Zulian, Patrick, Benson, Thomas, Villa, Umberto, Krause, Rolf, and Vassilevski, Panayot S. Scalable hierarchical PDE sampler for generating spatially correlated random fields using nonmatching meshes: Scalable hierarchical PDE sampler using nonmatching meshes. United States: N. p., 2018. Web. doi:10.1002/nla.2146.
Osborn, Sarah, Zulian, Patrick, Benson, Thomas, Villa, Umberto, Krause, Rolf, & Vassilevski, Panayot S. Scalable hierarchical PDE sampler for generating spatially correlated random fields using nonmatching meshes: Scalable hierarchical PDE sampler using nonmatching meshes. United States. doi:10.1002/nla.2146.
Osborn, Sarah, Zulian, Patrick, Benson, Thomas, Villa, Umberto, Krause, Rolf, and Vassilevski, Panayot S. Tue . "Scalable hierarchical PDE sampler for generating spatially correlated random fields using nonmatching meshes: Scalable hierarchical PDE sampler using nonmatching meshes". United States. doi:10.1002/nla.2146. https://www.osti.gov/servlets/purl/1438783.
@article{osti_1438783,
title = {Scalable hierarchical PDE sampler for generating spatially correlated random fields using nonmatching meshes: Scalable hierarchical PDE sampler using nonmatching meshes},
author = {Osborn, Sarah and Zulian, Patrick and Benson, Thomas and Villa, Umberto and Krause, Rolf and Vassilevski, Panayot S.},
abstractNote = {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. Here, 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 1.9·109 unknowns.},
doi = {10.1002/nla.2146},
journal = {Numerical Linear Algebra with Applications},
number = 3,
volume = 25,
place = {United States},
year = {2018},
month = {1}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Citation Metrics:
Cited by: 2 works
Citation information provided by
Web of Science

Figures / Tables:

Fig. 1 Fig. 1: (left) The initial crooked pipe mesh, a quarter cylinder shape with radius equal to 2 and height equal to 7, with 14370 hexahedral elements and (right) the larger, regular bounding box $\bar{D}$ = (0, 3)× (0, 3)× (0, 8) with 15360 hexahedral elements.

Save / Share:

Works referenced in this record:

The multi-level Monte Carlo finite element method for a stochastic Brinkman Problem
journal, March 2013

  • Gittelson, Claude J.; Könnö, Juho; Schwab, Christoph
  • Numerische Mathematik, Vol. 125, Issue 2
  • DOI: 10.1007/s00211-013-0537-5

Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients
journal, January 2011

  • Cliffe, K. A.; Giles, M. B.; Scheichl, R.
  • Computing and Visualization in Science, Vol. 14, Issue 1
  • DOI: 10.1007/s00791-011-0160-x

The Construction of the Coarse de Rham Complexes with Improved Approximation Properties
journal, January 2014

  • Lashuk, Ilya V.; Vassilevski, Panayot S.
  • Computational Methods in Applied Mathematics, Vol. 14, Issue 2
  • DOI: 10.1515/cmam-2014-0004

P3DFFT: A Framework for Parallel Computations of Fourier Transforms in Three Dimensions
journal, January 2012

  • Pekurovsky, Dmitry
  • SIAM Journal on Scientific Computing, Vol. 34, Issue 4
  • DOI: 10.1137/11082748X

A Multilevel, Hierarchical Sampling Technique for Spatially Correlated Random Fields
journal, January 2017

  • Osborn, Sarah; Vassilevski, Panayot S.; Villa, Umberto
  • SIAM Journal on Scientific Computing, Vol. 39, Issue 5
  • DOI: 10.1137/16M1082688

Nonlinear multigrid solvers exploiting AMGe coarse spaces with approximation properties
journal, October 2018

  • Christensen, Max la Cour; Vassilevski, Panayot S.; Villa, Umberto
  • Journal of Computational and Applied Mathematics, Vol. 340
  • DOI: 10.1016/j.cam.2017.10.029

OpenCL Based Parallel Algorithm for RBF-PUM Interpolation
journal, April 2017

  • Cavoretto, Roberto; Schneider, Teseo; Zulian, Patrick
  • Journal of Scientific Computing, Vol. 74, Issue 1
  • DOI: 10.1007/s10915-017-0431-x

Fast and Exact Simulation of Stationary Gaussian Processes through Circulant Embedding of the Covariance Matrix
journal, July 1997


Bayesian Spatial Modelling with R - INLA
journal, January 2015

  • Lindgren, Finn; Rue, Håvard
  • Journal of Statistical Software, Vol. 63, Issue 19
  • DOI: 10.18637/jss.v063.i19

Element agglomeration coarse Raviart-Thomas spaces with improved approximation properties: COARSE RAVIART-THOMAS SPACES WITH IMPROVED APPROXIMATION PROPERTIES
journal, January 2012

  • Lashuk, I. V.; Vassilevski, P. S.
  • Numerical Linear Algebra with Applications, Vol. 19, Issue 2
  • DOI: 10.1002/nla.1819

A Computational Framework for Infinite-Dimensional Bayesian Inverse Problems Part I: The Linearized Case, with Application to Global Seismic Inversion
journal, January 2013

  • Bui-Thanh, Tan; Ghattas, Omar; Martin, James
  • SIAM Journal on Scientific Computing, Vol. 35, Issue 6
  • DOI: 10.1137/12089586X

Perfect spatial hashing
journal, July 2006


Spatial variability and uncertainty in groundwater flow parameters: A geostatistical approach
journal, April 1979


Multilevel Techniques Lead to Accurate Numerical Upscaling and Scalable Robust Solvers for Reservoir Simulation
conference, February 2015

  • Christensen, Max la Cour; Villa, Umberto; Vassilevski, Panayot
  • SPE Reservoir Simulation Symposium
  • DOI: 10.2118/173257-MS

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

  • Charrier, J.; Scheichl, R.; Teckentrup, A. L.
  • SIAM Journal on Numerical Analysis, Vol. 51, Issue 1
  • DOI: 10.1137/110853054

Randomized algorithms for generalized Hermitian eigenvalue problems with application to computing Karhunen-Loève expansion: RANDOMIZED ALGORITHMS FOR GHEP
journal, November 2015

  • Saibaba, Arvind K.; Lee, Jonghyun; Kitanidis, Peter K.
  • Numerical Linear Algebra with Applications, Vol. 23, Issue 2
  • DOI: 10.1002/nla.2026

In order to make spatial statistics computationally feasible, we need to forget about the covariance function: SPDES, GMRFS, AND KERNEL METHODS
journal, October 2011

  • Simpson, Daniel; Lindgren, Finn; Rue, Håvard
  • Environmetrics, Vol. 23, Issue 1
  • DOI: 10.1002/env.1137

PFFT: An Extension of FFTW to Massively Parallel Architectures
journal, January 2013

  • Pippig, Michael
  • SIAM Journal on Scientific Computing, Vol. 35, Issue 3
  • DOI: 10.1137/120885887

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

  • Pasciak, Joseph E.; Vassilevski, Panayot S.
  • SIAM Journal on Scientific Computing, Vol. 30, Issue 5
  • DOI: 10.1137/070698178

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

  • Lindgren, Finn; Rue, Håvard; Lindström, Johan
  • Journal of the Royal Statistical Society: Series B (Statistical Methodology), Vol. 73, Issue 4
  • DOI: 10.1111/j.1467-9868.2011.00777.x

Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients
journal, April 2011

  • Barth, Andrea; Schwab, Christoph; Zollinger, Nathaniel
  • Numerische Mathematik, Vol. 119, Issue 1
  • DOI: 10.1007/s00211-011-0377-0

Upscaling of Mixed Finite Element Discretization Problems by the Spectral AMGe Method
journal, January 2016

  • Kalchev, D. Z.; Lee, C. S.; Villa, U.
  • SIAM Journal on Scientific Computing, Vol. 38, Issue 5
  • DOI: 10.1137/15M1036683

A Parallel Approach to the Variational Transfer of Discrete Fields between Arbitrarily Distributed Unstructured Finite Element Meshes
journal, January 2016

  • Krause, Rolf; Zulian, Patrick
  • SIAM Journal on Scientific Computing, Vol. 38, Issue 3
  • DOI: 10.1137/15M1008361

Scheduling Massively Parallel Multigrid for Multilevel Monte Carlo Methods
journal, January 2017

  • Drzisga, D.; Gmeiner, B.; Rüde, U.
  • SIAM Journal on Scientific Computing, Vol. 39, Issue 5
  • DOI: 10.1137/16M1083591

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

  • la Cour Christensen, Max; Villa, Umberto; Engsig-Karup, Allan P.
  • SIAM Journal on Scientific Computing, Vol. 39, Issue 1
  • DOI: 10.1137/140988991

Random numbers for large-scale distributed Monte Carlo simulations
journal, June 2007


A Note on Preconditioning for Indefinite Linear Systems
journal, January 2000

  • Murphy, Malcolm F.; Golub, Gene H.; Wathen, Andrew J.
  • SIAM Journal on Scientific Computing, Vol. 21, Issue 6
  • DOI: 10.1137/S1064827599355153

A Block Circulant Embedding Method for Simulation of Stationary Gaussian Random Fields on Block-Regular Grids
journal, January 2015


    Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.