# A two-level stochastic collocation method for semilinear elliptic equations with random coefficients

## Abstract

In this work, we propose a novel two-level discretization for solving semilinear elliptic equations with random coefficients. Motivated by the two-grid method for deterministic partial differential equations (PDEs) introduced by Xu, our two-level stochastic collocation method utilizes a two-grid finite element discretization in the physical space and a two-level collocation method in the random domain. In particular, we solve semilinear equations on a coarse mesh $$\mathcal{T}_H$$ with a low level stochastic collocation (corresponding to the polynomial space $$\mathcal{P}_{P}$$) and solve linearized equations on a fine mesh $$\mathcal{T}_h$$ using high level stochastic collocation (corresponding to the polynomial space $$\mathcal{P}_p$$). We prove that the approximated solution obtained from this method achieves the same order of accuracy as that from solving the original semilinear problem directly by stochastic collocation method with $$\mathcal{T}_h$$ and $$\mathcal{P}_p$$. The two-level method is computationally more efficient, especially for nonlinear problems with high random dimensions. Numerical experiments are also provided to verify the theoretical results.

- Authors:

- Publication Date:

- Research Org.:
- Pacific Northwest National Lab. (PNNL), Richland, WA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1339789

- Report Number(s):
- PNNL-SA-103740

Journal ID: ISSN 0377-0427; KJ0401000

- DOE Contract Number:
- AC05-76RL01830

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational and Applied Mathematics; Journal Volume: 315

- Country of Publication:
- United States

- Language:
- English

- Subject:
- Semilinear problems; random coefficients; two-grid; stochastic collocation

### Citation Formats

```
Chen, Luoping, Zheng, Bin, Lin, Guang, and Voulgarakis, Nikolaos.
```*A two-level stochastic collocation method for semilinear elliptic equations with random coefficients*. United States: N. p., 2017.
Web. doi:10.1016/j.cam.2016.10.030.

```
Chen, Luoping, Zheng, Bin, Lin, Guang, & Voulgarakis, Nikolaos.
```*A two-level stochastic collocation method for semilinear elliptic equations with random coefficients*. United States. doi:10.1016/j.cam.2016.10.030.

```
Chen, Luoping, Zheng, Bin, Lin, Guang, and Voulgarakis, Nikolaos. Mon .
"A two-level stochastic collocation method for semilinear elliptic equations with random coefficients". United States.
doi:10.1016/j.cam.2016.10.030.
```

```
@article{osti_1339789,
```

title = {A two-level stochastic collocation method for semilinear elliptic equations with random coefficients},

author = {Chen, Luoping and Zheng, Bin and Lin, Guang and Voulgarakis, Nikolaos},

abstractNote = {In this work, we propose a novel two-level discretization for solving semilinear elliptic equations with random coefficients. Motivated by the two-grid method for deterministic partial differential equations (PDEs) introduced by Xu, our two-level stochastic collocation method utilizes a two-grid finite element discretization in the physical space and a two-level collocation method in the random domain. In particular, we solve semilinear equations on a coarse mesh $\mathcal{T}_H$ with a low level stochastic collocation (corresponding to the polynomial space $\mathcal{P}_{P}$) and solve linearized equations on a fine mesh $\mathcal{T}_h$ using high level stochastic collocation (corresponding to the polynomial space $\mathcal{P}_p$). We prove that the approximated solution obtained from this method achieves the same order of accuracy as that from solving the original semilinear problem directly by stochastic collocation method with $\mathcal{T}_h$ and $\mathcal{P}_p$. The two-level method is computationally more efficient, especially for nonlinear problems with high random dimensions. Numerical experiments are also provided to verify the theoretical results.},

doi = {10.1016/j.cam.2016.10.030},

journal = {Journal of Computational and Applied Mathematics},

number = ,

volume = 315,

place = {United States},

year = {Mon May 01 00:00:00 EDT 2017},

month = {Mon May 01 00:00:00 EDT 2017}

}