# Model's sparse representation based on reduced mixed GMsFE basis methods

## Abstract

In this paper, we propose a model's sparse representation based on reduced mixed generalized multiscale finite element (GMsFE) basis methods for elliptic PDEs with random inputs. A typical application for the elliptic PDEs is the flow in heterogeneous random porous media. Mixed generalized multiscale finite element method (GMsFEM) is one of the accurate and efficient approaches to solve the flow problem in a coarse grid and obtain the velocity with local mass conservation. When the inputs of the PDEs are parameterized by the random variables, the GMsFE basis functions usually depend on the random parameters. This leads to a large number degree of freedoms for the mixed GMsFEM and substantially impacts on the computation efficiency. In order to overcome the difficulty, we develop reduced mixed GMsFE basis methods such that the multiscale basis functions are independent of the random parameters and span a low-dimensional space. To this end, a greedy algorithm is used to find a set of optimal samples from a training set scattered in the parameter space. Reduced mixed GMsFE basis functions are constructed based on the optimal samples using two optimal sampling strategies: basis-oriented cross-validation and proper orthogonal decomposition. Although the dimension of the space spanned bymore »

- Authors:

- Institute of Mathematics, Hunan University, Changsha 410082 (China)
- College of Mathematics and Econometrics, Hunan University, Changsha 410082 (China)

- Publication Date:

- OSTI Identifier:
- 22622295

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 338; Other Information: Copyright (c) 2017 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGORITHMS; APPROXIMATIONS; COMPUTERIZED SIMULATION; DECOMPOSITION; DEGREES OF FREEDOM; EFFICIENCY; FINITE ELEMENT METHOD; FLOW MODELS; FUNCTIONS; GRIDS; LEAST SQUARE FIT; MASS; PARTIAL DIFFERENTIAL EQUATIONS; POROUS MATERIALS; RANDOMNESS; SAMPLING; TRAINING; TWO-PHASE FLOW; VALIDATION; VELOCITY

### Citation Formats

```
Jiang, Lijian, E-mail: ljjiang@hnu.edu.cn, and Li, Qiuqi, E-mail: qiuqili@hnu.edu.cn.
```*Model's sparse representation based on reduced mixed GMsFE basis methods*. United States: N. p., 2017.
Web. doi:10.1016/J.JCP.2017.02.055.

```
Jiang, Lijian, E-mail: ljjiang@hnu.edu.cn, & Li, Qiuqi, E-mail: qiuqili@hnu.edu.cn.
```*Model's sparse representation based on reduced mixed GMsFE basis methods*. United States. doi:10.1016/J.JCP.2017.02.055.

```
Jiang, Lijian, E-mail: ljjiang@hnu.edu.cn, and Li, Qiuqi, E-mail: qiuqili@hnu.edu.cn. Thu .
"Model's sparse representation based on reduced mixed GMsFE basis methods". United States.
doi:10.1016/J.JCP.2017.02.055.
```

```
@article{osti_22622295,
```

title = {Model's sparse representation based on reduced mixed GMsFE basis methods},

author = {Jiang, Lijian, E-mail: ljjiang@hnu.edu.cn and Li, Qiuqi, E-mail: qiuqili@hnu.edu.cn},

abstractNote = {In this paper, we propose a model's sparse representation based on reduced mixed generalized multiscale finite element (GMsFE) basis methods for elliptic PDEs with random inputs. A typical application for the elliptic PDEs is the flow in heterogeneous random porous media. Mixed generalized multiscale finite element method (GMsFEM) is one of the accurate and efficient approaches to solve the flow problem in a coarse grid and obtain the velocity with local mass conservation. When the inputs of the PDEs are parameterized by the random variables, the GMsFE basis functions usually depend on the random parameters. This leads to a large number degree of freedoms for the mixed GMsFEM and substantially impacts on the computation efficiency. In order to overcome the difficulty, we develop reduced mixed GMsFE basis methods such that the multiscale basis functions are independent of the random parameters and span a low-dimensional space. To this end, a greedy algorithm is used to find a set of optimal samples from a training set scattered in the parameter space. Reduced mixed GMsFE basis functions are constructed based on the optimal samples using two optimal sampling strategies: basis-oriented cross-validation and proper orthogonal decomposition. Although the dimension of the space spanned by the reduced mixed GMsFE basis functions is much smaller than the dimension of the original full order model, the online computation still depends on the number of coarse degree of freedoms. To significantly improve the online computation, we integrate the reduced mixed GMsFE basis methods with sparse tensor approximation and obtain a sparse representation for the model's outputs. The sparse representation is very efficient for evaluating the model's outputs for many instances of parameters. To illustrate the efficacy of the proposed methods, we present a few numerical examples for elliptic PDEs with multiscale and random inputs. In particular, a two-phase flow model in random porous media is simulated by the proposed sparse representation method.},

doi = {10.1016/J.JCP.2017.02.055},

journal = {Journal of Computational Physics},

number = ,

volume = 338,

place = {United States},

year = {Thu Jun 01 00:00:00 EDT 2017},

month = {Thu Jun 01 00:00:00 EDT 2017}

}