# Scalable domain decomposition solvers for stochastic PDEs in high performance computing

## Abstract

Stochastic spectral finite element models of practical engineering systems may involve solutions of linear systems or linearized systems for non-linear problems with billions of unknowns. For stochastic modeling, it is therefore essential to design robust, parallel and scalable algorithms that can efficiently utilize high-performance computing to tackle such large-scale systems. Domain decomposition based iterative solvers can handle such systems. And though these algorithms exhibit excellent scalabilities, significant algorithmic and implementational challenges exist to extend them to solve extreme-scale stochastic systems using emerging computing platforms. Intrusive polynomial chaos expansion based domain decomposition algorithms are extended here to concurrently handle high resolution in both spatial and stochastic domains using an in-house implementation. Sparse iterative solvers with efficient preconditioners are employed to solve the resulting global and subdomain level local systems through multi-level iterative solvers. We also use parallel sparse matrix–vector operations to reduce the floating-point operations and memory requirements. Numerical and parallel scalabilities of these algorithms are presented for the diffusion equation having spatially varying diffusion coefficient modeled by a non-Gaussian stochastic process. Scalability of the solvers with respect to the number of random variables is also investigated.

- Authors:

- Carleton Univ., Ottawa, ON (Canada). Dept. of Civil and Environmental Engineering
- Sandia National Lab. (SNL-CA), Livermore, CA (United States)
- United States Naval Academy, Annapolis, MD (United States). Dept. of Aerospace Engineering
- Royal Military College of Canada, Kingston, ON (Canada). Dept. of Mechanical and Aerospace Engineering

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-CA), Livermore, CA (United States)

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

- OSTI Identifier:
- 1399891

- Report Number(s):
- [SAND-2016-10429J]

[Journal ID: ISSN 0045-7825; 649571; TRN: US1702857]

- Grant/Contract Number:
- [AC04-94AL85000]

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Computer Methods in Applied Mechanics and Engineering

- Additional Journal Information:
- [ Journal Volume: 335]; Journal ID: ISSN 0045-7825

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; 37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY; schur complement; parallel preconditioner; balancing domain decomposition by constraints; dual-primal finite element tearing and interconnect method; polynomial chaos expansion; coarse grid

### Citation Formats

```
Desai, Ajit, Khalil, Mohammad, Pettit, Chris, Poirel, Dominique, and Sarkar, Abhijit. Scalable domain decomposition solvers for stochastic PDEs in high performance computing. United States: N. p., 2017.
Web. doi:10.1016/j.cma.2017.09.006.
```

```
Desai, Ajit, Khalil, Mohammad, Pettit, Chris, Poirel, Dominique, & Sarkar, Abhijit. Scalable domain decomposition solvers for stochastic PDEs in high performance computing. United States. doi:10.1016/j.cma.2017.09.006.
```

```
Desai, Ajit, Khalil, Mohammad, Pettit, Chris, Poirel, Dominique, and Sarkar, Abhijit. Thu .
"Scalable domain decomposition solvers for stochastic PDEs in high performance computing". United States. doi:10.1016/j.cma.2017.09.006. https://www.osti.gov/servlets/purl/1399891.
```

```
@article{osti_1399891,
```

title = {Scalable domain decomposition solvers for stochastic PDEs in high performance computing},

author = {Desai, Ajit and Khalil, Mohammad and Pettit, Chris and Poirel, Dominique and Sarkar, Abhijit},

abstractNote = {Stochastic spectral finite element models of practical engineering systems may involve solutions of linear systems or linearized systems for non-linear problems with billions of unknowns. For stochastic modeling, it is therefore essential to design robust, parallel and scalable algorithms that can efficiently utilize high-performance computing to tackle such large-scale systems. Domain decomposition based iterative solvers can handle such systems. And though these algorithms exhibit excellent scalabilities, significant algorithmic and implementational challenges exist to extend them to solve extreme-scale stochastic systems using emerging computing platforms. Intrusive polynomial chaos expansion based domain decomposition algorithms are extended here to concurrently handle high resolution in both spatial and stochastic domains using an in-house implementation. Sparse iterative solvers with efficient preconditioners are employed to solve the resulting global and subdomain level local systems through multi-level iterative solvers. We also use parallel sparse matrix–vector operations to reduce the floating-point operations and memory requirements. Numerical and parallel scalabilities of these algorithms are presented for the diffusion equation having spatially varying diffusion coefficient modeled by a non-Gaussian stochastic process. Scalability of the solvers with respect to the number of random variables is also investigated.},

doi = {10.1016/j.cma.2017.09.006},

journal = {Computer Methods in Applied Mechanics and Engineering},

number = ,

volume = [335],

place = {United States},

year = {2017},

month = {9}

}