## Abstract

Solving stochastic partial differential equations (SPDEs) can be a computationally intensive task, particularly when the underlying parametrization of the stochastic input field involves a large number of random variables. Direct Monte Carlo (MC) sampling methods are well suited for this type of situation since their cost is independent of the input complexity. Unfortunately, MC sampling methods suffer from slow convergence. In this manuscript, we propose an acceleration framework for elliptic SPDEs that relies on domain decomposition techniques and polynomial chaos (PC) expansions of local operators to reduce the cost of solving a SPDE via MC sampling. The approach exploits the fact that, at the subdomain level, the number of variables required to accurately parametrize the input stochastic field can be significantly reduced, as covered in detail in the prequel (Part A) to this paper. This makes it feasible to construct PC expansions of the local contributions to the condensed problem (i.e., the Schur complement of the discretized operator). The approach basically consists of two main stages: (1) a preprocessing stage in which PC expansions of the condensed problem are computed and (2) a Monte Carlo sampling stage where random samples of the solution are computed. The proposed method its naturally parallelizable. Extensive numerical tests are used to validate the methodology and assess its serial and parallel performance.

Original language | English (US) |
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Pages (from-to) | C547-C580 |

Journal | SIAM Journal on Scientific Computing |

Volume | 40 |

Issue number | 4 |

DOIs | |

State | Published - 2018 |

### Bibliographical note

Funding Information:˚Submitted to the journal’s Software and High-Performance Computing section May 30, 2017; accepted for publication (in revised form) April 30, 2018, published electronically July 17, 2018. http://www.siam.org/journals/sisc/40-4/M113219.html Funding: This work was supported by the U.S. Department of Energy (DOE), Office of Science, Office of Advanced Scientific Computing Research, under award number DE-SC0010540. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525. Support of the King Abdullah University of Science and Technology is also acknowledged. :Duke University, Durham, NC 27708 (andres.contreras@duke.edu). ;CERFACS, Toulouse 31057, France (mycek@cerfacs.fr). §CNRS, LIMSI, UniversitéParis Saclay, Orsay, France (olm@limsi.fr). ¶Sandia National Laboratories, Livermore, CA 94550 (fnrizzi@sandia.gov, bjdebus@sandia.gov). }King Abdullah University of Science and Technology, Thuwal, Saudi Arabia (omar.knio@ kaust.edu.sa).

Funding Information:

This work was supported by the U.S. Department of Energy (DOE), Office of Science, Office of Advanced Scientific Computing Research, under award number DE-SC0010540. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-NA-0003525. Support of the King Abdullah University of Science and Technology is also acknowledged.

Publisher Copyright:

© 2018 Society for Industrial and Applied Mathematics.

## Keywords

- Domain decomposition
- Monte Carlo method
- Polynomial chaos expansion
- Stochastic elliptic equations

## ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics