# Local polynomial chaos expansion for linear differential equations with high dimensional random inputs

## Abstract

In this paper we present a localized polynomial chaos expansion for partial differential equations (PDE) with random inputs. In particular, we focus on time independent linear stochastic problems with high dimensional random inputs, where the traditional polynomial chaos methods, and most of the existing methods, incur prohibitively high simulation cost. Furthermore, the local polynomial chaos method employs a domain decomposition technique to approximate the stochastic solution locally. In each subdomain, a subdomain problem is solved independently and, more importantly, in a much lower dimensional random space. In a postprocesing stage, accurate samples of the original stochastic problems are obtained from the samples of the local solutions by enforcing the correct stochastic structure of the random inputs and the coupling conditions at the interfaces of the subdomains. Overall, the method is able to solve stochastic PDEs in very large dimensions by solving a collection of low dimensional local problems and can be highly efficient. In our paper we present the general mathematical framework of the methodology and use numerical examples to demonstrate the properties of the method.

- Authors:

- Purdue Univ., West Lafayette, IN (United States)
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
- ETH Zurich, Zurich (Switzerland)
- Univ. of Utah, Salt Lake City, UT (United States)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1225870

- Report Number(s):
- SAND-2015-1493J

Journal ID: ISSN 1064-8275; 567323

- DOE Contract Number:
- AC04-94AL85000

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: SIAM Journal on Scientific Computing; Journal Volume: 37; Journal Issue: 1

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; generalized ploynomial chaos; domain decomposition; stochastic differential equation; uncertainty quantification

### Citation Formats

```
Chen, Yi, Jakeman, John, Gittelson, Claude, and Xiu, Dongbin.
```*Local polynomial chaos expansion for linear differential equations with high dimensional random inputs*. United States: N. p., 2015.
Web. doi:10.1137/140970100.

```
Chen, Yi, Jakeman, John, Gittelson, Claude, & Xiu, Dongbin.
```*Local polynomial chaos expansion for linear differential equations with high dimensional random inputs*. United States. doi:10.1137/140970100.

```
Chen, Yi, Jakeman, John, Gittelson, Claude, and Xiu, Dongbin. Thu .
"Local polynomial chaos expansion for linear differential equations with high dimensional random inputs". United States.
doi:10.1137/140970100.
```

```
@article{osti_1225870,
```

title = {Local polynomial chaos expansion for linear differential equations with high dimensional random inputs},

author = {Chen, Yi and Jakeman, John and Gittelson, Claude and Xiu, Dongbin},

abstractNote = {In this paper we present a localized polynomial chaos expansion for partial differential equations (PDE) with random inputs. In particular, we focus on time independent linear stochastic problems with high dimensional random inputs, where the traditional polynomial chaos methods, and most of the existing methods, incur prohibitively high simulation cost. Furthermore, the local polynomial chaos method employs a domain decomposition technique to approximate the stochastic solution locally. In each subdomain, a subdomain problem is solved independently and, more importantly, in a much lower dimensional random space. In a postprocesing stage, accurate samples of the original stochastic problems are obtained from the samples of the local solutions by enforcing the correct stochastic structure of the random inputs and the coupling conditions at the interfaces of the subdomains. Overall, the method is able to solve stochastic PDEs in very large dimensions by solving a collection of low dimensional local problems and can be highly efficient. In our paper we present the general mathematical framework of the methodology and use numerical examples to demonstrate the properties of the method.},

doi = {10.1137/140970100},

journal = {SIAM Journal on Scientific Computing},

number = 1,

volume = 37,

place = {United States},

year = {Thu Jan 08 00:00:00 EST 2015},

month = {Thu Jan 08 00:00:00 EST 2015}

}