# 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 Scientic 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. 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},

number = [5],

volume = [39],

place = {United States},

year = {2017},

month = {10}

}

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