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:
-
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Center for Applied Scienti c Computing
- 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:
- 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. https://doi.org/10.1137/16M1082688
Osborn, Sarah, Vassilevski, Panayot S., and Villa, Umberto. Thu .
"A Multilevel, Hierarchical Sampling Technique for Spatially Correlated Random Fields". United States. https://doi.org/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},
number = 5,
volume = 39,
place = {United States},
year = {Thu Oct 26 00:00:00 EDT 2017},
month = {Thu Oct 26 00:00:00 EDT 2017}
}
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