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 nonlinear problems with billions of unknowns. For stochastic modeling, it is therefore essential to design robust, parallel and scalable algorithms that can efficiently utilize highperformance computing to tackle such largescale 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 extremescale 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 inhouse implementation. Sparse iterative solvers with efficient preconditioners are employed to solve the resulting global and subdomain level local systems through multilevel iterative solvers. We also use parallel sparse matrix–vector operations to reduce the floatingpoint 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 nonGaussian 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. (SNLCA), 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. (SNLCA), Livermore, CA (United States)
 Sponsoring Org.:
 USDOE National Nuclear Security Administration (NNSA)
 OSTI Identifier:
 1399891
 Report Number(s):
 SAND201610429J
Journal ID: ISSN 00457825; 649571; TRN: US1702857
 Grant/Contract Number:
 AC0494AL85000
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Computer Methods in Applied Mechanics and Engineering
 Additional Journal Information:
 Journal Volume: 335; Journal ID: ISSN 00457825
 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; dualprimal 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:https://doi.org/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:https://doi.org/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 nonlinear problems with billions of unknowns. For stochastic modeling, it is therefore essential to design robust, parallel and scalable algorithms that can efficiently utilize highperformance computing to tackle such largescale 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 extremescale 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 inhouse implementation. Sparse iterative solvers with efficient preconditioners are employed to solve the resulting global and subdomain level local systems through multilevel iterative solvers. We also use parallel sparse matrix–vector operations to reduce the floatingpoint 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 nonGaussian 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}
}