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Title: A Multilevel, Hierarchical Sampling Technique for Spatially Correlated Random Fields

Abstract

In this paper, we propose an alternative method to generate samples of a spatially correlated random field with applications to large-scale problems for forward propagation of uncertainty. A classical approach for generating these samples is the Karhunen--Loève (KL) decomposition. However, the KL expansion requires solving a dense eigenvalue problem and is therefore computationally infeasible for large-scale problems. Sampling methods based on stochastic partial differential equations provide a highly scalable way to sample Gaussian fields, but the resulting parametrization is mesh dependent. We propose a multilevel decomposition of the stochastic field to allow for scalable, hierarchical sampling based on solving a mixed finite element formulation of a stochastic reaction-diffusion equation with a random, white noise source function. Lastly, numerical experiments are presented to demonstrate the scalability of the sampling method as well as numerical results of multilevel Monte Carlo simulations for a subsurface porous media flow application using the proposed sampling method.

Authors:
 [1];  [1];  [2]
  1. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Center for Applied Scienti c Computing
  2. Univ. of Texas, Austin, TX (United States). Institute for Computational Engineering and Sciences
Publication Date:
Research Org.:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1438756
Report Number(s):
LLNL-JRNL-696879; LLNL-JRNL-695979
Journal ID: ISSN 1064-8275
Grant/Contract Number:  
AC52-07NA27344
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
SIAM Journal on Scientific Computing
Additional Journal Information:
Journal Volume: 39; Journal Issue: 5; Journal ID: ISSN 1064-8275
Publisher:
SIAM
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; multilevel methods; PDEs with random input data; mixed nite elements; uncertainty quanti cation; multilevel Monte Carlo

Citation Formats

Osborn, Sarah, Vassilevski, Panayot S., and Villa, Umberto. A Multilevel, Hierarchical Sampling Technique for Spatially Correlated Random Fields. United States: N. p., 2017. Web. doi:10.1137/16M1082688.
Osborn, Sarah, Vassilevski, Panayot S., & Villa, Umberto. A Multilevel, Hierarchical Sampling Technique for Spatially Correlated Random Fields. United States. doi:10.1137/16M1082688.
Osborn, Sarah, Vassilevski, Panayot S., and Villa, Umberto. Thu . "A Multilevel, Hierarchical Sampling Technique for Spatially Correlated Random Fields". United States. doi:10.1137/16M1082688. https://www.osti.gov/servlets/purl/1438756.
@article{osti_1438756,
title = {A Multilevel, Hierarchical Sampling Technique for Spatially Correlated Random Fields},
author = {Osborn, Sarah and Vassilevski, Panayot S. and Villa, Umberto},
abstractNote = {In this paper, we propose an alternative method to generate samples of a spatially correlated random field with applications to large-scale problems for forward propagation of uncertainty. A classical approach for generating these samples is the Karhunen--Loève (KL) decomposition. However, the KL expansion requires solving a dense eigenvalue problem and is therefore computationally infeasible for large-scale problems. Sampling methods based on stochastic partial differential equations provide a highly scalable way to sample Gaussian fields, but the resulting parametrization is mesh dependent. We propose a multilevel decomposition of the stochastic field to allow for scalable, hierarchical sampling based on solving a mixed finite element formulation of a stochastic reaction-diffusion equation with a random, white noise source function. Lastly, numerical experiments are presented to demonstrate the scalability of the sampling method as well as numerical results of multilevel Monte Carlo simulations for a subsurface porous media flow application using the proposed sampling method.},
doi = {10.1137/16M1082688},
journal = {SIAM Journal on Scientific Computing},
issn = {1064-8275},
number = 5,
volume = 39,
place = {United States},
year = {2017},
month = {10}
}

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